J. Phys. Chem. 1994,98, 44224427
4422
Substituent Effects in Silicon Hydrides: Implications for Models of Surface Sites Sharmila Pai and Douglas Doren’ Department of Chemistry and Biochemistry, University of Delaware, Newark, Delaware I971 6 Received: February 4, 1994”
To establish guidelines for using small silicon-containing molecules to model silicon surfaces, a series of substituted silanes have been studied using post-HartreeFock and density functional theories. The two theories differ in their detailed predictions, but similar trends are found with both methods. S i S i and Si-H bond lengths vary slightly among the compounds studied. Mulliken charges on the substituted silicon atom are altered significantly, but Mulliken charges on the other atoms remain unaffected. The S i S i bond energy decreases by about 0.5 kcal/mol with each silyl group that replaces hydrogen. Force constants change by a few percent. The most dramatic effect of substitution is that the energy of Si-H bonds at the substituted Si decrease by about 3 kcal/mol with each successive replacement of hydrogen by silyl groups. Of the properties calculated, only the S i S i bond strength correlates with substituent electronegativity. The effects of model structure on surface ionization potentials have also been determined for comparison to earlier work. In general, substitution a t a surface site alters some model properties, but more distant substitutions have little effect.
I. Introduction In electronic structure calculations of interatomic forces or chemical reactions on silicon surfaces, clusters of a few silicon atoms are often used to model the surface.’-’’ Bonds to the bulk solid are represented by bonds to hydrogenic atoms. Oneversion of this approach uses model compounds,in which the terminating atoms are H atoms with standard basis sets and there are no constraints on bond lengths or angles. An alternative uses “pseudoatoms” with a minimal basis set and usually with some geometry constraints. Each method has been used successfully to predict some surface properties. Sauer has reviewed the advantages and shortcomings of each approach.’ Clearly, no one-electron atom can completely reproduce the electronic environment of a silicon surface. To explore how the properties of cluster models depend on changes in the electronic environment, we have studied a series of substituted silanes with ab initio electronic structure methods. Several prior studies have tested how predictions of surface properties depend on the choice of model. For example, Chabal and Raghavachar? used Si& to model the monohydride phase on Si( 100)-2X1. The predicted hydrogen vibrationalfrequencies, isotope dependence and dynamic dipole moments agree with experiment. They examined two larger clusters and Si9H14) and concluded that the hydridevibrational spectrum can be adequately predicted from the smallest cluster. Tully et aL3 used the same model to derive an anharmonic force field for molecular dynamics and used it to rationalize the measured infrared line width of H vibrations on Si(100)-2X1. Similar models with three or four Si atoms have been used to model vibrations on other silicon hydride surfaces, achieving quantitative agreement with e~periment.~ In a study of Si surface etching by hydrofluoric acid, Trucks et al. showed that the mechanism could be understood using a cluster with only two Si atoms.5 When they replace one of the terminating H atoms by a SiH3 group, the activation barrier decreases by less than 2 kcal/mol. Geometric constraints in the cluster have a smaller effect on the activation barrier. An influential pseudoatom approach was developed by Redondo et a1.6 They noted that H atoms are more electronegativethan Si and proposed modified one-electron atoms to model the surface environment. These pseudoatoms are called siligens and denoted H. They are assigned a single basis function with the oribtal a Abstract
published in Advance ACS Abstracts, April 1, 1994.
0022-3654/94/2098-4422%04.50/0
exponent chosen so that the central Si in Si(SiH& has no Mulliken charge when the siligens are fixed at the positions of bulk Si atoms. Redondo et a1 showed that the ionization energy of small siligen-terminatedclusters with a lone dangling bond (SiH3 and Si(SiH,),) is weakly size dependent. In the corresponding hydrogen-terminated clusters, the ionization energies are very different. This suggests that the propertiesof siligen-terminated models converge more quickly with respect to cluster size. Carter and co-workers7v8have pursued this idea and used siligens to model reactions on Si( 100). They have shown that qualitatively similar orbitals are obtained from a large cluster model (Si9H12) and a smaller cluster (Si2S) obtained by “geometry mapping” from the large cluster. They have predicted adsorbate binding and reaction energies with this approach, achieving reasonable agreement with experiment for the binding energy of H2 on Si(loo).* Calculations by Jing and Whitten9 and Nachtigall et al.l0 on large hydrogen-terminated clusters produce similar binding energies in this case. We are not aware of other quantitative comparisons between H-terminated and siligenterminated models. This previous work raises several questions. Some properties of Si surfaces seem to be reproduced in small model compounds with H-termination, while others appear more sensitive to the size of the model and the method of termination. To optimize the size of the model and the method of termination, the effects of substituents on chemical properties must be known. We have done electronic structure calculations for a series of substituted silanes, as indicated in Figure 1, and related radicals. These calculations demonstratethe changes that result when terminating H atoms are replaced by a range of substituents. We examine substituents that are both more electronegative (F, C1, CH3) and more electropositive (Li, SiH3) than hydrogen atoms. Replacing H by SiH3 amounts to using a larger cluster model, in which more silicon atoms are used to represent the surface. We describe how various cluster propertiesdepend on the electronegativityof substituents and number of silicon atoms. This provides some insight into the type of model needed to represent surface sites. In the next section, we describe the methods used. Most of the results described here are based on Hartree-Fock (HF) and MP2 calculations, though we also compare results from density functional theory (DFT). Section I11 describes the effects of substituents on cluster geometries, bond energies, charge distributions, S i S i force constants, and ionization energies. In section IV we compare predictions of local and nonlocal DlT to 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 4423
Substituent Effects in Silicon Hydrides
H
I
I
3 Silyl
s c
Methyl
P 3
Chlorine 2
C;
.
5.5
3
4 7.5
3
1
1
9.5
Total Electronegativity on Si,
I
J
n
Figure 1. Model silicon hydride. All molecules considered here are substituted at Si. only. Substituents (RJ include F, C1, CH3, H, SiH3, and Li.
the Hartree-Fock-based calculationsin section 111. Both methods show that cluster properties are not very sensitive to substituents. We also find that the changes caused by substituents do not correlate with substituent electronegativity. Thus, in contrast to thesuggestionof Redondoet al., thedifferenceinelectronegativity between silicon and hydrogen is not the dominant source of difference between model compound properties and surface properties.1J In section V we revisit predictions of the surface ionization potential from cluster models. Finally, we summarize our findings and give an example of their implications for models of surface reactions.
II. Methods Hartree-Fuck-Based Methods. Calculations were done with SPARTAN 2.0.12 Geometries were optimized at the HartreeFock (HF) level using the 6-31G** basis set." No symmetry restrictions were imposed on the calculations, though final geometries had some approximate symmetry elements. Total energies, Mulliken charges, and dipole moments (Tables 1 and 2) were determined from a single-point calculation at the MP2 level with a 6-31G** basis set. Vibrational frequencies were determined at the HF/6-3 1G* level of theory (with the geometry determined at the same level) and then corrected with a scaling factor of 0.8929. All frequenciescalculated were real. The scaled frequencies were used to calculate zero-point energies, the vibrational contribution to the enthalpy, and S i S i force constants (Table 3). Si.-Si,g and Si.-H bond energies (Tables 4 and 5 ) were calculated from the MP2 total energies and scaled H F vibrational frequencies. Bond energies are consistently lower than the experimental values by a few kilocalories per mole. This level of calculation is not adequate to determine Mfwithout empirical corrections.14 Our atomization enthalpy for disilane differs from the accurate value by about 50 kcal/mol. Theoretical atomization enthalpies with errors on the order of 1 kcal/mol have been calculated for some of these compounds at a similar level of theory by including empirical bond additivity corrections14 or using homodesmic reactions.15 The higher level G2 theory predicts atomization enthalpies for silane and disilane that are in error by about 5 kcal/mol.16 The success of bond additivity corrections demonstrates that the level of theory used here reproduces the correct trends in bond energies, with a relatively small error per bond. Results of these calculations are described in section 111. Density F ~ n c t i 0 ~Th 1eory. Calculations were done using DMol, version 2.3.'' Basis functions are given numerically on an atom-centeredspherical-polarmesh. A double numerical basis
Figure 2. Si,-Sip bond length (A) versus the total electronegativity on Si, (total electronegativity is the sum of the electronegativitiesfor RI, R2, R3). Squaresrepresent silyl substituents,circlescorrespondto methyl substituents, and triangles correspond to chlorine substituents. The numbers next to the symbols represent the number of substituents that have replaced hydrogen.
set with polarization functions on all atoms (comparable to 6-31G**) and a fine mesh were used for all calculations. Calculations within the local density approximation (LDA) used the Hedin-Lundquist/ Janak-Morruzi-Williams local correlation functional.18 We have also examined the effects of nonlocal corrections. Nonlocal exchange effects were treated with the Becke 1988 gradient-corrected exchange functional (B-88).19 Nonlocal exchange and correlation were also considered together using the B-88 exchange functional and the Lee-Yang-Parr (1988) nonlocal correlation functional (B-LYP).ZO In several cases the minimized structure had one or more imaginary frequencies which could not be eliminated by further attempts to minimize the energy. As geometries are essentially the same as for the minima from H F theory, this probably reflectsnumerical artifacts. For bond energy calculations, the zero-point energy was taken from the HFcalculation. Results of these calculations are described in section IV. 111. Substituent Effects: Hartree-Fock-Based Calculations
In this section we describe the results of Hartree-Fock-based calculations on the stable species in Table 1 and radicals in Table 2. Geometries. The few available comparisons to experimental geometries (for Si2Hs and S ~ ~ H S Fshow ) ~ ' that HF/6-31GS* theory can reproduce experimental bond lengths to within 0.02 A. Substituents have a small effect on the cluster geometries. Some examples are shown in Figure 2. Among all of the compounds studied, the variation in Si,Sip bond lengths is less than 0.02 A. The interatomic distance in bulk silicon is 2.352 A, which is slightly shorter than any of the values shown for silyl-substitutedcases in Figure 2. Including electron correlation at the MP2 level in the geometry optimizationll shortens the disilane bond length to 2.334 A, so that this change in the level of theory has as big an effect as changing the cluster size. Changes in bond length do not correlate with substituent electronegativity. To quantify this, we use Pauling electronegativities for atoms and Boyd-Edgecomb22 group electronegativities for CH3 and SiH,. Though derived by different methods, these two scales give consistent results.22 Figure 2 shows the relationship between the Si,Sip bond length and the combined substituent group electronegativity (the sum of the electronegativities for R1, R2, and R3), where the substituents are C1, CH3, or SiH3. As the hydrogens on Si, are replaced by these functional groups, the Si,-SiB bond length changes linearly with successive substitutions of the same functional group. However, comparing different functionalgroups makes it clear that changesin geometry are not simply related to group electronegativity. Both methyl and chlorine are more electronegative than hydrogen, but methyl
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Pai and Doren
TABLE 1: Molecular Species Mulliken Charges and Dipole Moments' mulliken charges tot dipole electroneg Hon Hon moment molecule onSi,* Si, Si8 Si,c Si@ (D) Si2H6
6.3
0.475 0.475 -0.158 -0.158
0.00
S~~HJF 8.2 0.930 0.422 -0.190 -0.155 1.44 Si2H5CI 7.2 0.616 0.499 -0.147 -0.150 2.22 Si2HsCH3 6.8 0.589 0.471 -0.172 -0.163 0.77 Si2HsSiH3 6.1 0.227 0.510 -0.150 -0.158 0.04 SizHsLi 5.2 0.134 0.458 -0.185 -0.175 7.76 H3SiSiHC12 8.1 0.685 0.532 -0.132 -0.144 2.52 H,SiSiH(CH,)2 7.2 0.704 0.468 -0.185 -0.167 0.99 H$iSiH(SiH3)2 5.9 -0.066 0.539 -0.137 -0.157 0.08 H3SiSiCI3 9.0 0.689 0.576 -0.139 2.54 H,SiSi(CH& 7.7 0.819 0.469 -0,170 1.13 HsSiSi(SiHp)f 5.7 -0,426 0.574 -0.156 0.01 a Calculated at MP2 level using geometry optimized at HF/6-31GS* level. Sum of Paulingelectronegativitiesfor atomsand Boyd-Edgecomb electronegativities for functional groups. Average over all hydrogens onSi,.d Averageover all hydrogens onSip. Calculatedat HF/6-31G** level.
Radical Mulliken Charges and Dipole Moments' mulliken charges dipole moment radical Si, H on Si, (D) 0.20 SiH3 0.432 -0.144 0.851 -0.172 FSiH2 1.58 ClSiH2 0.617 -0,133 2.08 0.532 -0.161 CHaSiH2 0.69 SiH3SiH2 0.239 -0.137 0.12 (SiH3)zSiH 0.005 -0.128 0.04 a Calculated at MP2 level using geometry optimized at HF/6-31G** level.
-' Silyl
Methyl 5.5
b) Sip
7.5
9.5
Total Electronegativity on S,i
TABLE 2
substitution increases the Si,-Sip bond length, while chlorine substitution decreases this bond length. Silyl substituents, which are less electronegative than hydrogen, increase the bond length. Si-H bond lengths vary linearly with successive substitutions at Si,, though the changes are never more than 0.02 A for Si,-H and 0.01 8, for S i r H . MullikenCharges and Dipole Moments. Mulliken charges and dipole moments of the clusters and related radicals are tabulated in Tables 1 and 2. Few experimental measurements of the dipole moments of substituted disilanes are available. The only molecule in Table 1 for which a dipole moment has been measured is Si2H5F. Our calculated result of 1.44 D is slightly higher than the experimenta121 result (1.26 f 0.01 D). To establish further comparisonswith experiment,we have calculated dipole moments for SiH3CH2CH3 (present theory, p = 0.74 D; experiment,21p = 0.81 D) and SiH2(CH3)2 (present theory, ~1 = 0.67 D; experiment,21 p = 0.71 D). Accurate calculations of dipole moments are difficult because the dipole moment is sensitive to subtle features of the wave function.l3 The largest error among these examples is only 0.2 D or 14'36, indicating that the level of theory used provides a good description of the charge distribution and that calculated dipole moments should be qualitatively reliable for these compounds. Figure 3 shows how the Mulliken charges of Si, and Sip depend on the electronegativity of substituents on Si,. With silyl substituents, the Mulliken charge on Si,decreases due to donation of electron density from the electropositive silyl. The charge on Sip increases slightly. Chlorine increases the Mulliken charge on both Si, and Sip, Methyl substituents increase the Mulliken charge on Si, and decrease the charge on Sip In all cases the variations in the Mulliken charges on Sip are much smaller than the changes in the Mulliken charges on Si,. Mulliken charges on the H atoms bound to Si, are also insensitive to substituents. Table 1 shows that these charges never change by more than a few hundredths of a unit, regardless of the substituent. Thus, the effect of substituents on the electron distribution is strongly
Chlorine
Figure 3. Mulliken charges on (a) Si, and (b) Si0 as a function of the total electronegativityon Si,. Squares represent the silyl substituents, circles correspondto methyl substituents, and triangles correspondto the chlorinesubstituents. The unmarkedpoint at the intersectionof the lines represents unsubstituted disilane.
TABLE 3: Si-Sib Force Constants force constant molecule (hartree/au2) molecule H3SiSiH3 0.267 H,SiSi(CH3)3 H3SiSi(SiH& 0.252 HaSiSiCll
force constant (hartree/auZ) 0.269 0.262
TABLE 4
Si,-Sie Bond Energies DFTbond energies MP2 (kcal/mol) expta bond energies molecule (kcal/mol) JMW B-88 B-LYP (kcal/mol) HlSiSiH3 70.0 81.6 63.3 64.6 74+ 2 H3SiSiH2F 72.4 81.9 63.3 65.3 HpSiSiHzCl 70.4 80.3 61.6 64.0 H3SiSiHzCH3 70.8 81.9 62.8 H3SiSiHzSiH3 69.3 80.0 60.4 71f2 H3SiSiH(SiH3)2 68.8 78.5 58.4 HsSiSi(CH3)3 73.0 a Reference 23.
localized. Only the Mulliken charge of the silicon atom directly bound to a substituent is significantly affected. Similar trends are seen in the radical species. Force Constants. As noted above, the Si-H force constants for small model compounds give an excellent approximation to the force constants for surface hydrides.24 We have calculated a2E/ar(Si, - Sip)2,the force constant for stretching the Si,-Sip bond, in disilane and the triply substituted cases (Table 3). The largest difference from thedisilane forceconstant is 0.01 5 hartre/ au2, which is a 6% change. Since the frequency of a vibration is proportional to the square root of the forceconstant, the change in frequency is never more than 3%. For a typical SiSistretching frequency of 400-500 cm-1, this amounts to a shift of 10-15 cm-1. Thus, the curvature of the potential energy surface a t the minimum is weakly sensitive to substituents. Again, the changes do not correlate with substituent electronegativity. S i s i Bond Energies. Calculated Si,-& bond energies for some of the compounds are reported in Table 4 and Figure 4. With each successive silyl substitution the Si,-Sip bond energy decreases slightly (about 0.5 kcal/mol). Methyl, chlorine, and fluorine substitutions all increase the Si,Sip bond energy by a small amount. As previously known from experiments, methyl substitutions are unique since they increase both the bond length
.
The Journal of Physical Chemistry, Vol. 98, No. 16, I994 4425
Substituent Effects in Silicon Hydrides
-0
h
E
3C x
72
w
P 70 68
3 I
7.5
5.5
9.5
Total Electronegativity on Si,
Figure 4. Si,SiO bond energy versus total electronegativityon Si, (the sum of the electronegativitiesfor RI, R2, and R3). Annotations indicate the substituents on Si,. The point without annotation corresponds to unsubstituted disilane.
70 72 MPa6-31G”
74
Figure 5. Comparison of Si,Si# bond energies calculated from DFT and post-Hartres-Fock theory. Squares indicate results of local DFT, circles refer to the B-88 nonlocal DFT,and triangles indicate the B-LYP nonlocal DFT. The straight line shows where the MP2 results would fall. Generally the trends in the four theories are the same. To a good approximation they differ from each other by an added constant.
TABLE 5 S i - H Bond Energies molecule SiH3-H H3SiSiHrH (H3Si)zSiH-H (H3Si)sSi-H CH3SiHrH (CH3)zSiH-H (CH&Si-H Cl3Si-H F3Si-H a Reference 23.
MP2
DFT bond energies
bond energies (kcal/mol)
(kcal/mol) JMW B-88
86.4 83.2 80.0 79.0
91.1 88.6 85.3 83.9
83.5 79.3 78.6
expta (kcal/mol) 90.3 f 1 86.3 f 1 89.6 f 2 89.4 f 2 90.3 f 1 91.3 i 1 100.1 i 1
and the bond energy, in contrast to the common relationship between these q~antities.2~ The Si,Sia bond energies correlate well with the total electronegativity acting on Si, (Figure 4). Si-H Bond Energies. Table 5 reports Si,-H bond energies in model compounds where Sip is replaced by a H atom. Successive silyl substitutions on Si, weaken the Si,-H bond by about 3 kcal/ mol, while fluorine strengthens the bond. Other substituents have negligible effect. By itself, substituent electronegativity is not a useful predictor of Si-H bond energy.
IV. Comparison of Post-Hartree-Fock and Density Functional Methods Density functional calculations of chemical properties have received extensiverecent attention because they includecorrelation effects at lower cost than post-HartreeFockmethods. However, the accuracy of chemical predictions from DFT are still being tested. The most detailed work to date on silicon hydrides is that of Nachtigall et ai., who applied DFT and several other ab initio theories to a model of hydrogen adsorption on Si(100)-2X1. To test DFT predictions for silicon hydrides more generally, we have applied local and nonlocal density functional theories to the molecules described above. Because the information available from experiment is limited, we have compared the DFT results to those of Hartree-Fock-based theory. Of course, allowances must be made for the known errors in the latter method (notably, consistently small S i S i and Si-H bond energies). All of the conclusions drawn above about substituent effects are confirmed in the density functional calculations. Geometries from local and nonlocal density functionals compare well to both experiment (where available) and the HFresults. DFT predictions for S i S i bond lengths never differ from the H F results by more than 0.04& and bond anglesdiffer by less than 1O . DFT Mulliken charges follow the trends described in section 111. In all cases
I
78
03 MP2l6-3lG”
88
Figure 6. Comparison of SL-H bond energies calculated from DFT and post-Hartree-Fock theory. Squares indicate results of local DFT,and circles refer to the B-88 nonlocal DFT. The straight line shows where MP2 resultswould fall. The threetheoriesfollow thesame trends, differing approximately by a constant amount. wherecomparisons could be made, DFT zero-point energiesagree with the Hartree-Fock values to within 1 kcal/mol. Si-Si bond energies from LDA calculationsare consistently too high by about 10 kcal/mol. Witheither the B-88 or B-LYPnonlocalfunctionals, S i S i bond energies are about 10 kcal/mol too low. Figure 5 shows that all of the density functional theories are less accurate than the post-HartreeFock results (which are themselves a few kilocalories per mole too low). Density functional predictions of Si-H bond energies are comparable to the post-Hartree-Fock predictions. As shown in Figure 6, the LDA results are higher than MP2, while the B-88 functional gives virtually the same result as MP2. Regardless of absolute accuracy in bond energies, Figures 5 and 6 show that all of these theories predict the same trends in bond energies. To a good approximation, they differ by added constants. Bond additivity corrections are likely to be useful.14
V. Surface Ionization Potential Our results show that terminating a cluster model of a surface with hydrogen has a short-range effect on many cluster properties. This is consistent with the short screening length in bulk silicon.25 In their pioneering work, Redondo et al. indicated that the surface ionization potential converges slowly with respect to cluster size in hydrogen-terminated clusters. Our findings do not seem consistent with their conclusion, so we have reexamined the dependence of ionization energy on model structure to determine whether ionizationenergies are more sensitiveto cluster structure. Note that we do not expect to reproduce the measured ionization energy. The structures used by Redondo et al. were models of the unreconstructed Si(ll1) surface. Agreement with measurements made on reconstructed surfaces could only be coincidental.
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The Journal of Physical Chemistry, Vol. 98, No. 16, I9194
Pai and Doren
e
4(a
2
+ A-1)
(1)
where a is the distance of the charge from the surface and A-l is the Thomas-Fermi screening length of the solid.*' For Si, A-1 = 0.5 The approximation is accurate when a > (2A)-I. We takea to be the height ofthesurfaceSiatoniabove the bottommost Si in the cluster. Thus, a = 0 for the one-layer model, 1.36 A for the two-layer model, and 4.08 A in the four-layer model. The calculated corrections for the different models are included in Table 6. As noted by Redondo et al., when the correction is subtracted from the ionization energy of the two-layer model, the experimental result is recovered. However, the dielectric correction decreases rapidly with each added layer, while the cluster ionization energy decreases more slowly. Either a more sophisticated dielectric correction is needed or the agreement found with the two-layer model is accidental.
VI. Conclusions Figure 7. Models for Si(ll1)-1x1: (a) RI = Rz = R3 = H for the one-layer model and R1 = Rz = Rs = SiH3 for the two-layer model; (b) the four-layer model.
TABLE 6 (eV)
Ionization Potentials for Models of Si(ll1)-1x1
I-layer model 2-layer model 4-layer model (SiHd . (Si4Hd (SisHd MP2/6-3lG* 8.49 7.46 MP2/6-31G** 8.55 9.06 7.62 7.46 LDA dielectric continuum 7.48 1.95 0.65 correction (eq 1) Our goal is simply to understand better how the calculated energy depends on cluster size. We have examined ionization potentials in several H-terminated models of unreconstructed Si(111) surfaces. The first two are SiH3 (referred to as the one-layer model) and Si4H9 (the twolayer model), with the structure indicated in Figure 7a. These are the models studied by Redondoet al. The ionization potential in each of these models has been studied at the MP2 level and with local DFT. We have also studied the four-layer model in Figure 7b using local DFT. Structures of the neutral radicals were fully optimized as described before. The energy of the ionic final state was calculated at the geometry of the neutral. The ionization energy is the difference of the ion and the neutral electronic energies. Results of these calculations are in Table 6. For comparison, the peaks in photoelectron spectra assigned to dangling bonds26 are centered at 5.9 eV (relative to the vacuum level) on Si( 111)-2X1 and 5.6 eV on Si( 111)-7X7. All of these models predict ionization energies at least 1.5 eV above the experimental photoelectron peaks. The one-layer and two-layer models differ by over 1 eV. As expected, the electropositive SiHs stabilizes the positively charged final state more effectively than H. However, including more silicon atoms, as in the four-layer model, has a small effect. This suggests that the ionization energy has nearly converged with respect to cluster size in the two-layer model. It is consistent with the findings of section I11 that substitutions that are not adjacent to the dangling bond have a small effect on the ionization potential. Redondo et al. suggested that the errors in ionization energy can be corrected by using a continuum dielectric model to account for stabilization of the final state by the bulk dielectric relaxation. We have used our results to make a simple test of this idea. The image attraction between a charge q and a jellium surface is well approximated as
Our results demonstrate that substituents cause small changes in silicon cluster properties. Neither electron-withdrawing nor electron-donating substituents have a significant effect on the geometries of these clusters. The Mulliken charge on the substituted silicon atom (Si,) is sensitive to substitutions,especially by silyl. However, the effect of substituents on the charge distribution is highly localized: there is no significant change beyond the Si atom to which the substituent is bonded. Substituents alter the force constant for the S i S i stretch by a few percent. S i S i and Si-H bond energies show small but potentially significant variations. Of all these properties only t h e S i S i bond strength is well correlated with substituent electronegativity. If siligen-terminated clusters effectively reproduce the environment of bulk Si, it is not simply a consequence of their electronegativity. The difference in electronegativity between H and SiH3 is smaller than the difference between some other substituents considered here, yet SiHS has a more dramatic effect on cluster properties than the other substituents. The large differences in ionization energy between siligen-terminated and H-terminated clusters6 indicate that siligens model different physical effects than those obtained by simply increasing the number of Si atoms. Additional work is needed to understand the nature of these pseudoatoms. Geometries, Mulliken charges, and zero-point energies are similar in both density functional and HartreeFock-based methods. The level of post-HF theory used here underestimates S i S i bond energies by 3-4 kcal/mol. This is substantially better than the results from local or nonlocal DFT. Local and nonlocal DFT methods and the post-HF theory all provide similar accuracy for Si-H bond energies. In the few cases examined, local DFT and the MP2 theory give similar ionization potentials. Systematic improvements to the post-HF theory, though costly, should improve its predictions. Improved DFT results must await improved energy functionals. DFT methods reproduce the trends found in other theories and permit correlated calculations on clusters with more Si atoms than post-HF methods. It will be valuable to explore empirical corrections to DFT bond energies for silicon hydrides. Now consider using a small cluster to model a surface reaction. Quantitative predictions of cluster properties require a higher level of theory than used here, including electron correlation in the geometry optimization.ll Our results illustrate the effects of cluster size. If changing the number of silicon atoms in the cluster alters the bond energy or the electron distribution a t the reactive site, then the calculated activation barrier (and possibly the qualitative nature of the mechanism) will depend on the cluster size. We have shown that the charge distribution is not sensitive to cluster size as long as the reactive site is not adjacent to the cluster edge. This may still require large clusters, especially if
Substituent Effects in Silicon Hydrides more than one surface site is involved in the reaction. Only a few previous cluster models of silicon surface reactions have included enough atoms to satisfy this criterion.r9J0 It remainsto determine how large an error in activation barriers is caused by using too small a cluster. In theonly previous attempt to test thedependence of an activationbarrier on the structure of a hydrogen-terminated model, Trucks et ai. estimated an error of less than 2 kcal/mol due to the small cluster model in their study of etching.5 We can apply the results of this work to another example. Dobbs and Doren used SizHs to model reactions of atomic hydrogen with the silicon surface trihydride." This species is an intermediate in silicon surface etching by hydrogen and in the dissociativeadsorptionof silane on Si surfaces. Dobbs and Doren have reviewed the experimental and theoretical evidence for the role of trihydrideson hydrogen-saturatedSi surfaces.l* The atom denoted as Si6 in Figure 1 represents the reactive trihydride site, while Si, and the H atoms bound to it represent the remainder of the surface. Under attack by atomic H, two reactions compete: the S i S i bond may be broken to form SiH3 and SiH4, or the incident H atom may abstract a H atom to form H2 and Si2H5. Dobbs and Doren found that the barriers for the two reactions are similar (2.4 kcal/mol for abstraction and 3.0 kcal/ mol to break a S i S i bond). Our results imply that the reactivity of the Si-H bond in the surface trihydride should be well modeled by the S i r H bond in disilane. However, the Si,Sis bond on the surface will be weaker than the bond in disilane by a few kilocalories per mole. The electron density in the S i s i bond on the surface will also be higher than in disilane, due to donation from surrounding Si atoms. This should make attack by the electrophilic H atom more favorable. Both effects tend to make the activationbarrier for S i S i bond breaking lower on the surface than in the model compound. This will increase the rate of etching relative to that of abstraction and is consistent with the low activation barriers for surface etching observed by Abrefah and Olander29 and Gates et al.30
Acknowledgment. We would like to thank the other participants in our course in Modern ElectronicStructure Theory who worked on the calculations in the early part of this project. They are Robert Bear, August Calhoun, Hayes Williams, Carl Krauthauser, Ching-Lung Lin, Yasunori Nakajima, Philip Ross, and Ratna Shekhar. We also thank Biosym Technologies, Inc., for
The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 4421 their support in our use of DMol. This work was supported by the National Science Foundation under Grant CHE-9015368.
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