J . Phys. Chem. 1985,89, 1428-1432
1428
A Valence-Bond Potential-Energy Surface for Silylene Dissociation R. Viswanathan, D. L. Thompson, and L. M. Raff* Department of Chemistry, Oklahoma State University, Stillwater, Oklahoma 74078 (Received: October 1, 1984)
A semiempirical valence-bond calculation of the SiHz potential-energy surface is reported. The VB wave function is written
as a linear combination of the four bond eigenfunctions representing the canonical structures for the reactants and products of the two- and three-center dissociation channels. The matrix elements are evaluated semiempirically in a manner that incorporates the available experimental and ab initio CI data into the formalism. The effect of the silicon nonbonded electron pair is incorporated by addition of a bending potential term. Surface gradients are obtained by a generalization of the Hellmann-Feynman theorem. The resulting SiH2 surface gives the correct bond energies and equilibrium distances for all diatomic products. The calculated reaction endothermicitiesfor all dissociation channels are in agreement with the experimental results to within 6.7% or better. The computed equilibrium bond lengths and bond angle for SiHz are in fair accord with the corresponding experimental values. The errors in these quantities are +8.5% and -9.8%, respectively. The VB surface predicts a back-reaction barrier for Si + H2 SiH2of 0.30 eV, which is in agreement with scaled SCF calculations. There are no attractive wells along the predicted reaction coordinate for three-center dissociation.
-
I. Introduction Due to the importance of silicon solar cells, the chemical vapor deposition (CVD) of silicon from silane has been the subject of intensive experimental research. The homogeneous reactions that have been postulated to be of greatest importance in the silicon CVD process' include the unimolecular dissociation of SiH4 SiH, SiH,
-
SiH2 + HZ SiH3
+H
(R1) (R2)
the subsequent dissociation of SiHz formed in reaction R1
-
SiHz
SiHz
+ H2 SiH + H Si
- -
(R3) (R4)
the polymerization of SiH2 2SiHz
Si2H4
SizHz + H2
--
(R5)
and the nucleation of Si atoms formed in reaction R3
+ Si + M Siz + Si + M Si
Siz + M Si,
+M
(R6)
etc. Reactions R1 and R2 have been the focus of several experimental studies. Claassen and Bloem2 have shown that reaction R1 is the dominant SiH4 decomposition pathway in mixtures of SiH4Hz-N2. Single-pulse shock tube experiments by Newman et aL3 and bond dissociation energy measurements by Doncaster and Walch4 have shown that reaction R1 is the major reaction pathway for SiH, pyrolysis. We have recently reported the results of quasiclassical trajectory studies of reactions R1 and R25 and Monte Carlo transition-state theory calculations for reaction R2.6 Microcanonical rate coefficients, unimolecular falloff curves, and mechanisms were determined for both reactions. The results were in good accord with the shock tube data3 and confirm that reaction R1 is the dominant mode for SiH4 dissociation. (1) For example, see A. K.Praturi, 'Proceedings of the Sixth International Conference on Chemical Vapor Deposition", L. F. Donaghey and P. RaiChoudhury, Ed., The Electrochemical Society, Princeton, NJ, 1977, p 20. (2) W. A. P. Claassen and J. Bloem, J. Cryst. Growth, 51, 443 (1981). (3) C. G. Newman, H. E. O'Neal, M. A. Ring, F. Leska, and N. Shipley, Inf. J . Chem. Kinef., 11, 1167 (1979). (4) A. M. Doncaster and R. Walch, Znf.J . Chem. Kinef.,13, 503 (1981). (5) R. Viswanathan, D. L. Thompson, and L. M. Raff, J . Chem. Phys., 80, 4230 (1984). (6) R. Viswanathan, L. M. Raff, and D. L. Thompson, J . Chem. Phys., 81, 828 (1984).
0022-3654/85/2089-1428$01.50/0
Due to the reactive nature of SiH2, there is little direct experimental data related to reactions R3, R4, and R5. This is unfortunate since these reactions are almost certainly of central importance in the silicon CVD process. On the other hand, reactions R3 and R4 are much more amenable to theoretical investigation than the corresponding reactions involving SiH4. Accurate ab initio SCF-CI calculations for various Si-H electronic states have been reported.'** Similar calculations of the SiHz bending potential for several electronic states are also a ~ a i l a b l e . ~ The endothermicities for reactions R 3 and R4 are known4*I0as is the equilibrium SiH2 structure." However, these results have not yet been incorporated into a global potential-energy surface for the SiHz system. In this paper, we report the results of a valence-bond calculation of the SiH2 potential surface. Much of the available a b initio and experimental data are incorporated in a semiempirical way into the VB formalism. Although the final result is not in the form of an analytic global function, calculation of the SiH2potential energy for any arbitrary configuration is sufficiently simple that the surface may be easily used in transition-state theory calculations. Since the gradients of the surface may be obtained by using a generalization of the Hellmann-Feynman theorem,12 the formulation is also useful for trajectory calculations. 11. Method
The minimum requirement to be satisfied by a potential-energy surface to be used in dynamical studies is that it represent the energies of the reactant and product limits with sufficient accuracy. It is well-known that a simple VB model contains sufficient configuration interaction to ensure the appropriate atomic limits. Consequently, we have chosen to represent the SiH2surface with such a formalism. A complete a b initio treatment of the 16-electron system over the entire configuration space of importance in reactions R3 and R4 is not feasible at the present time. We therefore simplify the problem to its essential elements, the four bonding electrons, a nonpolarizable Si2+core, and some consideration of the effect of the nonbonded valence electron pair on the bending potential of the system. The minium set of four-electron bond eigenfunctions required to represent the reactant and product limits for reactions R3 and R4 are shown in Figure 1. (7) A. Mavridus and J. F. Harrison, J. Phys. Chem., 86, 1979 (1982). (8) M. Lewerenz, P. J. Bruna, S.D. Peyerimhoff, and R. J. Buenker, Mol. Phys., 49, 1 (1983). (9) J. E. Rice and N. C. Handy, Chem. Phys. Left.,107, 365 (1984). (10) (a) R. Walch, Acc. Chem. Res., 14, 246 (1981). (b) T. A. Carlson, N. Durie, P. Erman, and M. Larsson, J . Phys. B, 11, 3667 (1978). (11) J. M. Brown and A. D. Fackerell, Phys. Scr., 25, 351 (1982). (12) For example, see P. 0. Lowdin, J. Mol. Specfrosc.,3, 2232 (1962).
0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 8, 1985 1429
VB Calculation of SiH2 Potential-Energy Surface H
U
TABLE I: Ground- and Excited-State Parameters for H2and SiH parameter H,' SiH D,eV 4.7466 3.34 1.004435 a,au-' 0.7585128 1.402 2.8743 Re, au 1.9668 1.0933901 4, eV 6, au-l 1.000122 0.5702456 25.55301785 201.3031846 C, eV A , au 1 .o -1.664834951 (I, au-I 1.6756385 1.500326208 1.6 3.0 R*, au
H %I
H .C
a
a Si
si.'
0
e b
i:
are computed by using a modified form of the Heitler-London expression:
Qij = 0.5('Eij + 3Eij]exp(-o(Rij - R$)
H
H
\d
Reference 15.
*D
*C
Jij = 0.5{'E, - 'Eij]
Figure 1. Canonical structures for SiH2.
(ij= a, c or 6, d)
If we assume that the molecular wave functions can be adequately represented with a minimal basis set of one orbital per electron, the corresponding wave functions, which are simultaneous eigenfunctions of S2 and S,, can be written in the form NA(4I - 43 - 4 4
+A
+E
+ '$6)
= NB(dl - 43 + 4 4 - 461
+c = NC(4l + 43 - 4 4 - 461 #D
= ND(42 + 43 - 4 4 - 451
(1)
where the 4i are the Slater determinants 41 = labc'dl, 42 = labkdl, 43 = la'bcdl 44
(2)
= l~bk'dl, 45 = l ~ ' b ~ ' d (4, 6 = (a'b'cdl
The potential energy E is then obtained as the lowest root, EL, of the secular equation
IR- ESl = 0
(3)
where the matrix elements are Hap
=
(J/aIE.rl#@)
(a,B = A$, C, or
Sap = (+,I+p)
D)
(4)
The secul_ar equation may be rewritten as an eigenvalue problem, f?# = ES#, and solved by using the IMSL subroutine EIGZF. Finally, we write the total SiH2 potential as the sum VSiHz
=EL +
(6)
vb
(5)
where V, is a bending potential added to account for the effect of the nonbonded valence electron pair. The evaluation of the matrix elements in eq 4 requires the evaluation of integrals of the type ( 4 i l f l ~ j )We . assume that the one-electron basis functions are mutually orthogonal so that all integrals involving multiple exchanges vanish. Thus, the matrix element Hil is zero unless $i and differ only in the spins of two orbitals, in which case H,, is the negative of the corresponding exchange integral. The matrix element Hii is the sum of the diatomic coulomb integrals minus the sum of all exchange integrals between orbitals having the same spin. A more detailed description of the procedure can be found in many standard reference^.'^ With these assumptions, the evaluation of the matrix elements reduces to the evaluation of coulomb and exchange integrals between the one-electron basis functions. These integrals are obtained semiempirically in a manner that incorporates as much of the available experimental and a b initio data as possible. The two-center coulomb and exchange integrals between Si-H orbitals (13) For example, see H. Eyring, J. Walter, and G. E. Kimball, "Quantum Chemistry", Wiley, New York, 1944, Chapter 13.
where lEijand 3Eijare the ground-state bonding and excited-state antibonding energies, respectively, for the i-j diatomic system. The coulomb and exchange integrals between hydrogen orbitals are computed by using the Heitler-London expression, i.e., o = 0 in eq 6. The two-center coulomb integrals between orbitals centered on nonbonded atoms are expressed in terms of the corresponding two-center coulomb integrals between bonded atoms: Qad
= 0.5('Ebd + 3Ebd1 expb(Rbd -
Qbe
= 0*5('Eac 3 E a J exPb'(Rac - Q
31
(7)
The exchange integrals between pairs of nonbonded orbitals are set to zero. The modified forms for the Heitler-London expression for the coulomb integrals, eq 6 and 7, were employed to gain two additional parametric degrees of freedom, w and u, which are employed to fit the surface to the experimental endothermicities and back-reaction barrier for reaction R3 as discussed below. The choice of exponential functions with powers of 5 and 3 on the distance factors in eq 6 and 7, respectively, was made empirically by testing several possible functions each of which attenuates Qij more rapidly as Rij increases. The single-center coulomb and exchange integrals for the two orbitals centered on the silicon atom are assumed to be invariant with the configuration of the system and are assigned a common constant value: = Jab = A
(8) The ground-state bonding function is represented by a Morse function Qab
'Eij = D(exp(-2a(Rij - Re))- 2 exp(-a(Rij - Re)]] (9) The excited-state energies are fitted to the functional forms previously used for the triplet state of H2:14 'Eij = D3{exp(-2B(Rij- Re))+ 2 exp(-B(Rij - Re)]) for Rij 6 R* 3Eij= C(Rij+ A ) exp(-uRij) for Rij > R*
(IO)
The parameter values for the H2 diatomic have been determined elsewhere.15 The curvature and equilibrium distance parameters, a and Re, for the ground-state Si-H system are obtained by a nonlinear least-squares fit of the ab initio CI results reported by Mavridus and Harrison.' The Si-H well-depth parameter, D, is fitted to the experimental Si-H bond energy reported by Carlson (14) L. Pederson and R. N. Porter, J . Chem. Phys., 47, 4751 (1967). (15) L. M. Raff, L. Stivers, R. N. Porter, D. L. Thompson, and L. B. Sims, J . Chem. Phys., 52, 3449 (1970).
1430 The Journal of Physical Chemistry, Vol. 89, No. 8, 1985
,
I
I
Viswanathan et al.
1
\
f 0.8 cu
n
4
8
0.7
0.5 R1 1a.u.
I 2.0
I
4.0
6.0 RSi+/a.U.
8.0
10.0
Figure 2. Ground-statepotential curves for the Si-H bond: (0)ab initio results, ref 7; (A)least-squares fit of eq 9.
F i e 4. Attenuation of the bending force constant with the Si-H bond distance. The line is the least-squares fit of eq 12. The (0) points are ab initio SCF results.
-3'6t
3 0 -'%
-4.2
-6.6
-7.2
20
0
40
60
80
100
120 140 160 180
e Figure 5. Bending potential of SiH2as a function of the bond angle: (A) CI results from ref 9;( 0 )VB surface. .
)
6
.
TABLE IIk Bending Potential Parameters for SiHl
0.0 ' RSi-H/a.U.
Figure 3. Variation of 411 state energy with bond length for the Si-H bond: (A)CI results, ref 8; ( 0 )least-squares fit of eq 10. TABLE II: Matrix Element Parameters for VB Surface Darameter value w, au+ 0.0080 Y , au-3 0.60 0.20 A, eV
et al.Iob The Si-H excited-state parameters are obtained by a nonlinear least-squares fit of the MRD-CI results for the 411 state obtained by Lewerenz, Bruna, Peyerimhoff, and Buenker.8 These diatomic parameters are given in Table I. Comparisons between the results obtained from eq 9 and 10 and the ab initio values for the ground and excited states of Si-H are shown in Figures 2 and 3, respectively. The parameters w , Y, and X are obtained by fitting the surface to the experimental endothermicities for reactions R3 and R4 and to the back-reaction barrier for reaction R3. We obtain the back-reaction barrier from the results of ab initio SCF calculations using GAUSSIAN 80 with a 6-31G basis set. To compensate for the lack of configuration mixing in the S C F wave function, the S C F energies are scaled to reproduce. the experimental endothermicity. This procedure predicts a 0.30-eV barrier at RSi-H = 4.2 au for the reaction Si + H2 SiH2 along a symmetric abstraction pathway. The values obtained for w , v, and X are given in Table 11. -+
parameter
value
k:, eV/rad2 AI A2, Be, deg
3.075664 1.038057 0.8120513 92
The equilibrium SiH2 angle and bending potential is strongly influenced by the nonbonded electron pair on the silicon atom. Since this pair is not explicitly included in the VB formalism, we introduce a bending potential vb to account for its effect. We take vb
= 0.5kb(Rl,Rz)(O- 8,)'
for 6 - 6, > 0
(1 1)
where 8 is the H-Si-H angle and R , and R2 are the two Si-H interatomic distances. The value of kb for the equilibrium configuration of SiH, is obtained by a least-squares fit of the ab initio results. To determine the rate of attenuation of kb as the Si-H bond is stretched, we have carried out ab initio SCF calculations of the bending potential as a function of R 1 . These results are fitted by a nonlinear least-squares procedure to kb(R1,R2)= kboAi exP(iAz(R1 - Re)'
+ A2(R2 - Re)2)) (12)
The bending potential parameters so obtained are given in Table 111. Figure 4 shows a plot of kb/kboas a function of R,.The points are the ab initio SCF results and the line is the least-squares fit of eq 12. Figure 5 shows a compatison of the ab initio CI bending potential reported by Rice and Handy9 with that obtained from eq 5 . There is good agreement between the two results for 8 - Be > 0. For 8 - 6, < 0, the curvature of the VB result is less
The Journal of Physical Chemistry, Vol. 89, No. 8, 1985 1431
VB Calculation of SiH2 Potential-Energy Surface
~21au Fipre 6. SiH2 potential-energy contour map for the dissociation pathway shown in the inset to the diagram with a = 10'. Contour line values are given in electronvolts relative to equilibrium SiH2. 12.4
1 I .3
10.2 9.1
. a' 3
m
8.0 6.9
5.8 4.7
3.6
2.5
R2/au Figure 7. Same as Figure 6 except a = -loo. In this case, the dissociation coordinate corresponds to abstraction along lines that form an angle a with the C2axis of SiH? At the point where a symmetric abstractionof both hydrogen atoms would produce an H-H separation equal to the equilibrium H2 bond length, the dissociation coordinate is altered to a path that corresponds to motion parallel to the C, axis.
than that predicted by the CI calculations. Consequently, the VB surface will yield an equilibrium SiH2 bending frequency that is somewhat too small. The results obtained from eq 1-12 suffice for various types of transition-state theory calculations of rates for reactions R3 and R4. However, quasiclassical trajectory calculations require the gradients of the potential along the course of the trajectory. Such calculation is usually difficult if a closed-form analytic expression is not available for the potential-energy function. However, an extremely simple procedure for the evaluation of the surface derivatives arises from a generalization of the Hellmann-Feynman theorem.I2 The derivatives of E L with respect to R, are given by aEL/aRi
= $LtK'~/aRiI$~/$LtS$L
(13)
where +L is the_eigenvector corresponding to the lowest eigenvalue of eq 3 and aH/aR, is the terivative of the H matrix. Since all of the matrix elements of H are expressed in analytic form, eq 13 provides a simple numerical route to the surface gradients. A
similar approach has been used by Kroger and MuckermanI6 in calculating the surface derivatives in their trajectory study of hot-atom reactions and by NoorBatcha and Sathyamurthy" in their study of the H6system. 111. VB Surface for SiH2 The VB formalism described in section I1 predicts the correct experimental bond energies and equilibrium bond lengths for all diatomic product limits. The calculated endothermicities for reactions R 3 and R4 differ from the experimental v a l ~ e s ~by, ' ~ 0.18%and 6.7%, respectively. The computed equilibrium structure for SiHz corresponds to Si-H bond lengths of 3.120 au with an H-Si-H equilibrium angle of 83O. These results differ from the corresponding experimental values reported by Brown and (16) P.
M. Kroger and J. T. Muckerman, Radiochim. Acta, 28,
(1981). (17) I. NoorBatcha and N. Sathyamurthy, unpublished results.
215
Viswanathan et al.
1432 The Journal of Physical Chemistry, Vol. 89, No. 8, 1985
R2/au Figure 8. Same as Figure 7 except a = -30'.
TABLE I V Comparison of Reactant aod Product Limits of the VB Surface for SiH, with Experimcntpl aod CI Values quantity system VB surface expt or CI R,(B-H), au SiH 2.8143 2.8743" D, eV SiH 3.34 3.34b 3.120 2.874= R,(Si-H), au SiH2 83 92c SiH2 , ,e deg 1.402 1 .4026 R,(H-H), au H2 4.7466d HZ 4.7466 D, eV 2.16V SiH2 Si + H2 2.172 AE,eV 3.428 3.658/ SiH2 SiH + H AE,eV
--
'Reference 7. *Reference lob. cReference 9. d G . Herzberg, 'Molecular, Spectra and Molecular Structure", Vol. I, Van Nostrand, New York, p 530. 'Reference loa. 'Reference 4.
Fackerell" by +8.5% and -9.5%, respectively. A comparison of the results for all reactant and product limits with experimental and CI data is given in Table IV. Figures 6-8 show contour maps of the SiH2 surface. The dissociation coordinate for each map corresponds to abstraction of the two hydrogen atoms along lines that form an angle a with the C, axis of SiHi. At the point where a symmetric abstraction of both hydrogen atoms would produce an H-H separation equal to the equilibrium H2 bond length, the dissociation coordinate is altered to a path that corresponds to motion parallel to the C, axis. Figure 6 shows the potential contours for dissociation along the R , and Rz coordinates shown in the inset for the figure with a = loo. For this dissociation path, the upper left- and lower right-hand portions of the map correspond to dissociation to SiH
+
H. Motion along the figure diagonal leads to Si + 2H. As can be seen, there is no barrier to the back-reaction for reaction R4. Figure 7 shows the potential contours for dissociation along a pathway described by the insert in Figure 6 with a = -10'. This path leads to SiH + H products in the configuration space corresponding to the upper left and lower right of the figure and to Si H2 products for motion along a coordinate corresponding to the figure diagonal. There is a very large barrier (about 3.2 eV) to the back-reaction for the three-center elimination for dissociation along this pathway. The barrier to the back-reaction for reaction R3 decreases continuously as dissociation m u r s along pathways described by decreasing values of a. The contour map for dissociation along the path with a = -30' is shown in Figure 8. The barrier has now decreased to about 0.60 eV. The exact barrier along the reaction coordinate is obtained by minimizing the potential with respect to a at the various values of R , and Rzalong the reaction coordinate. This yields a minimum barrier to the (R3)back-reaction of 0.30 eV in agreement with the scaled SCF results. Such minimization also eliminates the attractive well in the product valley observed in Figure 8. For the minimum-energy dissociation coordinate, there are no local minima.
+
Acknowledgment. The authors thank the Air Force Office of Scientific Research for financial support under Grant No. AFOSR-82-0311. We also thank Dr.I. NoorBatcha for his helpful comments throughout the course of this work. Registry No. SiH,, 13825-90-6.
