Langmuir 1985, 1, 33-39 We have already proposed3 that the pivotal factor in the primary chemistry is the effective charge state of O(a). If this is not too high, electrophilic attack on C2H4 is possible, leading eventually to C2H40formation.
Surface C1 or dissolved 0 compete for Ag lattice electrons and act to reduce the electronic charge on O(a)-the enhancement of selectivity induced by these species is thereby explained. Direct combustion of ethylene is thought to involve a mechanism which is initiated by the stripping of hydrogen from the h y d r ~ g e n . ~ . ~ ~
X This requires the oxygen atom to act as a Bransted base, a process that should be enhanced by increasing its charge state. Cs strongly donates valence charge to the solid;27 much of this ends up on the O(a) favoring reaction 1 at the expense of reaction 2. The positive effect of Cs on C02 production is thus understandable in terms of its role in
33
decreasing the selectivity of the primary stage. It is therefore proposed that the effect of Cs in promoting the yield of ethylene oxide is the result of its action on the secondary chemistry. The further oxidation of ethylene oxide is thought to be initiated by its isomerization to CH3CH0,which then undergoes rapid combustion.24p25A possible mechanism for the isomerization can be suggested as in Scheme I. The crucial ring-opening step is followed by a facile 1,2 hydride shift.2s In this ringopening step, the lowest unoccupied molecular orbital, LUMO, of C2H40 will be lower in energy than for the gas-phase molecule because of orbital relxation following adsorption and the polar nature of the adsorbed species.29 Thus adsorbed ethylene oxide is more prone to attack by nucleophiles having a low-energy highest occupied molecular orbital, HOMO, than the gas-phase molecule, i.e., hard nucleophiles such as Oh.30 Again, the presence of Cs raises the charge density on the oxygen and hence the energy of the corresponding orbital; the HOMO-LUMO energy match is adversely affected, and the ring-opening step is consequently inhibited.
Acknowledgment. R.B.G. thanks the SERC for the award of a Research Studentship and we acknowledge additional finalcial support from IC1 plc. We thank Johnson Matthey Ltd. for a loan of precious metals. Registry No. Cs, 7440-46-2; Ag, 7440-22-4; ethylene, 74-85-1.
Formation of Electron-Hole Pairs in a Semiconductor by Vibrationally Excited Molecules Isidore Last Applied Mathematics, Soreq Nuclear Research Center, Yavne, 70600 Israel
Thomas F. George" Department of Chemistry, University of Rochester, Rochester, New York 14627 Received May 23, 1984 Both one-dimensional and three-dimensional models are presented for the collision of a vibrationally excited molecule with a semiconductor surface, where the transfer of vibrational energy leads to the formation of electron-hole pairs. The transition probability P is calculated as a function of the molecule-surface distance for two real systems, HC1+ InSb and HC1+ PbSe, as well as for some model systems with different values of parameters. While P generally increases as the distance decreases, there are some minima at intermediate distances. The overall probability is obtained as m approximate integral of P over the distance, and for thermal collisions values of a few percent are obtained. Such values are high enough for an experimental observation of electrical conductivity due to electron-hole pair formation.
1. Introduction In the present study we consider the transfer of vibrational energy to the electrons of a semiconductor surface. This A vibrationally excited molecule near a solid surface can process differs from the case of a metallic surface due to transfer vibrational energy to either phonons or electrons the energy gap between the valence and conduction bands. (or both) of the solid. The situation involving excitation Here the transfer of vibrational energy can excite electrons of phonons has been actively and theoretical from the valence to the conduction band and hence result investigations of energy transfer to the electrons of a solid have been carried out for the case of a metallic s ~ r f a c e . ~ - ~ in the formation of electron-hole pairs which can be detected by measuring the semiconductor electrical conductivity.'O In section I1 we present a one-dimensional (1) Wolken, G., Jr. J. Chem. Phys. 1976,60, 2210. model that depends on a transition matrix element. This (2)Lin, Y.-W.; Wolken, G., Jr. J. Chem. Phys. 1976,65, 2634,3729. (3)Diebold, A. C.; Wolken, G., Jr. Surf. Sci. 1979,82,245. element is evaluated in section 111,and the results are used (4)Brus, L. E. J. Chem. Phys. 1980,73, 940. to determine the transition probability in section IV. The (5)Perrson, M.;Perrson, B. N. J. In "Vibrations at Surfaces"; Caupolarization of the solid state is considered in section V, dano, R., Gilles, J. M., Lucas, A. A., Eds.; Plenum: New York, 1982;pp 113-122. (6)Persson, B.N.J.; Lang,N. D. Phys. Reu. B 1982,26,5409. (7)Perrson, B. N. J.; Schaich, W. L. J. Phys. C 1981,14, 5583. (8)Gadzuk, J. W.; Metiu, H. In "Vibrations at Surfaces"; Caudano, R., Gilles, J. M., Lucas, A. A., Eds.; Plenum: New York, 1982;pp 519-540.
(9)Korzeniewski, G.;Maniv, T.; Metiu, H. Chem. Phys. Lett. 1980, 73, 212;J. Chem. Phys. 1982,76, 1564. (10)Last, I.; George, T. F.; Perry, D. S. Mater. Lett. 1984,2, 315.
0743-7463f 85f 24OI-OO33$01.50f 0 0 1985 American Chemical Society
34 Langmuir, Vol. 1, No. 1, 1985
rt
Last and George
1
Conduction band
Z
I
t
I 0
R
c
2
Figure 1. Location of the molecule with respect to the solid-state atoms in a one-dimensional chain: a , lattice constant;R, mole-
Figure 2. Band structure of the semiconductor: Z , vibrational coordinate; z, electronic coordinate, Ev-, top level of the valence band; Ec-, bottom level of the conduction band; E,, vibrational energy level spacing; Ev and Ec, levels involved in the transition.
and an extension to three dimensions is discussed in section VI. The results of numerical calculations, with specific applications to the HC1+ InSb and HCl + PbSe systems, are presented in section VII, and section VI11 is the Summary and Concluding Remarks.
this transition the Ec level is greater than Ev by the vibrational quantum E, Ec = Ev E, (3) The upper limit on the integral is the maximum electronic level of the valence band, and the lower limit is given by
cule-surface distance: d, border distance for the wave functions of eq 11 and 12.
11. The Model
Let us consider a diatomic molecule whose center of mass is located at some distance R from a semiconductor surface. Within the framework of a one-dimensional model, the molecule is assumed to be oriented perpendicular to the surface on the continuation of a line formed by a linear chain of lattice atoms (Figure I), where the interaction of the molecule with the solid is restricted to just the atoms on the chain. We assume the velocity of the molecule to be sufficiently low such that the influence of its translational motion can be neglected. While the vibrational motion of the molecule affects the motion of both the nuclei and electrons of the lattice, we shall consider its effect only on the electrons. The transfer of molecular vibrational energy to the chain electrons leads to the excitation of electrons from the valence band to the conduction band and therefore to the formation of electron-hole pairs. Such excitation is possible only when the vibrational energy level spacing E, = hw is larger than the semiconductor band-gap energy Eg (Figure 2) E, = h w > Eg (1) For simplicity we take the molecule to be in its first excited vibrational state and assume the influence of the solid on its vibrational motion to be negligible so that E, is constant. We further restrict ourselves to cases where E, is less than 2Eg so that only a one-electron transition is possible. Considering the coupling between the vibrational motion of the molecule and the solid electrons as a weak perturbation, we can use the golden rule to determine the transition probability, which can be written for the case of a one-electron transition between two continua as
+
&min
= E""
- (E, - Eg)
(4)
The matrix element Wo,is expressed as an integral over the electronic coordinate z and the vibrational coordinate Z (Figure 2)
Woi(Ev,Ev + Eg) = dZ + E , ' ~ ' ( ~ ) X ~ ( Z ) ~ ~ T ( ~ , Z ) ~ E ~ +(5) E,'~'(~)X~( where +(vand +(c) are the electronic wave functions in the valence and conduction bands, respectively, xo and x1are the ground and first exciied state vibrational wave functions, respectively, and W is the interaction between the electron and the molecule. At close distances, where the overlap between the molecular orbitals agd the electronic states of the solid cannot be neglected, W is complicated. At far enpugh distances where the overlap can be neglected, W becomes electrostatic. If the size of the molecule is smaller than the distance to the solid-state electron, then the field of the vibrating molecule can be expressed as a sum of dipole and higher order multipole potentials. We shall consider only the point dipole potential, as has been done for the case of a metal surface.6 If the polarization of the solid is neglected, i.e., setting the dielectric constant equal to one, then the interaction becomes (in atomic units)
where p ( Z ) is the dipole moment. Substituting eq 6 into 5, we obtain a separation of variables Woi(EvJv+E,) =
-
where L is the length of the chain and pol is the dynamic dipole moment for the 1 0 transition
P is the probability of transition per second, Ev and Ec are one-electron energies of the valence and conduction bands, respectively, f v is the density of electron states in the valence band, qc is the density of empty levels in the conduction band, and Wol is the matrix element for the transition from the excited vibrational state to the ground state with a simultaneous transition from the valence band to the conduction band. Due to energy conservation, for
The separation of variables is also achievable when other multipole moments are included in W . The dynamic dipole moment is known for many diatomic molecules from experiments and/or ab initio calculations. The determination of Wolis thus reduced to the evaluation of a one-
Langmuir, Vol. 1, No. 1, 1985 35
Formation of Electron-Hole Pairs dimensional integral over the solid-state electronic coordinate. In the model considered here, only polar molecules can transfer vibrational energy to the solid. However, at close distances, particularly for an adsorbed molecule, this transfer is possible also for nonpolar molecules due to the overlap between the molecular orbitals and the electronic states of the solid. 111. Transition Matrix Element In order to calculate the transition matrix element displayed in eq 7 , one must know the solid-state electronic wave functions #(m and #('). Forms for these are available in terms of the wavenumber k,11J2but we can reexpress these in unitless and more symmetrical forms in terms of the parameter x determined from the equations k = (g/2)(1 + x), -1 6 x
