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Laser interaction with the end atoms of a linear chain
Surface Science 0 North-Holland
102 (1981) Publishing
L466LSO Company
SURFACE SCIENCE LETTERS LASER INTERACTION
WITH THE END ATOMS OF A LINEAR CHAIN *
William C. MURPHY and Thomas F. GEORGE ** of Chemistry,University of Rochester, Rochester, New York 14627, USA
Department Received
15 July 1980; accepted
for publication
15 August
1980
The effects of laser radiation on a one-dimensional linear system are examined using classical mechanics. Laser-induced desorption of oxygen from silicon and hydrogen from lead are studied in detail.
Most theoretical studies [l] of laser influenced chemistry have been concerned with the gas phase. Recently, however, interest [2] in reactions at the gas-solid interface under the influence of laser radiation has been growing. Motivation for this development has been provided by an experimental investigation by Djidjoev and coworkers [3] on the influence of laser radiation on various surface processes. It was discovered that the laser radiation may substantially affect the rate of desorption from the surface and selectively promote surface reactions. Furthermore, these effects have been found to occur at very low laser power, 10 W/cm2, as compared to the powers of lo6 W/cm2 or greater characteristic of gasphase processes. To attempt to understand these results, we must first consider the difference between a solid and a gas. In particular, the vibrational energy of a solid is divided into complicated bands in contrast to the discrete states of a gas. Interaction of these bands with the adsorbed species and the laser radiation should be important in explaining the experimental results. Although the band structure of even a simple three-dimensional finite solid can be very complicated [4], most major features of this band structure can be reproduced by a simple one-dimensional linear chain [5]. Consequently, the effects of laser radiation on such a linear chain will be examined. Furthermore, we will limit our calculations to classical mechanics since this is the preferred method for lattice
* This research was supported in part by the Office of Naval Research and by the Air Force Office of Scientific Research (AFSC), United States Air Force, under Contract No. F4962078C-0005. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon. ** Alfred P. Sloan Research Fellow; Camille and Henry Dreyfus TeacherrScholar.
L46
W.C. Murphy, T.F. George /Laser interaction with the end atoms
LA7
dynamical calculations [4,6]. Our system consists of a harmonic chain of N - 1 identical atoms with one atom of a different species attached on the end. Charge transfer, 4, will be assumed between the adatom and the surface atom. A laser field with the electric field parallel to the chain will be coupled to this dipole. Under these circumstances, the bond equations of motion are
(lb) 2 CY Sit l?Sj=-GO[Sj+M(Sj_I
&,-_I •t rk,-,
+Sj+l),
2 (Y = - - (YSN-_1 •t - SN-2 , A4
M
3GjGN-2,
UC) (14
where Sj is the displacement of the bond length between atoms j and j + 1 from equilibrium; r is a damping factor; ~1is the reduced mass of the adatom-surface atom bond; M is the mass of the lattice atom; Q!is the coupling constant for the lattice; and E is the electric field due to the laser. The potential, V(sl), coupling the adatom to the bulk, has been treated as a harmonic potential, and more realistically as a Morse oscillator. The equation of motion for the center-of-mass translation has been ignored.
ka
Fig. 1. Dispersion relation for a onedimensional semi-infinite chain with an adatom on one end. Isolated point at n/2 is due to the presence of the adatom. Fig. 2. Minimum laser intensity needed for the desorption of an oxygen atom from a semiinfinite chain of silicon within the truncated harmonic oscillator approximation.
L48
W.C. Murphy,
T.F. George /Laser
interaction
with the end atoms
The first object of this study is to determine the band structure of the system. For this purpose, the surface potential is treated harmonically, with the field absent (E = 0). The solutions of eq. (1) are of the form Si = s ei(wf+kia) >
(2)
where a is the equilibrium separation between lattice atoms. For this solution the dispersion relationship is shown in fig. 1. The discrete state at k = raa/2 corresponds to vibrations localized in the surface area; k has a large imaginary part which leads to exponential damping. This state occurs at w = WS = (cys/Z.Q1’2,
(3)
where o, is the adatom-surface oidal with its peak occurring at w -.\/zwr_
atom diatomic
frequency.
Also, the band is sinus-
= (4cX/M)“2 ,
(4)
where wL is the lattice atom diatomic frequency. We now introduce an electric field into the system of the form E = (87rZ/~)“~ sin(ot)
,
(5)
where Z is the intensity and c the speed of light. In addition to solving eq. (1) for sr , we are also interested in determining the energy of the first bond: &
=I!5 j2 *P 1 cycle
j-
dt$+3 p
dts:,
(6)
1 cycle
where we have averaged over one laser cycle, t,, to reduce short-term oscillations of the energy. After much algebra, eqs. (l), (5) and (6) yield the minimum intensity necessary to obtain a given energy, g :
(7a) (7b) (7c) cos(0)
=
(L$_- cd2+ il?W)/ot .
(74
As shown in eq. (7) the energy level and the charge only appear in IO. Consequently, Z is linear in the energy and inversely quadratic with the charge. The behavior of the intensity with respect to the other parameters is not as obvious. For a semi-infinite system, however, we have found that the minimum intensity needed
W.C. Murphy,
T.F. George /Laser interaction with the end atoms
L49
to reach a given energy occurs when @a> @b) (8~) To continue further, we have numerically examined oxygen on silicon and hydrogen on lead, with the physical parameters taken from the gas-phase values of Herzberg [7]. These two systems contrast nicely, since oxygen is bound very tightly to silicon and has a small dipole moment, whereas hydrogen is weakly bound to lead but has a large dipole moment. At o = ws and r = 0, we have examined the intensity to achieve dissociation for various chain lengths. For the oxygen on silicon system, we have obtained I@=
3)/Z, = 7.417 x lo4
z(N-+ “)/lo
= 7.582 X lo4
)
(9) .
As can be seen, little effect is achieved by changing the length. A similar result has also been obtained for hydrogen on lead. Finally, we have studied the variation of the minimum intensity needed for desorption with respect to the laser frequency. The results for oxygen on silicon, given in fig. 2, are again qualitatively similar to the results for hydrogen on lead. Not surprisingly, the deep minimum occurs when the laser is in resonance with the surface frequency. For oxygen on silicon, this minimum occurs at Z = 3.42 X 10’ 3 W/cm’, and for hydrogen on lead at Z = 1.48 X 10’ W/cm’. However, the other sharp minimum occurs at the top of the lattice frequency band. Eq. (1) was also studied numerically with V(s,) being modeled by a Morse potential. The results obtained were consistent with the harmonic oscillator results. Typical time to achieve a steady-state condition was several thousand cycles of the laser at high intensity. This, of course, led to very large execution times and also made it virtually impossible to examine the possibility of desorption at very low intensities. Although the above results preclude laser enhanced desorption for resonance radiation at very low intensities, the minimum in the intensity at the band frequency would hint at the possibility of indirect energy transfer to the surface bond. Laser heating of the surface may consequently be an effective way to achieve enhanced desorption. Furthermore, in a real system, quantum mechanics tells us that there will be a zero-point vibrational energy. Also, at any finite temperature, there will be thermal heating of the solid. These two effects will raise the energy of the surface bond and thus reduce the necessity for high laser intensity. This simple model only considers the phonon spectra. Most catalysis, however,
L50
W.C. Murphy,
T.F. George /Laser
interaction
with the end atoms
seems to take place on metal surfaces [8]. The nearly free electrons of the metal will consequently be an important artifact in surface catalysis. A more complete description of desorption from a metal would therefore have to consider electron excitations and electron-phonon interactions in addition to effects on the phonon spectra. Finally, it should be pointed out that there are many limitations involved in onedimensional systems. In a real finite three-dimensional solid, the band structure can be quite complex 143, both in the number of bands and in their crossings. A complete picture must attempt to account for this multiplicity of bands and their effect on laser stimulated surface desorption. In view of the above limitations, we conclude that an adequate physical representation for laser-stimulated desorption must include three~imensional effects.
References [l] [2] [3]
[4] [5] [6]
[ 71 [8]
See, e.g., J.H. Eberly and P. Lambropoulos, Eds., Multiphoton Processes (Wiley, New York, 1978). T.F. George, J. Lin, KS. Lam and C. Chang, Opt. Eng. 19 (1980) 100. MS. Djidjoev, R.V. Khokhtov, A.V. Kiselev, V.I. Lygin, V.A. Namiot, A.I. Osipov, V.I. Panchenko and B.I. Provotorov, in: Tunable Lasers and Applications, Eds. A. Mooradian, T. Jaeger and P. Stokseth (Springer, Berlin, 1976) p. 100. R.E. Allen, G.P. Alldredge and F.W. de Witte, Phys. Rev. B4 (1971) 1648, 1661. R.F. Waflis, Phys. Rev. 105 (1957) 540. L. Andersson, in: Surface Science, Vol. 1 (lnternat~onal Atomic Energy Agency, Vienna, 1975) p. 113. G. Horzberg, Electronic Spectra and Electronic Structure of Polyatomic Molecules (Van Nostrand, Princeton, 1966). J. Sinfelt, Rev. Mod. Phys. 51 (1979) 569.
102 (1981) Publishing
L466LSO Company
SURFACE SCIENCE LETTERS LASER INTERACTION
WITH THE END ATOMS OF A LINEAR CHAIN *
William C. MURPHY and Thomas F. GEORGE ** of Chemistry,University of Rochester, Rochester, New York 14627, USA
Department Received
15 July 1980; accepted
for publication
15 August
1980
The effects of laser radiation on a one-dimensional linear system are examined using classical mechanics. Laser-induced desorption of oxygen from silicon and hydrogen from lead are studied in detail.
Most theoretical studies [l] of laser influenced chemistry have been concerned with the gas phase. Recently, however, interest [2] in reactions at the gas-solid interface under the influence of laser radiation has been growing. Motivation for this development has been provided by an experimental investigation by Djidjoev and coworkers [3] on the influence of laser radiation on various surface processes. It was discovered that the laser radiation may substantially affect the rate of desorption from the surface and selectively promote surface reactions. Furthermore, these effects have been found to occur at very low laser power, 10 W/cm2, as compared to the powers of lo6 W/cm2 or greater characteristic of gasphase processes. To attempt to understand these results, we must first consider the difference between a solid and a gas. In particular, the vibrational energy of a solid is divided into complicated bands in contrast to the discrete states of a gas. Interaction of these bands with the adsorbed species and the laser radiation should be important in explaining the experimental results. Although the band structure of even a simple three-dimensional finite solid can be very complicated [4], most major features of this band structure can be reproduced by a simple one-dimensional linear chain [5]. Consequently, the effects of laser radiation on such a linear chain will be examined. Furthermore, we will limit our calculations to classical mechanics since this is the preferred method for lattice
* This research was supported in part by the Office of Naval Research and by the Air Force Office of Scientific Research (AFSC), United States Air Force, under Contract No. F4962078C-0005. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon. ** Alfred P. Sloan Research Fellow; Camille and Henry Dreyfus TeacherrScholar.
L46
W.C. Murphy, T.F. George /Laser interaction with the end atoms
LA7
dynamical calculations [4,6]. Our system consists of a harmonic chain of N - 1 identical atoms with one atom of a different species attached on the end. Charge transfer, 4, will be assumed between the adatom and the surface atom. A laser field with the electric field parallel to the chain will be coupled to this dipole. Under these circumstances, the bond equations of motion are
(lb) 2 CY Sit l?Sj=-GO[Sj+M(Sj_I
&,-_I •t rk,-,
+Sj+l),
2 (Y = - - (YSN-_1 •t - SN-2 , A4
M
3GjGN-2,
UC) (14
where Sj is the displacement of the bond length between atoms j and j + 1 from equilibrium; r is a damping factor; ~1is the reduced mass of the adatom-surface atom bond; M is the mass of the lattice atom; Q!is the coupling constant for the lattice; and E is the electric field due to the laser. The potential, V(sl), coupling the adatom to the bulk, has been treated as a harmonic potential, and more realistically as a Morse oscillator. The equation of motion for the center-of-mass translation has been ignored.
ka
Fig. 1. Dispersion relation for a onedimensional semi-infinite chain with an adatom on one end. Isolated point at n/2 is due to the presence of the adatom. Fig. 2. Minimum laser intensity needed for the desorption of an oxygen atom from a semiinfinite chain of silicon within the truncated harmonic oscillator approximation.
L48
W.C. Murphy,
T.F. George /Laser
interaction
with the end atoms
The first object of this study is to determine the band structure of the system. For this purpose, the surface potential is treated harmonically, with the field absent (E = 0). The solutions of eq. (1) are of the form Si = s ei(wf+kia) >
(2)
where a is the equilibrium separation between lattice atoms. For this solution the dispersion relationship is shown in fig. 1. The discrete state at k = raa/2 corresponds to vibrations localized in the surface area; k has a large imaginary part which leads to exponential damping. This state occurs at w = WS = (cys/Z.Q1’2,
(3)
where o, is the adatom-surface oidal with its peak occurring at w -.\/zwr_
atom diatomic
frequency.
Also, the band is sinus-
= (4cX/M)“2 ,
(4)
where wL is the lattice atom diatomic frequency. We now introduce an electric field into the system of the form E = (87rZ/~)“~ sin(ot)
,
(5)
where Z is the intensity and c the speed of light. In addition to solving eq. (1) for sr , we are also interested in determining the energy of the first bond: &
=I!5 j2 *P 1 cycle
j-
dt$+3 p
dts:,
(6)
1 cycle
where we have averaged over one laser cycle, t,, to reduce short-term oscillations of the energy. After much algebra, eqs. (l), (5) and (6) yield the minimum intensity necessary to obtain a given energy, g :
(7a) (7b) (7c) cos(0)
=
(L$_- cd2+ il?W)/ot .
(74
As shown in eq. (7) the energy level and the charge only appear in IO. Consequently, Z is linear in the energy and inversely quadratic with the charge. The behavior of the intensity with respect to the other parameters is not as obvious. For a semi-infinite system, however, we have found that the minimum intensity needed
W.C. Murphy,
T.F. George /Laser interaction with the end atoms
L49
to reach a given energy occurs when @a> @b) (8~) To continue further, we have numerically examined oxygen on silicon and hydrogen on lead, with the physical parameters taken from the gas-phase values of Herzberg [7]. These two systems contrast nicely, since oxygen is bound very tightly to silicon and has a small dipole moment, whereas hydrogen is weakly bound to lead but has a large dipole moment. At o = ws and r = 0, we have examined the intensity to achieve dissociation for various chain lengths. For the oxygen on silicon system, we have obtained I@=
3)/Z, = 7.417 x lo4
z(N-+ “)/lo
= 7.582 X lo4
)
(9) .
As can be seen, little effect is achieved by changing the length. A similar result has also been obtained for hydrogen on lead. Finally, we have studied the variation of the minimum intensity needed for desorption with respect to the laser frequency. The results for oxygen on silicon, given in fig. 2, are again qualitatively similar to the results for hydrogen on lead. Not surprisingly, the deep minimum occurs when the laser is in resonance with the surface frequency. For oxygen on silicon, this minimum occurs at Z = 3.42 X 10’ 3 W/cm’, and for hydrogen on lead at Z = 1.48 X 10’ W/cm’. However, the other sharp minimum occurs at the top of the lattice frequency band. Eq. (1) was also studied numerically with V(s,) being modeled by a Morse potential. The results obtained were consistent with the harmonic oscillator results. Typical time to achieve a steady-state condition was several thousand cycles of the laser at high intensity. This, of course, led to very large execution times and also made it virtually impossible to examine the possibility of desorption at very low intensities. Although the above results preclude laser enhanced desorption for resonance radiation at very low intensities, the minimum in the intensity at the band frequency would hint at the possibility of indirect energy transfer to the surface bond. Laser heating of the surface may consequently be an effective way to achieve enhanced desorption. Furthermore, in a real system, quantum mechanics tells us that there will be a zero-point vibrational energy. Also, at any finite temperature, there will be thermal heating of the solid. These two effects will raise the energy of the surface bond and thus reduce the necessity for high laser intensity. This simple model only considers the phonon spectra. Most catalysis, however,
L50
W.C. Murphy,
T.F. George /Laser
interaction
with the end atoms
seems to take place on metal surfaces [8]. The nearly free electrons of the metal will consequently be an important artifact in surface catalysis. A more complete description of desorption from a metal would therefore have to consider electron excitations and electron-phonon interactions in addition to effects on the phonon spectra. Finally, it should be pointed out that there are many limitations involved in onedimensional systems. In a real finite three-dimensional solid, the band structure can be quite complex 143, both in the number of bands and in their crossings. A complete picture must attempt to account for this multiplicity of bands and their effect on laser stimulated surface desorption. In view of the above limitations, we conclude that an adequate physical representation for laser-stimulated desorption must include three~imensional effects.
References [l] [2] [3]
[4] [5] [6]
[ 71 [8]
See, e.g., J.H. Eberly and P. Lambropoulos, Eds., Multiphoton Processes (Wiley, New York, 1978). T.F. George, J. Lin, KS. Lam and C. Chang, Opt. Eng. 19 (1980) 100. MS. Djidjoev, R.V. Khokhtov, A.V. Kiselev, V.I. Lygin, V.A. Namiot, A.I. Osipov, V.I. Panchenko and B.I. Provotorov, in: Tunable Lasers and Applications, Eds. A. Mooradian, T. Jaeger and P. Stokseth (Springer, Berlin, 1976) p. 100. R.E. Allen, G.P. Alldredge and F.W. de Witte, Phys. Rev. B4 (1971) 1648, 1661. R.F. Waflis, Phys. Rev. 105 (1957) 540. L. Andersson, in: Surface Science, Vol. 1 (lnternat~onal Atomic Energy Agency, Vienna, 1975) p. 113. G. Horzberg, Electronic Spectra and Electronic Structure of Polyatomic Molecules (Van Nostrand, Princeton, 1966). J. Sinfelt, Rev. Mod. Phys. 51 (1979) 569.