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Calculation of neutral fraction for ion neutralization at metal surfaces by bulk-plasmon excitation
ARTICLE IN PRESS
Vacuum 81 (2006) 150–154 www.elsevier.com/locate/vacuum
Calculation of neutral fraction for ion neutralization at metal surfaces by bulk-plasmon excitation Abdalaziz A. Almulhem Physics Department, King Faisal University PO Box 9149 Al-Ahssa, 31982, Saudi Arabia
Abstract The excitation of bulk plasmons during the neutralization of protons scattered from metal surfaces was analyzed theoretically in previous work. Also, experimental evidence for this mechanism is now discussed in the literature. In this work the neutral fraction generated by this mechanism is calculated. The calculation for the neutral fraction in the scattering of protons from aluminum surface shows that it is quite small compared to that for surface-plasmon-assisted neutralization. The characteristic velocity for bulk-plasmonsassisted neutralization is 0.0009 (in atomic units). Its value for surface-plasmon-assisted neutralization is 0.75. It is concluded that, within the assumption of our theory that there is no penetration of protons into the surface (implying low-energy protons), the mechanism of surface-plasmons-assisted neutralization would be expected to be more important than that of bulk-plasmon-assisted neutralization. r 2006 Elsevier Ltd. All rights reserved. Keywords: Plasmon-assisted neutralization; Ion neutralization; Charge fraction; Ion–solid interaction
1. Introduction Ion surface collisions have been used as an analytical tool for surfaces for a long time. Ion-scattering spectrometry with time-of-flight (TOF-ISS) analysis can resolvee the contributions to the backscattering yield coming from different atomic layers and from the different atoms present at the surface. A thorough understanding of the neutralization processes is important for the quantitative analysis in low-energy ion-scattering spectroscopy (ISS or LEIS) and secondary ion mass spectroscopy (SIMS) [1]. Different mechanisms of ion neutralization seem to compete with each other in ion scattering from metal surfaces. Plasmon excitation and decay competes with the other two mechanisms for potential electron emission at metal surfaces: Auger neutralization and resonance neutralization. Since the work by Baragiola and Dukes [9], suggesting that electron emission from plasmon decay is more important than Auger neutralization, several investigations have been performed to study potential electron emission from free electron metals [2–8]. Tel.: +966 3 575 6645; fax: +966 3 580 5182.
E-mail address: [email protected]. 0042-207X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.vacuum.2006.03.019
The neutralization energy released by electron capture of slow ions at metal surfaces provides potential energy that can excite plasmons which are quantized collective oscillations of the metal charge density. When the potential energy transferred to the solid is equal to the energy of those plasmons plasmon excitation is probable. Electron emission is then expected as a result of exciting valence electrons of the solid by the decaying plasmons. Surface electronic excitations have been the object of much theoretical and experimental work. Specifically for surface plasmons significant progress has been made over recent years as a result of considerable experimental and theoretical work. It has been recently shown that plasmonassisted neutralization is an important process [10]. In plasmon-assisted neutralization an electron is transferred from the valence band of the metal to the ground state of the ion and the excess energy is utilized in plasmon excitation. A number of theoretical studies have been devoted to ion-induced plasmon excitation. This has been considered in terms of direct excitation in fast ion scattering [2–8], non-radiative de-excitation of excited atoms leaving a surface [11,12] and neutralization of incident ions by electrons from the metal valence band [2–8]. Electrons or photons resulting from plasmon decay
ARTICLE IN PRESS A.A. Almulhem / Vacuum 81 (2006) 150–154
are used to study ion-induced plasmon excitation. Most recent experimental studies have focused on the identification of small structures observable in derivatives of secondary electron spectra, ascribable to plasmon decay [13–16]. These structures have been attributed to surface and bulk plasmon excitation. Most of the studies of potential plasmon excitation have considered ion impact on Al surfaces. Because of the overlap in the plasmon and Auger structures in the electron energy spectra for the Al surface, experimental analysis is difficult. For low-energy ions, the plasmon structure observed in the spectra is attributed to the decay of multiple surface plasmons excited by the potential energy released by the neutralization at the surface of incoming ions. Bulk-plasmon excitation is expected at higher impact energies. It is well understood that charged particles interacting with solid surfaces can create electronic collective excitations in the solid (surface and bulk plasmons). The scalar electric potential due to bulk plasmons vanishes outside the surface [17] in the absence of electron-gas dispersion. This implies that probes exterior to the solid can only generate surface excitations. However, external probes have been shown to interact with bulk plasmons by Barton [18] and Eguiluz [19], and more recently, by Nazarov and Luniakov [20]; this is if electron-gas dispersion is assumed. The fact that within a non-local description of screening, bulk plasmons do give rise to a potential outside the solid has been ignored over the years [21,22]. Recently, Baragiola and Dukes [9] have studied the emission spectra produced by slow ions that were incident at grazing angles; their data indicate that the bulk plasmon is importantly involved in the emission process, though the projectiles are not expected to have penetrated into the solid. Bulk-plasmon excitation in electron emission spectra produced by slow multiply charged ions has also been investigated [25], with projectiles that may enter the solid. The excitation of surface and bulk plasmons during the neutralization of slow ions scattered from surfaces of freeelectron metals has been the subject of intense investigations from both the theoretical and the experimental point of view [8,9,23–27]. Plasmon excitation can be experimentally identified by the characteristic structure produced in the energy distribution of electrons emitted from the solid. This is possible because the plasmon decays by valence-electron excitation. Plasmon structures have been observed for the first time by Baragiola and Dukes in the energy distributions of electrons emitted by the freeelectron metals Al and Mg under slow noble gas ions impact [28]. Since then, several investigations [24–26,29] have been performed to study plasmon excitation by slow ions. Direct or potential excitation of plasmons can take place if the available ion neutralization energy Epot exceeds the sum of plasmon energy (15.3 and 10.9 eV for bulk and surface plasmons, respectively) and Al work function (4.3 eV). With the experimental conditions in the work of
151
Eder et al. [29] excitation of bulk plasmons was considerably stronger than of surface plasmons. The excitation of bulk plasmons was reported in experiments of bombardment of Al and Mg surfaces with keV ions. Recently it has been proposed theoretically [30] that bulk plasmons can be excited by charged particles moving outside a metal surface with a probability that decreases strongly with increasing ion surface distance. Riccardi et al. [31] have reported studies of plasmon excitation in Mg under the impact of slow He+ ions in the incident energy range 0.1–0.5 keV. Consistent with studies on Al surfaces, a transition from surface to bulk-plasmon excitation occurs as the energy of the ion is increased. Riccardi et al. have also reported experimental studies of electron emission from decay of bulk plasmons excited in the interaction of slow single charged Ne and Ar ions with polycrystalline Al surfaces [23]. The question of whether bulk plasmons can be excited by charged particles moving outside the surface was answered by performing measurements as a function of emission angle. Their experiments show that bulk plasmon excitation occurs inside the solid for incident ion energies above a threshold of 1 keV. Barone et al. [32] have studied electron emission from decay of plasmons excited in the interactions of slow single charged Ne ions with Al surfaces as a function of incident energy ranging from 0.7 to 8 keV. Bulk plasmon excitation was found to occur above a threshold incident energy of 1 keV and initially coexists with potential excitation of surface modes. The excitation of bulk plasmons in Al is determined by fast electrons travelling inside the solid. 2. Plasmons The mechanism of surface plasmon assisted ion neutralization is now a well established idea both theoretically [33] and experimentally [8,9,23–27]. for the case of bulkplasmon-assisted ion neutralization the idea must be investigated further. this is because in almost all theoretical work on bulk-plasmon-assisted neutralization; it is assumed that no penetration of the surface occurs; since the potential of the bulk plasmons does not interact with charged particles outside the surface in the free-electron gas model. However, if electron gas dispersion is assumed; the interaction between the potential of the bulk plasmon and charged particles moving outside the surface is possible. within the quantized hydrodynamic model of the bounded electron gas, an explicit expression for the probability of bulk-plasmon excitation by external charged particles moving parallel to a jellium surface can be derived. A sharp electron density profile at the surface is assumed. The ion is assumed to be moving with velocity v outside the metal surface. The ionic background is represented by the jellium model (the jellium occupying the half-space zo0), and a sharp electron-density profile at the surface is assumed. The dispersion relations for bulk and surface
ARTICLE IN PRESS A.A. Almulhem / Vacuum 81 (2006) 150–154
152
plasmons frequencies are given by o2B ¼ o2p þ b2 ðq2 þ p2 Þ, o2S ¼
1 2
h
i o2p þ b2 q2 þ b qð2 o2p þ b2 q2 Þ1=2 ,
(1) (2)
where, op is the plasma frequency, q is the wave vector and b represents the speed of propagation of hydrodynamic pffiffiffiffiffiffiffiffi disturbances in the electron system. The value b ¼ 3=5 is chosen. Bulk dielectric function can be derived within the hydrodynamic approximation. The scalar electric potential due to bulk and surface plasmons is given by [32] h i X ** FB ¼ V1 f B ei q R bb y þ bb ; (3) q40
FS ¼ A1
X q
* * y f S ei q R b c . c þb
(4)
Here, bb y ; bb are creation and annihilation operators for bulk plasmons. Similarly, b c y; b c are creation and annihilation operators for surface plasmons. And f B , f S are bulk and surface coupling functions, respectively: pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p=oB op p eq z B f ¼h (5) i1=2 , p4 þ p2 ðq2 þ o2p =b2 Þ þ o4p =ð4b4 Þ S
f ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p g=oS op eq z ½q ðq þ 2 gÞ1=2
.
(6)
In the last equation g represents the inverse decay length of surface plasmon charge fluctuations: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 g¼ bq þ 2 o2p þ b2 q2 . (7) 2b The quantized hydrodynamic model of the bounded electron gas proves to be useful to demonstrate that bulk plasmons undergo real excitations, even in the case of charged particles that do not penetrate into the solid. 3. Neutral fraction calculations The theoretical work for the calculation of the transition rate for bulk-plasmon-assisted neutralization was developed in previous work by this author [33]. There the transition rate was calculated using a second-quantized Hamiltonian that was acted upon by a unitary transformation. It was shown in that paper that the transition rate for this mechanism depends exponentially on the ion distance from the surface. Also, the transition rate for bulkplasmon-assisted neutralization was lower by two orders of magnitude than the transition rate for surface-plasmonassisted neutralization. Further work on surface-plasmonassisted neutralization was published [34] which does not rely on the fixed ion approximation. Furthermore, the neutral fraction for surface-plasmon-assisted ion neutralization was discussed in a recent work [35]. In this paper the
neutral fraction, furthermore ion neutralization at metal surfaces assuming the excitation of bulk plasmons during the neutralization will be calculated. The results of the transition rate from earlier work [33] will be used. The neutralization rate for bulk and surface-plasmonassisted neutralization respectively was given by ZZZ * * V PBulk ¼ (8) dK dkz dqz Q qz jBj2 . op p2 The matrix element B is evaluated K 2 ¼ ef 12 k20 ¼ Eð1sÞ þ op 12 k2z . Z V g2 dkz jSj2 . PSurface ¼ 2p2
at
1 2
(9) *2
Similarly, the matrix element S here is evaluated at 12 K ¼ ef 12 k20 ¼ Eð1sÞ þ oS 12 k2z . In the above equations, * Q is the plasmon wave vector in the parallel to the surface * * direction and K is the component of k parallel to the surface. Since experimental work represents the results of ion neutralization by scattering from metal surfaces in the form of a particle count (charged and neutral), theoretical neutral fraction calculation would be appropriate. The neutral fraction is defined as the ratio of particles neutralized during scattering over the total number (charged and neutralized) of particles scattered. Within the approximations usually assumed in this field (classical trajectory for the charged particle and specular reflection with constant charged particle velocity v? in the perpendicular direction) the neutral fraction f 0 assumes a simple relation with the transition given by Z 1 2 f 0 ¼ 1 exp Pðs0 Þ d s0 . (10) v? s0 In the above equation Pðs0 Þ is the transition rate for either bulk or surface plasmon, and s0 is the distance from the surface. The limits of the integral are given by s0 and 1 where s0 is the distance of closest approach. The value of the integral represents a measuring parameter vc, usually called characteristic velocity. Z 1 Pðs0 Þ d s0 . (11) vc ¼ 2 s0
Large values for vc (characteristic velocity) correspond to strong neutralization. Theoretical calculations of the neutral fraction are applied for the system of proton scattering from an aluminum surface. Aluminum metal is a good example of a semi-free electron metal where the approximations assumed in our theory are appropriate. Also the experimental evidence of surface and bulk plasmons is available in the literature. Fig. 1 shows the neutral fraction for protons scattered from an aluminum surface assuming bulk-plasmon excitation during the neutralization. The x-axis represents the component of the proton velocity measured in atomic units in the direction perpendicular to the surface. Atomic units for the velocity can be deducted
ARTICLE IN PRESS A.A. Almulhem / Vacuum 81 (2006) 150–154 0.10
Neutral fraction
0.08
0.06
0.04
0.02
0.00 0.0
0.2
0.4 0.6 Proton velocity
0.8
1.0
Fig. 1. Neutral fraction for bulk-plasmon-assisted neutralization against proton velocity in the direction perpendicular to the surface v? (atomic units for velocities are used).
1.0
0.8 Bulk
153
higher velocities do not reveal greater probability of bulkplasmon excitation. Clearly if penetration of the surface was assumed (that would be more probable for more energetic protons) then bulk plasmons would have a higher probability of excitation. The characteristic velocities vc, as defined in Eq. (11), for bulk-plasmon-assisted proton neutralizations is extracted from the figure to be 0.0009. This corresponds to an energy of 0.02 eV. This is a very small value compared with vc for surface plasmon-assisted ion neutralization where vc equals 0.75 and corresponding to an energy of 14 keV. In fact the small value of the characteristic velocity for bulk-plasmon-assisted neutralization compared to that for surface-plasmon-assisted neutralization is a mathematical interpretation of the low probability of this mechanism. In order to compare between bulk-plasmon-assisted and surface-plasmon-assisted neutralization, Fig. 2 is plotted. The figure shows the neutral fractions for both mechanisms. The figure shows clearly that the neutral fraction for surface-plasmon-assisted neutralization is much greater than that for the bulk-plasmon-assisted neutralization. This is definitely expected with our assumption of no ion penetration of the surface. It is expected that any theory that takes into account both mechanisms simultaneously would give the same result.
Surface
4. Conclusion
Neutral fraction
0.6
0.4
0.2
0.0
0.0
0.2
0.4 0.6 Proton velocity
0.8
1.0
In conclusion, the neutral fraction for ion neutralization by bulk plasmons was calculated. It was seen that this neutral fraction is very small, implying a very low probability for the neutralization via this mechanism. When compared to the neutral fraction for the mechanism of neutralization via surface plasmons, it was seen that this mechanism can be neglected. This might be understood and acceptable since in our theory it was assumed that no ion penetration to the surface is taken into account. A characteristic velocity was deduced for ion neutralization by both bulk and surface plasmons. This characteristic velocity that appears in the equation for the neutral fraction can be used as a measure of the importance of the neutralization mechanism. Higher values for this characteristic velocity vc correspond to strong neutralization probability. References
Fig. 2. Neutral fraction for plasmon-assisted neutralization against proton velocity in the direction perpendicular to the surface v? (atomic units for velocities are used). The broken curve represents the case of surface plasmon-assisted neutralization. The solid curve represents the case of bulk-plasmon-assisted neutralization.
from the relation that for velocity 1 atomic unit corresponds to 2.187 106 m/s. It is clear from the figure that the neutral fraction is so small that it might be assumed zero for velocities greater than 0.4 (in atomic units). Let us not forget that no penetration is assumed. This is why
[1] Guillemot D, Staicu-Casagrande EM, Lacombe S, Esaulov VA, Pasquali L, Nannarone S, et al. Nucl Instrum Methods B 2001; 182:41–8. [2] Almulhem AA, Girardeau MD. Surf Sci 1989;210:138–62. [3] Lucas AA, Sunjic M. Surf Sci 1972;32:439–58. [4] Guttieres FA. Surf Sci 1997;370:77–84. [5] Lorente N, Monreal R. Surf Sci 1997;370:324–38. [6] Monreal R, Lorente N. Phys Rev B 1995;52:4760–3. [7] Alducin M. Phys Rev A 1996;53:4222–7. [8] Ritzau SM, Baragiola RA, Monreal RC. Phys Rev B 1999;59: 15506–12. [9] Baragiola RA, Dukes CA. Phys Rev Lett 1996;76:2547–50.
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[10] Bandurin Y, Lacombe S, Guillemot L, Esaulov VA. Surf Sci Lett 2002;513:L413–8. [11] Zimny R. Nucl Instrum Methods B 1993;78:153–8. [12] Gerstein JI, Tsoar N. Phys Rev B 1974;9:4038–41. [13] Spierings G, Urazgil’din I, Zeijlmans van Emmichoven PA, Neihaus A. Nucl Instrum Methods B 1996;115:220–3. [14] Riccardi P, Barone P, Camarca M, Mlandarino N, Xu F, Oliva A, et al. Nucl Instrum Methods B 2001;182:84–8. [15] Stolterfoht N, Bremer JH, Hoffman V, Ro¨sler M, Baragiola R, De Gortari I. Nucl Instrum Methods B 2001;182:89–95. [16] Winter HP, Eder H, Aumayer F, Lorincik J, Sroubek Z. Nucl Instrum Methods B 2001;182:15–22. [17] Mahan GD, in: Devreese JT, Kunz AB, Collins TC, editors. Elementary Excitations in Solids, Molecules and Atoms. London: Plenum; 1974. p. 93. [18] Barton G. Rep Prog Phys 1979;42:963–1016. [19] Eguiluz AG. Solid State Commun 1980;33:21–5. [20] Nazarov VU, Luniakov YV. Phys Rev B 1995;52:12414–8. [21] Gersten JI, Tzoar N. Phys Rev B 1973;8:5671–83. [22] Garcı´ a de Abajo FJ, Echenique PM. Phys Rev B 1992;46:2663–75.
[23] Baragiola RA, Dukes CA, Riccardi P. Nucl Instrum Methods B 2001;182:73–83. [24] Riccardi P, Barone P, Bonanno A, Oliva A, Baragiola RA. Phys Rev Lett 2000;84:371–8. [25] Niemann D, Grether M, Ro¨sler M, Stolterfoht N. Phys Rev Lett 1998;80:3328–31. [26] Monreal R. Surf Sci 1997;388:231–41. [27] Van Someren B, Zeijlmans van Emmichoven PA, Urazgil’din IF, Niehaus A. Phys Rev A 2000;61:032902–8. [28] Baragiola RA, Dukes CA. Phys Rev Lett 1996;76:14–7. [29] Eder H, Aumayr F, Berlinger P, Sto¨ri H, Winter HP. Surf Scie 2001;472:195–204. [30] Bergara A, Pitarke JM, Ritchie RH. Phys Lett A 1999;256:405–10. [31] Riccardi P, Sindona A, Barone P, Bonanno A, Oliva A, Baragiola RA. Nucl Instrum Methods B 2003;212:339–45. [32] Barone P, Baragiola RA, Bonanno A, Camarca M, Oliva A, Riccardi P, Xu F. Surf Sci Lett 2001;480:L420–6. [33] Almulhem AA. Brazilian J Phys 2002;32:916–20. [34] Almulhem AA. Chin J Phys 2003;41:121–9. [35] Almulhem AA. Chin J Phys 2005;43:52–7.
Vacuum 81 (2006) 150–154 www.elsevier.com/locate/vacuum
Calculation of neutral fraction for ion neutralization at metal surfaces by bulk-plasmon excitation Abdalaziz A. Almulhem Physics Department, King Faisal University PO Box 9149 Al-Ahssa, 31982, Saudi Arabia
Abstract The excitation of bulk plasmons during the neutralization of protons scattered from metal surfaces was analyzed theoretically in previous work. Also, experimental evidence for this mechanism is now discussed in the literature. In this work the neutral fraction generated by this mechanism is calculated. The calculation for the neutral fraction in the scattering of protons from aluminum surface shows that it is quite small compared to that for surface-plasmon-assisted neutralization. The characteristic velocity for bulk-plasmonsassisted neutralization is 0.0009 (in atomic units). Its value for surface-plasmon-assisted neutralization is 0.75. It is concluded that, within the assumption of our theory that there is no penetration of protons into the surface (implying low-energy protons), the mechanism of surface-plasmons-assisted neutralization would be expected to be more important than that of bulk-plasmon-assisted neutralization. r 2006 Elsevier Ltd. All rights reserved. Keywords: Plasmon-assisted neutralization; Ion neutralization; Charge fraction; Ion–solid interaction
1. Introduction Ion surface collisions have been used as an analytical tool for surfaces for a long time. Ion-scattering spectrometry with time-of-flight (TOF-ISS) analysis can resolvee the contributions to the backscattering yield coming from different atomic layers and from the different atoms present at the surface. A thorough understanding of the neutralization processes is important for the quantitative analysis in low-energy ion-scattering spectroscopy (ISS or LEIS) and secondary ion mass spectroscopy (SIMS) [1]. Different mechanisms of ion neutralization seem to compete with each other in ion scattering from metal surfaces. Plasmon excitation and decay competes with the other two mechanisms for potential electron emission at metal surfaces: Auger neutralization and resonance neutralization. Since the work by Baragiola and Dukes [9], suggesting that electron emission from plasmon decay is more important than Auger neutralization, several investigations have been performed to study potential electron emission from free electron metals [2–8]. Tel.: +966 3 575 6645; fax: +966 3 580 5182.
E-mail address: [email protected]. 0042-207X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.vacuum.2006.03.019
The neutralization energy released by electron capture of slow ions at metal surfaces provides potential energy that can excite plasmons which are quantized collective oscillations of the metal charge density. When the potential energy transferred to the solid is equal to the energy of those plasmons plasmon excitation is probable. Electron emission is then expected as a result of exciting valence electrons of the solid by the decaying plasmons. Surface electronic excitations have been the object of much theoretical and experimental work. Specifically for surface plasmons significant progress has been made over recent years as a result of considerable experimental and theoretical work. It has been recently shown that plasmonassisted neutralization is an important process [10]. In plasmon-assisted neutralization an electron is transferred from the valence band of the metal to the ground state of the ion and the excess energy is utilized in plasmon excitation. A number of theoretical studies have been devoted to ion-induced plasmon excitation. This has been considered in terms of direct excitation in fast ion scattering [2–8], non-radiative de-excitation of excited atoms leaving a surface [11,12] and neutralization of incident ions by electrons from the metal valence band [2–8]. Electrons or photons resulting from plasmon decay
ARTICLE IN PRESS A.A. Almulhem / Vacuum 81 (2006) 150–154
are used to study ion-induced plasmon excitation. Most recent experimental studies have focused on the identification of small structures observable in derivatives of secondary electron spectra, ascribable to plasmon decay [13–16]. These structures have been attributed to surface and bulk plasmon excitation. Most of the studies of potential plasmon excitation have considered ion impact on Al surfaces. Because of the overlap in the plasmon and Auger structures in the electron energy spectra for the Al surface, experimental analysis is difficult. For low-energy ions, the plasmon structure observed in the spectra is attributed to the decay of multiple surface plasmons excited by the potential energy released by the neutralization at the surface of incoming ions. Bulk-plasmon excitation is expected at higher impact energies. It is well understood that charged particles interacting with solid surfaces can create electronic collective excitations in the solid (surface and bulk plasmons). The scalar electric potential due to bulk plasmons vanishes outside the surface [17] in the absence of electron-gas dispersion. This implies that probes exterior to the solid can only generate surface excitations. However, external probes have been shown to interact with bulk plasmons by Barton [18] and Eguiluz [19], and more recently, by Nazarov and Luniakov [20]; this is if electron-gas dispersion is assumed. The fact that within a non-local description of screening, bulk plasmons do give rise to a potential outside the solid has been ignored over the years [21,22]. Recently, Baragiola and Dukes [9] have studied the emission spectra produced by slow ions that were incident at grazing angles; their data indicate that the bulk plasmon is importantly involved in the emission process, though the projectiles are not expected to have penetrated into the solid. Bulk-plasmon excitation in electron emission spectra produced by slow multiply charged ions has also been investigated [25], with projectiles that may enter the solid. The excitation of surface and bulk plasmons during the neutralization of slow ions scattered from surfaces of freeelectron metals has been the subject of intense investigations from both the theoretical and the experimental point of view [8,9,23–27]. Plasmon excitation can be experimentally identified by the characteristic structure produced in the energy distribution of electrons emitted from the solid. This is possible because the plasmon decays by valence-electron excitation. Plasmon structures have been observed for the first time by Baragiola and Dukes in the energy distributions of electrons emitted by the freeelectron metals Al and Mg under slow noble gas ions impact [28]. Since then, several investigations [24–26,29] have been performed to study plasmon excitation by slow ions. Direct or potential excitation of plasmons can take place if the available ion neutralization energy Epot exceeds the sum of plasmon energy (15.3 and 10.9 eV for bulk and surface plasmons, respectively) and Al work function (4.3 eV). With the experimental conditions in the work of
151
Eder et al. [29] excitation of bulk plasmons was considerably stronger than of surface plasmons. The excitation of bulk plasmons was reported in experiments of bombardment of Al and Mg surfaces with keV ions. Recently it has been proposed theoretically [30] that bulk plasmons can be excited by charged particles moving outside a metal surface with a probability that decreases strongly with increasing ion surface distance. Riccardi et al. [31] have reported studies of plasmon excitation in Mg under the impact of slow He+ ions in the incident energy range 0.1–0.5 keV. Consistent with studies on Al surfaces, a transition from surface to bulk-plasmon excitation occurs as the energy of the ion is increased. Riccardi et al. have also reported experimental studies of electron emission from decay of bulk plasmons excited in the interaction of slow single charged Ne and Ar ions with polycrystalline Al surfaces [23]. The question of whether bulk plasmons can be excited by charged particles moving outside the surface was answered by performing measurements as a function of emission angle. Their experiments show that bulk plasmon excitation occurs inside the solid for incident ion energies above a threshold of 1 keV. Barone et al. [32] have studied electron emission from decay of plasmons excited in the interactions of slow single charged Ne ions with Al surfaces as a function of incident energy ranging from 0.7 to 8 keV. Bulk plasmon excitation was found to occur above a threshold incident energy of 1 keV and initially coexists with potential excitation of surface modes. The excitation of bulk plasmons in Al is determined by fast electrons travelling inside the solid. 2. Plasmons The mechanism of surface plasmon assisted ion neutralization is now a well established idea both theoretically [33] and experimentally [8,9,23–27]. for the case of bulkplasmon-assisted ion neutralization the idea must be investigated further. this is because in almost all theoretical work on bulk-plasmon-assisted neutralization; it is assumed that no penetration of the surface occurs; since the potential of the bulk plasmons does not interact with charged particles outside the surface in the free-electron gas model. However, if electron gas dispersion is assumed; the interaction between the potential of the bulk plasmon and charged particles moving outside the surface is possible. within the quantized hydrodynamic model of the bounded electron gas, an explicit expression for the probability of bulk-plasmon excitation by external charged particles moving parallel to a jellium surface can be derived. A sharp electron density profile at the surface is assumed. The ion is assumed to be moving with velocity v outside the metal surface. The ionic background is represented by the jellium model (the jellium occupying the half-space zo0), and a sharp electron-density profile at the surface is assumed. The dispersion relations for bulk and surface
ARTICLE IN PRESS A.A. Almulhem / Vacuum 81 (2006) 150–154
152
plasmons frequencies are given by o2B ¼ o2p þ b2 ðq2 þ p2 Þ, o2S ¼
1 2
h
i o2p þ b2 q2 þ b qð2 o2p þ b2 q2 Þ1=2 ,
(1) (2)
where, op is the plasma frequency, q is the wave vector and b represents the speed of propagation of hydrodynamic pffiffiffiffiffiffiffiffi disturbances in the electron system. The value b ¼ 3=5 is chosen. Bulk dielectric function can be derived within the hydrodynamic approximation. The scalar electric potential due to bulk and surface plasmons is given by [32] h i X ** FB ¼ V1 f B ei q R bb y þ bb ; (3) q40
FS ¼ A1
X q
* * y f S ei q R b c . c þb
(4)
Here, bb y ; bb are creation and annihilation operators for bulk plasmons. Similarly, b c y; b c are creation and annihilation operators for surface plasmons. And f B , f S are bulk and surface coupling functions, respectively: pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p=oB op p eq z B f ¼h (5) i1=2 , p4 þ p2 ðq2 þ o2p =b2 Þ þ o4p =ð4b4 Þ S
f ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p g=oS op eq z ½q ðq þ 2 gÞ1=2
.
(6)
In the last equation g represents the inverse decay length of surface plasmon charge fluctuations: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 g¼ bq þ 2 o2p þ b2 q2 . (7) 2b The quantized hydrodynamic model of the bounded electron gas proves to be useful to demonstrate that bulk plasmons undergo real excitations, even in the case of charged particles that do not penetrate into the solid. 3. Neutral fraction calculations The theoretical work for the calculation of the transition rate for bulk-plasmon-assisted neutralization was developed in previous work by this author [33]. There the transition rate was calculated using a second-quantized Hamiltonian that was acted upon by a unitary transformation. It was shown in that paper that the transition rate for this mechanism depends exponentially on the ion distance from the surface. Also, the transition rate for bulkplasmon-assisted neutralization was lower by two orders of magnitude than the transition rate for surface-plasmonassisted neutralization. Further work on surface-plasmonassisted neutralization was published [34] which does not rely on the fixed ion approximation. Furthermore, the neutral fraction for surface-plasmon-assisted ion neutralization was discussed in a recent work [35]. In this paper the
neutral fraction, furthermore ion neutralization at metal surfaces assuming the excitation of bulk plasmons during the neutralization will be calculated. The results of the transition rate from earlier work [33] will be used. The neutralization rate for bulk and surface-plasmonassisted neutralization respectively was given by ZZZ * * V PBulk ¼ (8) dK dkz dqz Q qz jBj2 . op p2 The matrix element B is evaluated K 2 ¼ ef 12 k20 ¼ Eð1sÞ þ op 12 k2z . Z V g2 dkz jSj2 . PSurface ¼ 2p2
at
1 2
(9) *2
Similarly, the matrix element S here is evaluated at 12 K ¼ ef 12 k20 ¼ Eð1sÞ þ oS 12 k2z . In the above equations, * Q is the plasmon wave vector in the parallel to the surface * * direction and K is the component of k parallel to the surface. Since experimental work represents the results of ion neutralization by scattering from metal surfaces in the form of a particle count (charged and neutral), theoretical neutral fraction calculation would be appropriate. The neutral fraction is defined as the ratio of particles neutralized during scattering over the total number (charged and neutralized) of particles scattered. Within the approximations usually assumed in this field (classical trajectory for the charged particle and specular reflection with constant charged particle velocity v? in the perpendicular direction) the neutral fraction f 0 assumes a simple relation with the transition given by Z 1 2 f 0 ¼ 1 exp Pðs0 Þ d s0 . (10) v? s0 In the above equation Pðs0 Þ is the transition rate for either bulk or surface plasmon, and s0 is the distance from the surface. The limits of the integral are given by s0 and 1 where s0 is the distance of closest approach. The value of the integral represents a measuring parameter vc, usually called characteristic velocity. Z 1 Pðs0 Þ d s0 . (11) vc ¼ 2 s0
Large values for vc (characteristic velocity) correspond to strong neutralization. Theoretical calculations of the neutral fraction are applied for the system of proton scattering from an aluminum surface. Aluminum metal is a good example of a semi-free electron metal where the approximations assumed in our theory are appropriate. Also the experimental evidence of surface and bulk plasmons is available in the literature. Fig. 1 shows the neutral fraction for protons scattered from an aluminum surface assuming bulk-plasmon excitation during the neutralization. The x-axis represents the component of the proton velocity measured in atomic units in the direction perpendicular to the surface. Atomic units for the velocity can be deducted
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Neutral fraction
0.08
0.06
0.04
0.02
0.00 0.0
0.2
0.4 0.6 Proton velocity
0.8
1.0
Fig. 1. Neutral fraction for bulk-plasmon-assisted neutralization against proton velocity in the direction perpendicular to the surface v? (atomic units for velocities are used).
1.0
0.8 Bulk
153
higher velocities do not reveal greater probability of bulkplasmon excitation. Clearly if penetration of the surface was assumed (that would be more probable for more energetic protons) then bulk plasmons would have a higher probability of excitation. The characteristic velocities vc, as defined in Eq. (11), for bulk-plasmon-assisted proton neutralizations is extracted from the figure to be 0.0009. This corresponds to an energy of 0.02 eV. This is a very small value compared with vc for surface plasmon-assisted ion neutralization where vc equals 0.75 and corresponding to an energy of 14 keV. In fact the small value of the characteristic velocity for bulk-plasmon-assisted neutralization compared to that for surface-plasmon-assisted neutralization is a mathematical interpretation of the low probability of this mechanism. In order to compare between bulk-plasmon-assisted and surface-plasmon-assisted neutralization, Fig. 2 is plotted. The figure shows the neutral fractions for both mechanisms. The figure shows clearly that the neutral fraction for surface-plasmon-assisted neutralization is much greater than that for the bulk-plasmon-assisted neutralization. This is definitely expected with our assumption of no ion penetration of the surface. It is expected that any theory that takes into account both mechanisms simultaneously would give the same result.
Surface
4. Conclusion
Neutral fraction
0.6
0.4
0.2
0.0
0.0
0.2
0.4 0.6 Proton velocity
0.8
1.0
In conclusion, the neutral fraction for ion neutralization by bulk plasmons was calculated. It was seen that this neutral fraction is very small, implying a very low probability for the neutralization via this mechanism. When compared to the neutral fraction for the mechanism of neutralization via surface plasmons, it was seen that this mechanism can be neglected. This might be understood and acceptable since in our theory it was assumed that no ion penetration to the surface is taken into account. A characteristic velocity was deduced for ion neutralization by both bulk and surface plasmons. This characteristic velocity that appears in the equation for the neutral fraction can be used as a measure of the importance of the neutralization mechanism. Higher values for this characteristic velocity vc correspond to strong neutralization probability. References
Fig. 2. Neutral fraction for plasmon-assisted neutralization against proton velocity in the direction perpendicular to the surface v? (atomic units for velocities are used). The broken curve represents the case of surface plasmon-assisted neutralization. The solid curve represents the case of bulk-plasmon-assisted neutralization.
from the relation that for velocity 1 atomic unit corresponds to 2.187 106 m/s. It is clear from the figure that the neutral fraction is so small that it might be assumed zero for velocities greater than 0.4 (in atomic units). Let us not forget that no penetration is assumed. This is why
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