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Charge equilibration of energetic He ions in the Si〈100〉 channel
Nuclear Instruments and Methods in Physics Research B 168 (2000) 321±328
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Charge equilibration of energetic He ions in the Sih1 0 0i channel G. de M. Azevedo a, J.R.A. Kaschny a, M. Behar S. Kalbitzer b a
a,*
, P.L. Grande a, Ch. Klatt b,
Instituto de Fõsica, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, Av. Bento Concalves 9500, 91501-970 Porto Alegre, RS, Brazil b Max-Planck-Institut f ur Kernphysik, Postfach 103980, D-69029 Heidelberg, Germany Received 19 July 1999; received in revised form 2 December 1999
Abstract We report experimental results on the charge equilibration process of both He and He ions at energies from 0.4 to 1.5 MeV in the Sih1 0 0i direction. The characteristic equilibration distance increases with energy from 3 nm to about 10 nm at the highest energy. The experimental ®ndings are compared with Monte Carlo calculations describing the particle motion in a continuum potential for the h1 0 0i channel. Ó 2000 Elsevier Science B.V. All rights reserved.
1. Introduction At high velocities, the electronic stopping power for bare ions of nuclear charge Z1 is well predicted by the Bethe theory, i.e., S / Z12 [1]. At intermediate and low velocities, however, the stopping power is a function of the projectile charge state [2] which inside a target material is dierent from the incident ion charge, since electron transfer phenomena take place in the solid. For random directions these processes are noticeable only at high energies, say at roughly E P 1 MeV/u. Here the equilibrium charge state is reached after the ions have passed a distance of the order of L 10 nm through the target [3±5]. For
*
Corresponding author. Tel.: +55-51-336-6551; fax: +55-51336-1762. E-mail address: [email protected] (M. Behar).
E 6 1 MeV/u charge equilibration requires equilibration distances of only L 1 nm [6±8] ± in many cases a process of minor practical importance. Under channeling conditions, however, the situation should be rather dierent. In fact, an ion beam incident along any major crystal direction of the target can experience strongly reduced charge exchange probabilities, if a large fraction of the beam is shielded from close encounters with the lattice atoms. Thus, noticeable eects on the preequilibrium charge transfer and the equilibration distances are expected. On a macroscopic level, the electronic stopping power may dier considerably under channeling conditions, as is well known. Recent measurements have determined a charge state equilibration distance of L 3 nm for 0.38 MeV 4 He ions along the Sih1 0 0i channel [9]. Clearly, it is of interest to investigate these phenomena at higher energies, especially for those energies in use for materials analysis.
0168-583X/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 9 ) 0 1 0 9 6 - 4
322
G. de M. Azevedo et al. / Nucl. Instr. and Meth. in Phys. Res. B 168 (2000) 321±328
Therefore, we have determined a characteristic equilibration distance for He ions in the Sih1 0 0i direction also at higher energies between 1 and 1.5 MeV. Ion beam analysis, by using the well-established processes of Rutherford backscattering/ channeling (RBS/C) has been applied to special targets prepared by SIMOX technology. In addition, we have performed Monte Carlo calculations in order to compare experimental results and theoretical predictions.
2. Experimental procedures Charge equilibration measurements have been carried out by a simple procedure based on the standard RBS/C techniques as described previously in some detail [10,11]. For this sake, SIMOX targets with a 200 2 nm Sih1 0 0i top layer on a 400 nm SiO2 buried layer have been analyzed at random and h1 0 0i channeling directions using 4 He and 4 He ions in backscattering geometry. The ion beams, at energies between 1 and 1.5 MeV and an angular divergence of less than 0:01°, have been produced by two tandem accelerators: the 3 MV Tandetron of the Instituto de Fõsica, Universidade Federal Rio Grande do Sul (IFUFRGS), Porto Alegre, Brazil, and the 3 MV Pelletron of the Max-Planck-Institut f ur Kernphysik (MPI-K), Heidelberg, Germany. Just before each run the respective sample was cleaned and etched with a 10% HF solution to remove the native oxide on the top Si layer of the SIMOX structure. Then, it was quickly mounted onto a 3-axis goniometer of 0:005° angular precision and the scattering chamber was evacuated immediately. At IF-UFRGS, the following experimental conditions have been provided. The vacuum system, consisting of an oil-free turbo-molecular drag pump with a liquid nitrogen trap, is capable of reaching a ®nal pressure of about 10ÿ7 mbar. For energy spectrometry two Si(Li) surface-barrier detectors were used at backscattering angles of 170° with respect to the beam direction. The energy resolution of the He ions was better than 10 keV (FWHM).
At MPI-K, the scattering chamber was evacuated by an oil-free cryogenic pumping system; a ®nal pressure of 10ÿ8 mbar has been the standard condition. An electrostatic analyzer (ESA) has been used to determine the energy of the He particles backscattered at 125:3°. The energy resolution has been about 4 keV (FWHM). At all ion energies a He ±He ±He ±He sequence of measurements has been performed in order to monitor possible changes on the sample surface, e.g., by deposits of hydrocarbonic compounds. This is a very stringent requirement, since only small shifts of the energetic position of the upper Si=SiO2 interface are expected for the He and He channeling incidence. As a further precaution, the beam spot position on the sample has always been shifted after certain periods of exposure. Changing the ion beam from He to He , or vice-versa, at constant particle energy requires a modi®cation of the accelerator voltage. In order to correct for unavoidable beam energy shifts, random spectra were taken prior to and after each voltage setting. In this way, by using the Si front edge position in random direction we were able to reproduce the particle energy at the 250 eV level. In all cases, these small dierences were taken into account when the channeled spectra were analyzed. As noted above, ions of dierent initial charge states will suer dierent energy losses during the charge equilibration process. Therefore, under channeling conditions for He and He ions, the dierent energetic positions of the upper Si=SiO2 interface will yield direct information on the equilibration process. 3. Theoretical description The charge exchange processes under axial channeling conditions have been simulated with a Monte Carlo program. It takes into account the dierent ion trajectories and electron loss and capture probabilities depending on the ion impact parameter. As outlined in Appendix A, we assume the He ions to populate only two possible charge states, He and He , since the He0 charge state
G. de. M. Azevedo et al. / Nucl. Instr. and Meth. in Phys. Res. B 168 (2000) 321±328
fraction is negligible at present energies. In order to calculate ion trajectories we have used the continuum potential model. Consequently, only the transverse motion across the Sih1 0 0i channel has to be determined in order to examine the charge transfer processes. For Sih1 0 0i axial channeling, the potential energy of four atomic strings was used to calculate the transverse positions of the ions [12,13]. Fig. 1 displays a crosssectional view on the obtained channel potential. The full lines represent the equipotential lines, where the numbers indicate the potential energies in eV. The dashed line indicates the projection on the plane of one possible trajectory. The Newton equations of motion are then solved for an ensemble of ions entering the channel with an initial charge state q. Thus, the two-dimensional transverse motion and the charge state population are obtained as a function of the penetrated depth. Here, we assume the ion trajectories to be independent of the He projectile charge state, since they are basically determined by a screened potential of Si.
Fig. 1. Equipotential contour lines of the 4-string continuum potential of the Si h1 0 0i channel, numbers corresponding to energies in eV. A projection of a possible He trajectory on the plane is shown.
323
The electron loss probability Pl b, describing a collision of He with Si atoms as a function of the impact parameter b, was taken from results of coupled-channel calculations and determined from the projectile ionization probabilities [14]. We point out that only non-perturbative approaches for loss probability can be employed to calculate the electron loss probability at these ion velocities. For example a ®rst-order calculation overestimates the electron loss cross-section by a factor of 3, as for collisions with Ne atoms at 1000 keV [15]. The capture of an electron from a Si atom into a He2 bound state cannot be taken directly from [14]. In addition, a direct calculation of the function of the impact parameter is a complicated task, even in a ®rst-order approach. Thus, we have used a linear ``ansatz'' for the electron capture probability Pc b aPl b;
1
where a, a constant, is determined from the charge equilibrium fractions hf1 i and hf2 i under channeling conditions as a hf1 i=hf2 i [16]. We have tried other functional expressions for the capture probability but they did not bring major dierences. Each ion history is followed and the charge state of the projectile is recorded as a function of its depth. The charge state fractions are ®nally obtained by averaging the charge states populations at a given depth over all incident ions. As shown in Appendix A, for random direction the well-known simple exponential law results with a decay constant given in terms of the electron loss and capture cross-sections for the two charge states. In Fig. 2, the calculated charge fractions in the h1 0 0i channel are displayed as a function of penetrated depth for incident He and He ions at 1.2 MeV. In all cases, stationary values are reached only after traversed distances of some 10 nm. The decay is governed by two equilibration constants, k1 1:2 nm and k2 8:2 nm, as Fig. 3 demonstrates. 4. Experimental results and discussion In Fig. 4(a) are shown the RBS random spectra corresponding to 1.2 MeV He and He obtained
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G. de M. Azevedo et al. / Nucl. Instr. and Meth. in Phys. Res. B 168 (2000) 321±328
Fig. 2. Charge state fractions of 1.2 MeV 4 He ions with depth in Si h1 0 0i obtained by Monte Carlo calculations with the potential of Fig. 1. The equilibrium values at a depth of 50 nm have been taken from [16] for ions emerging from the target.
with the ESA system at MPI-K. The agreement between both spectra indicates the reproducibility of the beam energy settings for the two dierent terminal voltages of the Pelletron. In Fig. 4(b) the corresponding channeling spectra are displayed. The Si/SiO2 interface position for He is shifted to lower energies as compared with He indicating a higher energy loss for He than for He . These features, although less prominent, were also observed at IF-UFRGS with the Si(Li) detector arrangement. Similar features were observed when the SIMOX sample was analyzed with 1.5 MeV He and He beams. Figs. 5(a) and (b), presenting again ESA spectra of MPI-K, also demonstrate that the energy settings are precise and that He suers higher energy losses than He . The evaluation of the equilibration distance, L, the quantity of physical signi®cance, proceeds as
Fig. 3. Analysis of the equilibration process in an exponential decay plot: note the presence of a fast and a slow component. The equilibrium value of 21.7% is subtracted from the nonequilibrium populations.
follows. Assuming only two populated charge states for He, namely He and He , the dierence between their energy losses due to their initial charge states, as detected after the backscattering event, is DE KL S ÿ S ;
2
where K contains the kinematic details of the backscattering process and S, for brevity, denotes the stopping power. The physical meaning of the characteristic equilibration distance L is discussed in Appendix A. The partial stopping powers S and S contribute to an equilibrium value hSi, hSi hf1 iS hf2 iS ;
3
where hf1 i and hf2 i are the corresponding equilibrium charge state fractions. It has been assumed that these fractions for h1 0 0i and h1 1 1i axial channeling are equal. Furthermore, they are nearly equal to the ones at a random direction [16].
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Fig. 4. RBS spectra taken with the ESA system with 1.2 MeV 4 He and 1 He ions: (a) the Si front edge of the SIMOX sample in random direction; (b) the Si=SiO2 top interface layer in the Sih1 0 0i channeling direction.
Finally the energy loss ratio was calculated by using the results of [14]. The ratio S =S n
4
is about 2, close to the prediction from the equipartition rule, where distant and close collisions contribute to the energy loss equally. In combining Eqs. (2)±(4) we obtain the relation for the characteristic equilibration distance, L DE hf1 i hf2 in= n ÿ 1KhSi:
5
In Table 1 are summarized all the results obtained in the present experiment together with the one at 380 keV previously reported [9]. There we quote: the He energies, the corresponding stopping powers [17], the fractions f1 and f2 , the stopping ratios (S =S ), the measured hDEi values and the corresponding equilibration distances L. As can be
325
Fig. 5. RBS spectra as in Fig. 4, but with He and He ion beams at 1.5 MeV.
observed from Fig. 6 there is a monotonic increase of the equilibration distance with the projectile energy, from about L 3 to 10 nm at E 0:38 to 1.5 MeV. It should be mentioned that similar experiments performed in the h1 1 0i and h1 1 1i Si directions have given similar results. In order to make a meaningful comparison between the theoretical predictions and the experimental ®ndings, it is necessary to perform a weighted average of the calculated k values which as described in Appendix A corresponds to hki a1 k1 a2 k2 : This average gives for 1.2 MeV He , a value hki 2:7 nm. It can be observed that the theoretical±experimental agreement can be considered as fair/poor (experimental, L 10 nm). At the present we do not have a plausible explanation for this behavior. However we can point out two possible reasons.
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Table 1 Summary of the present data E (MeV)
hSir (eV/nm)
hf1 i
hf2 i
S =S
DE (keV)
L (nm)
0.38a;b 1b 1.2b 1.2c 1.5b 1.5c
332 288 272 272 251 251
0.72 0.3 0.22 0.22 0.15 0.15
0.28 0.7 0.78 0.78 0.85 0.85
2.4 2.1 1.8 1.8 1.8 1.8
0.5 0:8 0:3 0:9 0:3 0:85 0:15 0:8 0:3 0:72 0:15
3.0 8:0 3 12:0 4 10 1:3 12 4 9:5 2
a
See [9]. Measurements at UFRGS: k 0:5657. c Measurements at MPI-K: k 0:6356. b
[15], deliver wrong results, due to the breakdown of the independent particle model, on which these calculations are based. As can be observed from the above discussion the theoretical±experimental disagreement is an open question which deserves further investigations from the theoretical point of view.
5. Conclusions
Fig. 6. The experimental charge equilibrium distance L as a function of incident He energy E. Experimental errors are quoted in Table 1.
1. The capture probability has not been calculated directly. It was adjusted to reproduce the experimental charge fractions. Furthermore, these experimental fractions, measured by transmission geometry, may not correspond to the charge fractions inside the solid. 2. The ionization probability coming from coupled channel calculations may, in some cases
In the present work we have studied the charge equilibration process and determined characteristic equilibration distances for 4 He ions incident in h1 0 0i Si channeling direction. With this aim, we have used a simple experimental setup based on the RBS/C spectrometry together with a SIMOX target. The results indicate that the equilibration distance L is a function of the He energy being of the order of 3 nm, for E 380 keV and going up to 10 nm in the 1±1.5 MeV region. These results are almost independent of the chosen channeling direction. In addition we have performed Monte Carlo calculations in order to reproduce the experimental results. As a consequence it was found that the He beam has two fractions with equilibration constants k1 1 nm for the peripherical component and k2 10 nm for the well channeled one. In order to make a meaningful theoretical±experimental comparison a mean hki value was determined which gives as a result hki 3 nm is some kind of fair/poor agreement with the experimental value. At the present we do not have an explana-
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327
tion for this behavior. However, two hypotheses were advanced. The present experimental results clearly indicate that in high precision near-surface channeling experiments, the charge equilibration eect on the stopping power/depth scale should be carefully taken into account. On the other hand, when the measurements are done in random direction caution should be exercised only when ultra-thin layers (t 1 nm) are analyzed.
For an incident beam of He at x 0, we have f1 0 1 and f2 0 0, and thus a1 rc = re rc and a2 re = rc re , so that again at all depths f1 x f2 x 1. The next step is to relate energy losses to equilibration constants. The energy losses DE and DE for incoming He and He are Z 1 Z 1 f1 x dx S f2 x dx; DE S
Appendix A
DE S
In the following, we present the theoretical framework used for calculations of the channeling equilibration process and the resulting expressions for the equilibration constant. In the investigated range of ion energies we can make the simplifying assumption that in the solid only two charge fractions f1 x and f2 x, of He and He , respectively, are present: f1 x f2 x 1:
A:1
The dierential equations for this two-channel equilibration process read
0
0
A:4a Z
1
0
f1 x dx S
Z
1 0
f2 x dx; A:4b
respectively, where the superscripts ``+'' and ``++'' refer to the incident He charge state. Thus, the measurable energy shift DE K DE ÿ DE is obtained by Z 1 f2 x ÿ f2 x dx DE K S 0 Z 1 f1 x ÿ f1 x dx ; A:5 S 0
with the additional relations for incident beams of both He and He ,
df1 N ÿ re f1 rc f2 ; dx
A:2a
f2 ÿ f2 eÿN rc re x ;
A:6a
df2 N re f1 ÿ rc f2 ; dx
A:2b
f1 ÿ f1 ÿeÿN rc re x ;
A:6b
where the cross-sections r for electron loss and capture are identi®ed by the subscripts e and c, respectively, and N is the atomic number density of the solid under consideration. The general solution is given by 1 1 f1 eÿN re rc x : A:3 a1 a2 re =rc ÿ1 f2 For an incident beam of He at the surface, at x 0, we have f1 0 0 and f2 0 1, and thus a1 rc = re rc and a2 ÿrc = re rc ; the equilibrium concentrations at x 1 are hf1 i rc = re rc and hf2 i re = re rc , so that at all depths f1 x f2 x 1 is satis®ed.
the energy shift is DE K DE ÿ DE Kk S ÿ S
A:7
with k
1 ; N re rc
A:8
where k is the equilibration constant. Substituting for S and S their ratio, S n; S
A:9
and introducing the equilibrium stopping power hSir hf1 iS hf2 iS ;
A:10
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G. de M. Azevedo et al. / Nucl. Instr. and Meth. in Phys. Res. B 168 (2000) 321±328
we ®nally obtain the decay constant of the nonequilibrium excess charge fractions, k
DE hf1 i hf2 in : KhSir n ÿ 1
A:11
The above description applies for a random direction case when only one exponential decay k appears. In the case of two beam components inside the channel equilibrating with two dierent decay constants k1 and k2 , as displayed in Fig. 3, the energy shift parameter will be Z 1 e ÿx=k1 dx DE K S ÿ S a1 0 Z 1 ÿx=k2 e dx ; A:12 a2 0
so that one may de®ne, in analogy to the single component case, an average value hki a1 k1 a2 k2 :
A:13
References [1] H. Bethe, Ann. Phys. 5 (1930) 325. [2] W. Brandt, M. Kitagawa, Phys. Rev. B 25 (1982) 5631.
[3] H. Ogawa, I. Kitagawa, I. Sugai, Y. Haruyama, M. Tosaki, A. Aoki, K. Yoshida, H. Ikagami, Phys. Lett. A 167 (1992) 487, and references therein. [4] N.E.B. Cowern, P.M. Read, C.J. Sco®eld, L.B. Bridwell, M.W. Lucas, Phys. Rev. A 30 (1984) 1682. [5] A.F. Burenkov, F.F. Komarov, S.A. Fedotov, Nucl. Instr. and Meth. B 67 (1992) 35. [6] T.M. Buck, G.H. Wheatley, L.C. Feldman, Surf. Sci. 35 (1973) 345. [7] A. Luvio, J.F. Ziegler, in: Beam Foil Spectroscopy, Vol. 2, Plenum, New York, 1976, p. 665. [8] S. Jans, P. Oberschachtsiek, S. Kalbitzer, M. Behar, Nucl. Instr. and Meth. B 83 (1993) 1. [9] J.H.R. dos Santos, P.L. Grande, H. Boudinov, M. Behar, in: S. Coa, G. Ferla, F. Priolo, E. Rimini (Eds.), Proceedings of the Conference on Ion Implantation Technology-94, Elsevier, Amsterdam, 1995, p. 771. [10] J.H.R. dos Santos, P.L. Grande, M. Behar, H. Boudinov, G. Schiwietz, Phys. Rev. B 55 (1997) 4332. [11] G. de M. Azevedo, J.R.A. Kaschny, J.F. Dias, P.L. Grande, M. Behar, Ch. Klatt, S. Kalbitzer, Nucl. Instr. and Meth. B 148 (1999) 168. [12] D.V. Morgan (Ed.), Channeling: Theory, Observation and Applications, Wiley, New York, 1973. [13] D.S. Gemmel, Rev. Mod. Phys. 46 (1974) 129. [14] P.L. Grande, G. Schiwietz, Nucl. Instr. and Meth. B 136± 138 (1998) 125. [15] P.L. Grande, G. Schiwietz, G.M. Sigaud, E.C. Montenegro, Phys. Rev. A 54 (1996) 2983. [16] R.J. Pretty, G. Dearnaley, Phys. Lett. A 50 (1974) 273. [17] D. Niemann, P. Oberschachtsiek, S. Kalbitzer, H.P. Zeindl, Nucl. Instr. and Meth. B 80/81 (1993) 37.
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Charge equilibration of energetic He ions in the Sih1 0 0i channel G. de M. Azevedo a, J.R.A. Kaschny a, M. Behar S. Kalbitzer b a
a,*
, P.L. Grande a, Ch. Klatt b,
Instituto de Fõsica, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, Av. Bento Concalves 9500, 91501-970 Porto Alegre, RS, Brazil b Max-Planck-Institut f ur Kernphysik, Postfach 103980, D-69029 Heidelberg, Germany Received 19 July 1999; received in revised form 2 December 1999
Abstract We report experimental results on the charge equilibration process of both He and He ions at energies from 0.4 to 1.5 MeV in the Sih1 0 0i direction. The characteristic equilibration distance increases with energy from 3 nm to about 10 nm at the highest energy. The experimental ®ndings are compared with Monte Carlo calculations describing the particle motion in a continuum potential for the h1 0 0i channel. Ó 2000 Elsevier Science B.V. All rights reserved.
1. Introduction At high velocities, the electronic stopping power for bare ions of nuclear charge Z1 is well predicted by the Bethe theory, i.e., S / Z12 [1]. At intermediate and low velocities, however, the stopping power is a function of the projectile charge state [2] which inside a target material is dierent from the incident ion charge, since electron transfer phenomena take place in the solid. For random directions these processes are noticeable only at high energies, say at roughly E P 1 MeV/u. Here the equilibrium charge state is reached after the ions have passed a distance of the order of L 10 nm through the target [3±5]. For
*
Corresponding author. Tel.: +55-51-336-6551; fax: +55-51336-1762. E-mail address: [email protected] (M. Behar).
E 6 1 MeV/u charge equilibration requires equilibration distances of only L 1 nm [6±8] ± in many cases a process of minor practical importance. Under channeling conditions, however, the situation should be rather dierent. In fact, an ion beam incident along any major crystal direction of the target can experience strongly reduced charge exchange probabilities, if a large fraction of the beam is shielded from close encounters with the lattice atoms. Thus, noticeable eects on the preequilibrium charge transfer and the equilibration distances are expected. On a macroscopic level, the electronic stopping power may dier considerably under channeling conditions, as is well known. Recent measurements have determined a charge state equilibration distance of L 3 nm for 0.38 MeV 4 He ions along the Sih1 0 0i channel [9]. Clearly, it is of interest to investigate these phenomena at higher energies, especially for those energies in use for materials analysis.
0168-583X/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 9 ) 0 1 0 9 6 - 4
322
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Therefore, we have determined a characteristic equilibration distance for He ions in the Sih1 0 0i direction also at higher energies between 1 and 1.5 MeV. Ion beam analysis, by using the well-established processes of Rutherford backscattering/ channeling (RBS/C) has been applied to special targets prepared by SIMOX technology. In addition, we have performed Monte Carlo calculations in order to compare experimental results and theoretical predictions.
2. Experimental procedures Charge equilibration measurements have been carried out by a simple procedure based on the standard RBS/C techniques as described previously in some detail [10,11]. For this sake, SIMOX targets with a 200 2 nm Sih1 0 0i top layer on a 400 nm SiO2 buried layer have been analyzed at random and h1 0 0i channeling directions using 4 He and 4 He ions in backscattering geometry. The ion beams, at energies between 1 and 1.5 MeV and an angular divergence of less than 0:01°, have been produced by two tandem accelerators: the 3 MV Tandetron of the Instituto de Fõsica, Universidade Federal Rio Grande do Sul (IFUFRGS), Porto Alegre, Brazil, and the 3 MV Pelletron of the Max-Planck-Institut f ur Kernphysik (MPI-K), Heidelberg, Germany. Just before each run the respective sample was cleaned and etched with a 10% HF solution to remove the native oxide on the top Si layer of the SIMOX structure. Then, it was quickly mounted onto a 3-axis goniometer of 0:005° angular precision and the scattering chamber was evacuated immediately. At IF-UFRGS, the following experimental conditions have been provided. The vacuum system, consisting of an oil-free turbo-molecular drag pump with a liquid nitrogen trap, is capable of reaching a ®nal pressure of about 10ÿ7 mbar. For energy spectrometry two Si(Li) surface-barrier detectors were used at backscattering angles of 170° with respect to the beam direction. The energy resolution of the He ions was better than 10 keV (FWHM).
At MPI-K, the scattering chamber was evacuated by an oil-free cryogenic pumping system; a ®nal pressure of 10ÿ8 mbar has been the standard condition. An electrostatic analyzer (ESA) has been used to determine the energy of the He particles backscattered at 125:3°. The energy resolution has been about 4 keV (FWHM). At all ion energies a He ±He ±He ±He sequence of measurements has been performed in order to monitor possible changes on the sample surface, e.g., by deposits of hydrocarbonic compounds. This is a very stringent requirement, since only small shifts of the energetic position of the upper Si=SiO2 interface are expected for the He and He channeling incidence. As a further precaution, the beam spot position on the sample has always been shifted after certain periods of exposure. Changing the ion beam from He to He , or vice-versa, at constant particle energy requires a modi®cation of the accelerator voltage. In order to correct for unavoidable beam energy shifts, random spectra were taken prior to and after each voltage setting. In this way, by using the Si front edge position in random direction we were able to reproduce the particle energy at the 250 eV level. In all cases, these small dierences were taken into account when the channeled spectra were analyzed. As noted above, ions of dierent initial charge states will suer dierent energy losses during the charge equilibration process. Therefore, under channeling conditions for He and He ions, the dierent energetic positions of the upper Si=SiO2 interface will yield direct information on the equilibration process. 3. Theoretical description The charge exchange processes under axial channeling conditions have been simulated with a Monte Carlo program. It takes into account the dierent ion trajectories and electron loss and capture probabilities depending on the ion impact parameter. As outlined in Appendix A, we assume the He ions to populate only two possible charge states, He and He , since the He0 charge state
G. de. M. Azevedo et al. / Nucl. Instr. and Meth. in Phys. Res. B 168 (2000) 321±328
fraction is negligible at present energies. In order to calculate ion trajectories we have used the continuum potential model. Consequently, only the transverse motion across the Sih1 0 0i channel has to be determined in order to examine the charge transfer processes. For Sih1 0 0i axial channeling, the potential energy of four atomic strings was used to calculate the transverse positions of the ions [12,13]. Fig. 1 displays a crosssectional view on the obtained channel potential. The full lines represent the equipotential lines, where the numbers indicate the potential energies in eV. The dashed line indicates the projection on the plane of one possible trajectory. The Newton equations of motion are then solved for an ensemble of ions entering the channel with an initial charge state q. Thus, the two-dimensional transverse motion and the charge state population are obtained as a function of the penetrated depth. Here, we assume the ion trajectories to be independent of the He projectile charge state, since they are basically determined by a screened potential of Si.
Fig. 1. Equipotential contour lines of the 4-string continuum potential of the Si h1 0 0i channel, numbers corresponding to energies in eV. A projection of a possible He trajectory on the plane is shown.
323
The electron loss probability Pl b, describing a collision of He with Si atoms as a function of the impact parameter b, was taken from results of coupled-channel calculations and determined from the projectile ionization probabilities [14]. We point out that only non-perturbative approaches for loss probability can be employed to calculate the electron loss probability at these ion velocities. For example a ®rst-order calculation overestimates the electron loss cross-section by a factor of 3, as for collisions with Ne atoms at 1000 keV [15]. The capture of an electron from a Si atom into a He2 bound state cannot be taken directly from [14]. In addition, a direct calculation of the function of the impact parameter is a complicated task, even in a ®rst-order approach. Thus, we have used a linear ``ansatz'' for the electron capture probability Pc b aPl b;
1
where a, a constant, is determined from the charge equilibrium fractions hf1 i and hf2 i under channeling conditions as a hf1 i=hf2 i [16]. We have tried other functional expressions for the capture probability but they did not bring major dierences. Each ion history is followed and the charge state of the projectile is recorded as a function of its depth. The charge state fractions are ®nally obtained by averaging the charge states populations at a given depth over all incident ions. As shown in Appendix A, for random direction the well-known simple exponential law results with a decay constant given in terms of the electron loss and capture cross-sections for the two charge states. In Fig. 2, the calculated charge fractions in the h1 0 0i channel are displayed as a function of penetrated depth for incident He and He ions at 1.2 MeV. In all cases, stationary values are reached only after traversed distances of some 10 nm. The decay is governed by two equilibration constants, k1 1:2 nm and k2 8:2 nm, as Fig. 3 demonstrates. 4. Experimental results and discussion In Fig. 4(a) are shown the RBS random spectra corresponding to 1.2 MeV He and He obtained
324
G. de M. Azevedo et al. / Nucl. Instr. and Meth. in Phys. Res. B 168 (2000) 321±328
Fig. 2. Charge state fractions of 1.2 MeV 4 He ions with depth in Si h1 0 0i obtained by Monte Carlo calculations with the potential of Fig. 1. The equilibrium values at a depth of 50 nm have been taken from [16] for ions emerging from the target.
with the ESA system at MPI-K. The agreement between both spectra indicates the reproducibility of the beam energy settings for the two dierent terminal voltages of the Pelletron. In Fig. 4(b) the corresponding channeling spectra are displayed. The Si/SiO2 interface position for He is shifted to lower energies as compared with He indicating a higher energy loss for He than for He . These features, although less prominent, were also observed at IF-UFRGS with the Si(Li) detector arrangement. Similar features were observed when the SIMOX sample was analyzed with 1.5 MeV He and He beams. Figs. 5(a) and (b), presenting again ESA spectra of MPI-K, also demonstrate that the energy settings are precise and that He suers higher energy losses than He . The evaluation of the equilibration distance, L, the quantity of physical signi®cance, proceeds as
Fig. 3. Analysis of the equilibration process in an exponential decay plot: note the presence of a fast and a slow component. The equilibrium value of 21.7% is subtracted from the nonequilibrium populations.
follows. Assuming only two populated charge states for He, namely He and He , the dierence between their energy losses due to their initial charge states, as detected after the backscattering event, is DE KL S ÿ S ;
2
where K contains the kinematic details of the backscattering process and S, for brevity, denotes the stopping power. The physical meaning of the characteristic equilibration distance L is discussed in Appendix A. The partial stopping powers S and S contribute to an equilibrium value hSi, hSi hf1 iS hf2 iS ;
3
where hf1 i and hf2 i are the corresponding equilibrium charge state fractions. It has been assumed that these fractions for h1 0 0i and h1 1 1i axial channeling are equal. Furthermore, they are nearly equal to the ones at a random direction [16].
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Fig. 4. RBS spectra taken with the ESA system with 1.2 MeV 4 He and 1 He ions: (a) the Si front edge of the SIMOX sample in random direction; (b) the Si=SiO2 top interface layer in the Sih1 0 0i channeling direction.
Finally the energy loss ratio was calculated by using the results of [14]. The ratio S =S n
4
is about 2, close to the prediction from the equipartition rule, where distant and close collisions contribute to the energy loss equally. In combining Eqs. (2)±(4) we obtain the relation for the characteristic equilibration distance, L DE hf1 i hf2 in= n ÿ 1KhSi:
5
In Table 1 are summarized all the results obtained in the present experiment together with the one at 380 keV previously reported [9]. There we quote: the He energies, the corresponding stopping powers [17], the fractions f1 and f2 , the stopping ratios (S =S ), the measured hDEi values and the corresponding equilibration distances L. As can be
325
Fig. 5. RBS spectra as in Fig. 4, but with He and He ion beams at 1.5 MeV.
observed from Fig. 6 there is a monotonic increase of the equilibration distance with the projectile energy, from about L 3 to 10 nm at E 0:38 to 1.5 MeV. It should be mentioned that similar experiments performed in the h1 1 0i and h1 1 1i Si directions have given similar results. In order to make a meaningful comparison between the theoretical predictions and the experimental ®ndings, it is necessary to perform a weighted average of the calculated k values which as described in Appendix A corresponds to hki a1 k1 a2 k2 : This average gives for 1.2 MeV He , a value hki 2:7 nm. It can be observed that the theoretical±experimental agreement can be considered as fair/poor (experimental, L 10 nm). At the present we do not have a plausible explanation for this behavior. However we can point out two possible reasons.
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Table 1 Summary of the present data E (MeV)
hSir (eV/nm)
hf1 i
hf2 i
S =S
DE (keV)
L (nm)
0.38a;b 1b 1.2b 1.2c 1.5b 1.5c
332 288 272 272 251 251
0.72 0.3 0.22 0.22 0.15 0.15
0.28 0.7 0.78 0.78 0.85 0.85
2.4 2.1 1.8 1.8 1.8 1.8
0.5 0:8 0:3 0:9 0:3 0:85 0:15 0:8 0:3 0:72 0:15
3.0 8:0 3 12:0 4 10 1:3 12 4 9:5 2
a
See [9]. Measurements at UFRGS: k 0:5657. c Measurements at MPI-K: k 0:6356. b
[15], deliver wrong results, due to the breakdown of the independent particle model, on which these calculations are based. As can be observed from the above discussion the theoretical±experimental disagreement is an open question which deserves further investigations from the theoretical point of view.
5. Conclusions
Fig. 6. The experimental charge equilibrium distance L as a function of incident He energy E. Experimental errors are quoted in Table 1.
1. The capture probability has not been calculated directly. It was adjusted to reproduce the experimental charge fractions. Furthermore, these experimental fractions, measured by transmission geometry, may not correspond to the charge fractions inside the solid. 2. The ionization probability coming from coupled channel calculations may, in some cases
In the present work we have studied the charge equilibration process and determined characteristic equilibration distances for 4 He ions incident in h1 0 0i Si channeling direction. With this aim, we have used a simple experimental setup based on the RBS/C spectrometry together with a SIMOX target. The results indicate that the equilibration distance L is a function of the He energy being of the order of 3 nm, for E 380 keV and going up to 10 nm in the 1±1.5 MeV region. These results are almost independent of the chosen channeling direction. In addition we have performed Monte Carlo calculations in order to reproduce the experimental results. As a consequence it was found that the He beam has two fractions with equilibration constants k1 1 nm for the peripherical component and k2 10 nm for the well channeled one. In order to make a meaningful theoretical±experimental comparison a mean hki value was determined which gives as a result hki 3 nm is some kind of fair/poor agreement with the experimental value. At the present we do not have an explana-
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327
tion for this behavior. However, two hypotheses were advanced. The present experimental results clearly indicate that in high precision near-surface channeling experiments, the charge equilibration eect on the stopping power/depth scale should be carefully taken into account. On the other hand, when the measurements are done in random direction caution should be exercised only when ultra-thin layers (t 1 nm) are analyzed.
For an incident beam of He at x 0, we have f1 0 1 and f2 0 0, and thus a1 rc = re rc and a2 re = rc re , so that again at all depths f1 x f2 x 1. The next step is to relate energy losses to equilibration constants. The energy losses DE and DE for incoming He and He are Z 1 Z 1 f1 x dx S f2 x dx; DE S
Appendix A
DE S
In the following, we present the theoretical framework used for calculations of the channeling equilibration process and the resulting expressions for the equilibration constant. In the investigated range of ion energies we can make the simplifying assumption that in the solid only two charge fractions f1 x and f2 x, of He and He , respectively, are present: f1 x f2 x 1:
A:1
The dierential equations for this two-channel equilibration process read
0
0
A:4a Z
1
0
f1 x dx S
Z
1 0
f2 x dx; A:4b
respectively, where the superscripts ``+'' and ``++'' refer to the incident He charge state. Thus, the measurable energy shift DE K DE ÿ DE is obtained by Z 1 f2 x ÿ f2 x dx DE K S 0 Z 1 f1 x ÿ f1 x dx ; A:5 S 0
with the additional relations for incident beams of both He and He ,
df1 N ÿ re f1 rc f2 ; dx
A:2a
f2 ÿ f2 eÿN rc re x ;
A:6a
df2 N re f1 ÿ rc f2 ; dx
A:2b
f1 ÿ f1 ÿeÿN rc re x ;
A:6b
where the cross-sections r for electron loss and capture are identi®ed by the subscripts e and c, respectively, and N is the atomic number density of the solid under consideration. The general solution is given by 1 1 f1 eÿN re rc x : A:3 a1 a2 re =rc ÿ1 f2 For an incident beam of He at the surface, at x 0, we have f1 0 0 and f2 0 1, and thus a1 rc = re rc and a2 ÿrc = re rc ; the equilibrium concentrations at x 1 are hf1 i rc = re rc and hf2 i re = re rc , so that at all depths f1 x f2 x 1 is satis®ed.
the energy shift is DE K DE ÿ DE Kk S ÿ S
A:7
with k
1 ; N re rc
A:8
where k is the equilibration constant. Substituting for S and S their ratio, S n; S
A:9
and introducing the equilibrium stopping power hSir hf1 iS hf2 iS ;
A:10
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we ®nally obtain the decay constant of the nonequilibrium excess charge fractions, k
DE hf1 i hf2 in : KhSir n ÿ 1
A:11
The above description applies for a random direction case when only one exponential decay k appears. In the case of two beam components inside the channel equilibrating with two dierent decay constants k1 and k2 , as displayed in Fig. 3, the energy shift parameter will be Z 1 e ÿx=k1 dx DE K S ÿ S a1 0 Z 1 ÿx=k2 e dx ; A:12 a2 0
so that one may de®ne, in analogy to the single component case, an average value hki a1 k1 a2 k2 :
A:13
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