Improved calculations of the electronic energy loss under channeling conditions

Beam Interactions with Materials A Atoms

ELSEVIER

Nuclear

Instruments

and Methods

in Physics Research B 136-138 (1998) 1255131

Improved calculations of the electronic energy loss under channeling conditions P.L. Grande ’ Insiituto

‘-*, G. Schiwietz

b

de Fisicu. Univrrsidade Federui do Rio Grunde do Sul. Av. Bento Conquiver 9500, 91501-970 Porte Aleqv. b Huhn-Meitner-lnstitut, Bereich Festkiirperph>~sik, GlieCker Str. 100. 14109 Berlin. Germanic

Bruzil

Abstract Ab-initio calculations of the electronic stopping power are important not only as benchmark tests for many simplified models but also because they provide fundamental insights into the underlying physics of the energy dissipation of ions penetrating matter. In this work we investigate the electronic energy loss processes by using the coupled-channel method. This first-principle calculation is based on the solution of the time-dependent Schrodinger equation in the framework of the independent particle model. A brief review of such calculations is presented and special emphasis is given on recent calculations of the energy dependence of the stopping power for He ions channeling along the Si 0 1998 Elsevier Science B.V. (1 0 0) direction, a system of interest for ion beam analysis. PACS: Krywo~&:

79.20.N~; 61.80.Mk; 34.120.+x Energy loss; Channeling; Stopping power

34.50.B~;

1. Introduction

Accurate values of stopping powers are important not only for depth profiling of elements in materials [I] but also for the determination of the deposited energy and defect production [2] by the bombarding ion. Experiments that involve ions channeling along the major directions of a crystalline target require an even more detailed description of the energy loss mechanisms as a function of the impact parameter. A full treatment including all basic energy loss processes in a large energy *Corresponding author. Tel.: +55 51 3167111; 3 191761: e-mail: [email protected].

fax: +55 51

0168-583X/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved. PIISO168-583X(97)00869-0

range is still missing although the first steps have recently been made for simple systems [3-51. Hence, empirical and semi-empirical procedures have mostly been adopted by the ion-beam community in order to estimate the energy loss for practical applications. First-principle calculations of the electronic energy loss are very useful since they serve as benchmarks for models on which semi-empirical procedures are based and because they provide physical insights for the basic energy loss processes [6]. Recently realistic calculations of the electronic stopping power have been performed using some traditional methods known from atomic physics investigations such as coupled-channel [4], distort-

ed-wave [7] and classical-trajectory Monte Carlo [8] calculations. These new techniques (to the stopping-power field) offer alternative ways to obtain more information on the energy loss processes at intermediate projectile energies. Especially at energies around the maximum of the electronic stopping power complete coupled-channel calculations (full quantum mechanical solution of the electronic Schrodinger equation) seem to be necessary to obtain accurate results. The first stoppingpower calculations of this type were only recently performed [4,5], since the computation of stopping cross-sections requires the treatment of many electronic states and is very time consuming. The coupled-channel method has been used successfully to obtain reliable values of the electronic stopping power [43. the energy loss as a function of the impact parameter and scattering angle [9] for simple systems as well as for He channeling along the Si (1 0 0) direction [lo]. These calculations allow an accurate determination of the importance of different processes leading to the energy loss of ions in gases or insulators, namely electron capture and loss of projectile electrons as well as target ionization/excitation for different projectile chargestates. Concerning only the inner-shell electrons in solids they go far beyond the traditional approaches. Of course, other models [l l-131 have to be adopted for conduction-band excitations in solid-state targets in order to obtain an accurate description of the energy loss due to quasi free electrons. However. up to now most of these approaches [12,13] neglect the electron capture and loss contribution completely as well as the energy gaps (band structure) and they are based on a first-order generalization of the homogeneous electrongas model (LDA). For random stopping powers only a few models avoid these severe approximations [ 141 and for channeling conditions so far only two approaches account for the valence band structure consistently [ 15.161. In this work we use the coupled-channel method to calculate the electronic energy loss for a system of practical interest, namely He ions channeling along Si (1 0 0), (1 1 0) and (1 1 1) directions. In the following, we shall give a brief demethod coupled-channel the scription of (Section 2.1) and the procedure used to calculate

the electronic energy loss under channeling conditions (Section 2.2). Finally, in Section 3 we present our results for the energy dependence of the energy loss for different Si channels and compare the calculations to existing experimental data.

2. Theory

Let us consider that all events involving innershell electrons can be adequately treated as localized atomic ones. Then. atomic methods such as the coupled-channel atomic orbital (AO) method can be used. The principle of this method is based on the impact parameter method [17]. The projectile following a classical trajectory provides a time dependent perturbation on the target electrons. Hence, the time dependent Schrodinger equation is solved by expanding the electronic wave function in a truncated basis of states, namely atomic orbitals. A set of first-order ordinary coupled differential equations for the coefficients originating from this expansion, the so-called coupled-channel equations, is integrated numerically along the classical trajectory of the projectile for a given impact parameter h. Thus. the amplitude LZ,_.,is calculated for a given transition from an initial occupied state i to an unoccupied bound or continuum state,f‘and the transition probability P, .,(h) = fimjN,_ ., (t)f

(1)

corresponding to atomic excitation or ionization is determined. The independent-particle model is adopted for one active electron moving in the electrostatic field due to both nuclei and the other electrons, which are included in a frozencore Hartree-Fock-Slater framework [ 181. In this way, the ground-state and excited-state wave functions as well as the energies E, and E, ., (where a hole in the ith shell is left) of the active electron are calculated. Since each excited or continuum state corresponds to a well defined energy transfer (A/Z, ., = E, _, - E,). the average electronic energy loss due to one electron in the ith shell can be written as

J+_,(h)AE

pp---‘(h) =

r--.,

>

i

where the superscript P and T correspond to the projectile and target, respectively. It is noted that the above sum has to be replaced by an integral in the case of continuum states. The subscript i indicates the occupied initial state and ,f’the unoccupied bound and continuum states. In the case of He’+ projectiles, the average electronic energy loss (neglecting valence excitations) reads

(3) where the initial states i correspond to Si K and L shells (i= Is, 2s 2p,., 2p,., 2~:). The factor of two corresponds to the occupation number of the initial orbital i. For the singly ionized He’ fraction of the beam, the energy loss stems not only from the Si inner-shell electrons. but also from projectile ionization and excitation. Consequently, the mean energy loss for the He+ fraction is @ (h) =

C2@e -y/l) + Qf: .f’e’(h) I

and the equilibrium energy loss is obtained by averaging the above expressions over the chargestate distribution. For bare incident ions, the active-electron projectile interaction is just the Coulomb potential. In the case where the projectile carries electrons, the potential seen by the active electron contains not only the Coulomb part due to the projectilenuclear charge but also the static potential produced by the projectile electrons that screen the projectile nuclear charge. It is emphasized that the calculation of the energy loss due to the projectile electron is performed in the frame where the projectile is at rest. Thus. the perturbing particle is the neutral Si atom in this case. Details of the coupled-channel calculation, e.g., the numerical treatment of continuum states, adopted basis set, treatment of screened projectiles, capture into projectile states, may be found in Refs. [4,5.19].

Under channeling conditions due to the Si inner-shell electrons

the energy loss is strongly sup-

pressed, since the flux distribution of He ions channeled along a Si major direction has a peak in the middle of the channel (flux peaking). Hence. in contrast to a random direction the mean energy lost by the projectile after passing a thickness t through the Si crystal is calculated according to

where A is the transversal area of the Si channel (here we are consider only axial channeling), ,? the transversal distance (z = 0 is the center of the channel). Y the entrance angle of the He beam, and Cp the ion flux distribution at the distance f? and depth X. The energy loss per traversed distance (dE/du)(@) may be divided into three contributions

\ ut

/

\.3/cncc

The first contribution corresponds to the energy loss due to inner-shell electrons of the Si atoms. Since these electrons are highly localized, the corresponding energy loss depends strongly on the transverse distance and is the main responsible for the reduction of the energy loss under channeling conditions. The second term in Eq. (6) corresponds to the energy loss due to projectile ionization/excitation. The incident ion He” may capture electrons from Si. This process leads to the formation of He+. The energy loss due to this capture process is accounted for by the first term of Eq. (3) since the captured electron comes preferentially from a Si inner shell. The singly ionized He+ carries one electron and hence. projectile ionization/excitation is also a source of energy loss. The energy transfer due to these charge-changing processes (capture and loss) depends strongly on the impact parameter of the He-Si collision. Typical energy loss in a capture event is the difference of the binding energies for the Si L shell and He K shell plus the translation factor mr’/2. For 800 keV, this is about 200 eV. Typical Si L-shell excitation energies are about 100 eV.

128

P.L. Grand@. G. Schiwirtr

I Nucl. Instr. and Meth. in Phyx

Finally, the last term is due to the valence electrons of Si (M-shell). These electrons are almost homogeneously distributed and therefore they will contribute almost equally for channeling and random directions. We assume this contribution to be independent of the ion transverse position. As it will be shown, this approximation holds true for all Si channels, except for the widest one. namely (1 1 0). The present calculations consist of taking the first two terms of Eq. (6) from the coupled-channel method (Eqs. (3) and (4)) after averaging them over the charge-state distribution [lo]. The sum of each energy-loss contribution due to the Si atoms located across the channel (4 for Si (1 0 0), 6 for Si (1 1 0) and 3 for Si (1 1 1)) directions is performed. The energy-loss term associated with the Si valence electrons is obtained here from the experimental stopping cross-section of Ref. [20] by subtracting the contributions from other processes. The He charge-state distribution is deterresults under from experimental mined channeling conditions from Ref. [21] (see insert in Fig. 1). For energies larger than 1. I MeV, extrapolated values have been used. In order to obtain the ion flux distribution, we use the string potential model [22]. Consequently, only the transverse motion has to be determined. For Si axial channeling, Moliere string potentials [22] were added to calculate the potential energy as a function of position p’. We solve Newton’s equations of motion numerically for an ensemble of ions impinging on the channel with entrance angle Y. Thus, the two-dimensional transverse motion and the ion flux distribution @ as a function of the penetration depth are calculated. Other methods yield similar flux distributions for Y = 0 [23,24]. Further details of the present energy-loss model may be found in [lo].

3. Results and discussion According to the last section, we have calculated the electronic energy loss for 4He ions channeling along the major axes in Si. The results are presented in Figs. 1 and 2 in comparison with experimental data. Fig. 1 shows the stopping power

Rrs. B 136-138 (1998)

l-75-131

.

J

300

1000

5000

Fig. 1. Electronic stopping power as a function of the 4He energy for the (I 0 0) Si direction. The solid line corresponds to the random stopping power taken from Ref. [20] and the symbols represent the experimental data from Ref. [25] (full squares) and from Ref. [26] (open circle). The results of the present calculation (AO) are indicated by a dashed line. The He chargestate fractions (for He’+ and He’ ) are shown in the inset.

of He ions moving through the Si crystal in the (1 0 0) channeling direction. The symbols correspond to recent experimental data [25,26] for the channeled energy-loss and the solid line represents accurate experimental stopping values for a random direction [20]. Experiments at 800 keV with He+ and He2+ ions [25] show that chafge equilibrium is reached at a depth of about 50 A. We expect this distance to increase by an order of magnitude for 5 MeV He ions. Since the mean charge state of fast ions is close to 2 and the measurements above 1000 keV were performed with HeZ+ ions there should be no significant deviation from the assumed equilibrium charge-distribution. The results of the A0 calculations (dashed-line) for the projectile-energy dependence of the electronic stopping power under channeling conditions agree with the data to within the experimental uncertainty. For ion energies above 1.2 MeV (see insert in Fig. 1 for the He chargestate fractions), the He’+ fraction is dominant and the main physical process responsible for the reduction of the energy loss under channeling conditions is the suppressed inner-shell ionization (Lshell) of Si atoms. The energy region from 1.2 MeV up to 5 MeV is close to the maximum of the stopping cross section due to Si L-shell elec-

P.L. Grade.

G. Schitciet:

I Nucl. Instr. and Meth. in Phys. Rex B 136-138

trons and only non-perturbative calculations (including many higher-order terms) are reliable in this energy region. By comparing the A0 results with first-order ones at 2 MeV we obtain a difference of about 40% for b = 1.3 A (middle of (1 0 0) channel). For energies below 1.2 MeV, the influence of charge-changing processes begins to be significant, and at least two charge states of the He projectile have to be considered (He’+ and He*). Near the stopping power maximum at 800 keV, the reduction of the stopping power is basically due to the suppression of projectile electron loss and electron capture from Si inner shell. For this case, the coupled-channel method was able to reproduce the experimental channeled energy loss for different entrance angles by excluding multiple-electron processes involving the Si innershell electrons [lo]. This energy represents the lower bound for the use of the present A0 method, since the electron capture is barely described in a basis expansion including only target-centered the independent-particle states. Furthermore, model (IPM) on which the present calculations are based breaks down. At and below 800 keV corrections beyond the IPM have to be introduced [lo] which do not warrant the application of the present model for lower projectile energies. The present energy-loss results as a function of the projectile energy are most sensitive to the computation of the inner-shell contribution at random direction, since it determines the contribution of the valence excitations under channeling conditions. The inner-shell contribution under channeling condition is strongly suppressed (by 75% at 5 MeV). Thus, the comparison in Fig. 1 is less sensitive to the impact parameter dependence of the electronic energy loss. On the other side, a comparison with the angle dependent energy-loss results provides more information about the dependence of the energy loss on the impact parameter [IO]. The ratio CIbetween the channeled and random energy loss for 4He ion channeling along the Si (1 1 1) and (1 1 0) directions is shown in Fig. 2 as a function of the ion energy. The solid line corresponds to the experimental data of Eisen et al. [27] for the most probable energy loss value. Here, our calculations are represented by open circles

1.0

4He -

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E (MeV) Fig. 2. Ratio r between channeled and random [20] stopping powers for He channeling in Si (1 1 I) and (1 1 0) directions as function of the incident energy. The solid lines correspond to the experimental data from Eisen et al. [27]. The open circles represent our energy-loss model (AO). (impact-parameter dependent stopping power due to Si inner-shell and He electrons as well as uniform contribution of the valence electrons). The dotted line includes additionally the effect of the valence electron-density distribution on the channeled stopping power (see text). The insets show the projection of the Si atoms in the (I I I) and (1 1 0) directions.

with error bars (representing the numerical uncertainty) The contribution of the valence electrons used in the calculations is also shown. For the Si (1 1 1) direction the ratio c( is well reproduced by the present model (coupled-channel calculation of the energy loss arising from the Si inner-shell electrons and constant contribution of the valence electrons). However, for the widest channel of Si, namely (1 1 0), the present model strongly overestimates the channeled stopping power given by Eisen et al. [27]. This comes from the fact that we

have neglected the influence of the Si valence electrons. We can estimate the effect of the valence electrons on the channeled stopping power by considering the equipartition rule of the electronic stopping at high energies and the integrated electronic density along the ion path. According to the equipartition rule, half of the energy loss stems from distant collisions (non-local contribution) and therefore it is not suppressed under channeling conditions. The effect of the remaining local contribution can be estimated from the electron density integrated along the ion path. For the (1 0 0) axial channeling, a reduction of about 50% of the integrated density compared to its mean value is found [B]. A reduction of only 8% is found for the Si (1 1 1) channel. The dotted lines in Fig. 2 show the corresponding effect of the valence electrons on the ratio r for (1 1 1) and (1 1 0) channels. Hence, the variation of the valence-electron density can be neglected for the Si (1 1 1) and (1 0 0) channels but not for (1 1 0). the widest channel of the Si crystal.

higher energies, a suppression of inner-shell ionization is responsible for the reduced energy loss. The contribution of the Si inner-shell electrons to the total random stopping power becomes larger with increasing projectile energy and approaches 7 1% (10 of 14 electrons are inner-shell electrons) at asymptotically high energies. This contribution is almost completely suppressed under channeling conditions and thus, the ratio SI(channeling to random) decreases significantly at larger energies. A very good agreement with existing experimental data has been achieved without using any free parameters. Detailed treatments as, e.g., described in this work can provide a reliable scenario of the energy loss processes under channeling conditions.

Acknowledgements This work was partially (Conselho National Cientifico e Tecnologico).

supported by the CNPq de Desenvolvimento

References 4. Conclusions [II J.W. Mayer. E. Rimini (Eds.), Ion Beam Handbook We have calculated the electronic energy loss under channeling conditions for He slowing down in Si using the A0 method. .4 full analysis of the channeled stopping power as a function of the projectile energy was performed by considering the impact-parameter-dependent energy loss, projectile charge states and ion flux distribution. The energy-loss calculations are based on a numerical solution of the coupled-channel equations for each Si L-shell electron (the contribution of the K shell electrons is negligible for the energy range studied here, up to 10 MeV). Also the energy loss due to the projectile electron, in the case of He+ projectiles, has been taken into account. The independent particle model was used to obtain the total energy loss. Near the stopping power maximum, the reduced stopping power under channeling conditions is basically due to the impact-parameter dependence of projectile-electron loss and capture processes involving Si inner-shell electrons. At

VI [31

[41 [51

[61 [71 [81 I91 [lOI [l lI [l21 U31

for Material Analysis. Pergamon. Press. Oxford. 1978. K.B. Winterbon. P. Sigmund. J.B. Sanders. K. D&n. Vidensk. Sclsk. Mat. Fys. Medd. 37 (14) (1970). P. Bauer. F. Kastner, A. Arnau, A. Salin. P.D. Fainstein. V.H. Ponce. P.M. Echrnique. Phys. Rev. Lett. 69 (1992) 1137. G. Schiwietz. Phys. Rev. A 42 (1990) 296. P.L. Grande. G. Schiwietz. Phys. Rev. A 44 ( 1991) 2984; G. Schiwietz. P.L. Grande, Nucl. Instr. and Meth. B 69 (1992) IO: P.L. Grande. G. Schiwietz. Phys. Rev. A 47 (1993) 1119. P. Sigmund, Nucl. Instr. and Meth. B 85 (1994) 541. P.D. Fainstein, V.H. Ponce. A.E. Martinez, Phys. Rev. A 47 (1993) 3055. R.E. Olson. Radiat. Eff. Def. Solids I IO (1989) I. G. Schiwietz. P.L. Grande. C. Auth. H. Winter. A. Salin. Phys. Rev. Lett. I4 (1994) 2159. J.H.R. dos Santos. P.L. Grande, M. Behar. H. Boudinov. G. Schiwietz. Phys. Rev. B 55 (1997) 4332. L.R. Logan. C.S. Murthy. G.R. Srinivasan. Phys. Rev. A 46 (1992) 5754. A.F. Burenkov. F.F. Komarov. M.A. Kumakhov. Zh. Eksp. Tear. Fiz. 78 (1980) 1474. J.D. Meyer. R.W. Michelmann. F. Ditr6i. K. Bethge. Nucl. Instr. and Meth. B 89 (1994) 186; F. Ditrbi. J.D.

[14] [IS] [16] [17]

[18] [19] [20]

Meyer, R.W. Michelmann, K. Bethge. Nucl. Instr. and Meth. B 99 (1995) 182. A. Amau. M. Penalba. P.M. Echenique. F. Flores, R.H. Ritchie, Phys. Rev. Lett. 65 (1990) 1024. J.J. Dorado. F. Flares. Phys. Rev. A 47 (1993) 3062 J.M. Pitarke. private communication (1997). J. Bang. J.M. Hansteen, Kgl. Dan. Vidensk. Selsk. Mat. Fys. Medd. 31 (13) (1959); L. Wilets. S.J. Wallace. Phys. Rev. 169 (1968) 84: M.R. Flannery. K.J. MacCann. Phys. Rev. A 8 (1973) 2915. F. Herman, S. Skillmann. in: Atomic Structure Calculations, Prentice-Hall, Englewood ClilTs. New Jersey, 1963. P.L. Grdnde, G. Schiwietz. G.M. Sigaud. E.C. Montenegro, Phys. Rev. A 54 (1996) 2983. D. Niemann. P. Oberschachtsiek. S. Kalbitzer. H.P. Zeindl. Nucl. Instr. and Meth. B 80/81 (1993) 37.

[21] R.J. Petty, G. Dedrnaley. Phys. Lett. A 50 (1974) 273. [22] D.V. Morgan (Ed.). Channeling: Theory. Observation and Applications. Wiley. New York. 1973: D.S. Gemmel, Rev. Mod. Phys. 46 (1974) 129. [23] J. Lindhard. K. Dan. Vidensk. Selsk. Mat. Fya. Medd. 34 (14) (1965). [24] J.H. Barret. Phys. Rev. B 3 (lY71) 1527. [25] J.H.R. dos Santos. P.L. Grande. H. Boudinov. M. Behar. R. Stall, Chr. Klatt. S. Kalbitzer. Nucl Instr. and Meth. B 106 (1994) 51. [26] H.S. Jin, W.M. Gibson. Nucl. Instr. and Meth. B 13 (1986) 76. [27] F.H. Eisen. G.J. Clark. J. Rottiger. J.M. Poarc. Radiat. Eff. 13 (1972) 93. [28] A. Dygo. M.A. Boshart. L.E. Seiberling. N.M. Kabachnik. Phys. Rev. A 50 (1994) 4979.