Nuclear Instruments and Methods in Physics Research B 89 (1994) 186-190 North-Holland

NIOMI B Beam Interactions

with Materials&Atoms

Investigation of the electronic energy loss of light ions for materials analysis

J.D. Meyer a,*, R.W. Michelmann a, F. Ditr6i b,**, K. Bethge a ’ Institute for Nuclear Physics, J. W. Goethe University, August-Euler Str. 6, D-60486 Frankfurt/M, Germany b Institute of Nuclear Research of the Hung. Acad. of Sci., H-4001 Debrecen, P.O. Box 51, Hungary

The energy loss, straggling and multiple scattering of light ions in materials was investigated. The interaction of charged particles with crystalline materials was calculated on the basis of Lindhard’s dielectric response function. The contribution of local and global electronic excitation was taken into account. Simulations and experimental results of other authors in random and low indexed crystal directions are compared with the model calculations.

1. Introduction

The ion Channeling method [1,2] combined with other techniques (RBS [3], NRA [4-61, PIXE [7], CPAA [8]> has been widely used to study disorders, contami- nations and interfaces in single crystalline or epitaxially layered material. Especially enormous development was achieved in the field of investigation of semiconductors [4,9] (Si, GaAs, Ge) and superconductive crystalline samples. Early experimental investigations have proved, that well channeled ions have anomalous energy loss compared with that in amorphous material [lO,ll]. The description and discussion of the channeling process including the discussion of energy loss are collected in several review papers [1,2,9,12-181. The angular de- pendence of channeling has also been investigated experimentally [19], and some efforts have been made to handle the energy loss process taking into account the interaction between the energetic ions and the valence and core electrons of the atoms bound in the crystal lattice [20-271.

Energetic ions traversing a crystal in directions near to low index crystal axes are steered by the atomic potentials of the crystal in such a way, that they are passing through lower electron densities than the aver- age electron density in the crystal. Such ions have longer range and an inhomogeneous lateral density distribution (flux-peaking) after penetrating some depth in the lattice. Furtheron, following the path of the individual ions in the crystal one recognizes, that they are probing different electron densities (or stopping powers) on their way up to a given depth. This anoma-

* Corresponding author. ** Humboldt Research Fellow in Frankfurt

lous energy distribution of the ions plays an important role for the analysis of impurities in materials by NRA or CPAA experiments. Because of the strong energy dependence of the differential cross section in NRA experiments (e.g. 0i6(d, p)O”, C12(d, p)C13) the en- ergy of the ions which exhibit the nuclear reaction at a certain depth has to be known exactly. The same argument applies for Charged Particle Activation Analysis experiments under channeling conditions. Only with the knowledge of the flux distribution and the corresponding energy distribution of the probing ions an exact depth and lattice position dependent value for the impurity concentration in a monocrys- talline matrix can be given.

In this paper a Monte Carlo model #’ is presented which describes the motion of ions through a crystal. The quantitative dependence of the trajectory, the energy loss, the energy straggling and multiple scatter- ing from the ion channeling process is incorporated into the calculations. The model can be used generally for materials analysis, especially it is applicable for the evaluation of RBS, NRA and CPAA experiments. In this paper it is tested on a high precision experiment of protons transmitted through a monocrystalline (110) Si sample [28].

2. Theory

To describe the interactions of the bombarding ions with the thermally vibrating lattice atoms a screened Coulomb potential V(r) = Z,Zie24(r>/r was chosen (in cgs units). Using the impulse approximation which

#i MABIC, Materials Analysis by Ion Channeling.

0168-583X/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved SSDI 0168-583X(93)E0992-P

J.D. Meyer et al. / Nucl. Instr. and Meth. in Phys. Res. B 89 (1994) 186-190 187

is valid for small scattering angles, the scattering angle 19 was calculated:

6= Z,Z,e’ z -/, (4(r)+) cos 0 de.

Eb (1)

b is the impact parameter, and E is the ion energy. (b(r) is the “universal” screening function [29], Z,e is the charge of the bombarding ion and Zie is the charge of the target atom.

For ions of charge Z,e with energy E moving parallel to the z axis in a matter of uniform density p the local electronic energy loss is given by

dE 27~ (Z,e2)’ _-=

dz - --pL(p, E), me/M, E

(2)

where m, and Mi are the electron and ion masses and L(p, E) is the so called stopping number as a function of electron density and ion energy (or velocity). After Lindhard L can be written as [30,31]:

where o,, is the frequency of the electron plasma and l ,(k, o) the longitudinal dielectric constant which is a function of ‘the wave number k and frequency o of the continuum electron states.

Beside the local density contribution to the energy loss, also a non-local part, arising from collective plasma excitations, exists. This part, Leo, = ln(2mu2/ho) [21]

The numerical integration of Eq. (3) inserted into Eq. (2) yields the stopping power as a function of the electron density and the ion energy. Since the valence

is a generalisation of the Bethe formula with the re-

electron density in a crystal depends on the distance from the atoms, Eq. (3) can be used as an approxima- tion, if the valence electron density changes weakly [32]. This local density approximation (LDA) [34] bas- ing on Eq. (2) agrees well with experimental results for the stopping power in the high and medium energy region [35]. Two physical processes are contributing to the stopping number L, a non-resonant part, which resembles the contribution of close single-particle colli- sions, and a resonant part from collective excitations of the electron gas [36]. Mathematically, the resonance comes from the vanishing of the real and imaginary part of E, in the k-o plane. Integrating these two parts of Eq. (3) separately it is possible to calculate the normalized ratio cw(p, E) between single and collective excitations. This procedure is not possible if an analyti- cal approximation [31] is applied for the stopping num- ber L.

placement of the mean excitation potential I by the plasmonic excitation energy ho.

Summing up the two contributions to the energy loss of a valence electron gas the following formula for the energy loss is found:

dE 2a (Z,e’)’ _-=--

dt me/M, E

x [WL(PY E) + (1 - ~bvLco,b, ~91. (4)

The factor (Y determines that part of the strength of the single particle excitation contribution to the stop- ping power which depends on the local density. For high ion velocity Bohr’s equipartition rule holds with (Y = 0.5, giving equal contributions from close single and collective excitations. Several attempts were made in the literature to determine a, in the mean energy range of protons in silicon (Y = 0.5 was found [37]. The mean valence electron density per atom pv only enters the formula in the prefactor of the collective excitation part, since only a nearly constant valence electron density exhibits collective excitations. Similar formulas are used by several authors to calculate the energy loss under channeling conditions ([22,21] and references herein).

The nuclear energy loss resulting from scattering of ions by lattice atoms can be written as

AE, Mi

- = xrazT E A

where MA is the mass of the lattice atom, and 19 is the scattering angle after an ion-atom scattering given by Eq. (1). In the final calculations of AE, the well known fact 1381 turns out, that the nuclear energy loss be- comes negligible compared to the electronic loss if E/M, 2 100 keV/amu.

According to the statistical nature of the energy loss processes, the energy of ions moving a certain distance in matter shows a distribution, centered around the average energy. This straggling phenomenon is taken into account in our LDA calculations by adopting Lindhard’s straggling equation for the homogeneous electron gas 00,311:

It is well known from experimental results, even the best channeled ions (moving in the middle of

that

a channel) can leave the channel because of multiple scattering with electrons. This scattering phenomenon is responsible for the dechanneling in the case of

(6)

V. BACKSCATTERING/CHANNELING

188 J.D. Meyer et al. /Nucl. Instr. and Meth. in Phys. Rex B 89 (1994) 186-190

perfect crystals. The width of the scattering angle dis- tribution is calculated according to the formula given in refs. [30,31]:

d’P2 ZZeZh - = -2ik dz

~m~~‘Ud~~). (7)

For the numerical integrations in Eqs. (31, (41, (6) and (7) the electron density distribution within the crystal has to be known as a function of the position. In several papers from the literature [21,28,22] the va- lence-electron density as evaluated from X-ray struc- ture factor data is used in the calculations. In this work we use an approach which is applicable to density calculations for a wide range of crystal structures.

(1) For the total electron density, tabulated values from Ziegler [29] are used up to the Muffin Tin radius rMT (0.147 nm in the case of Si). These data are spherical symmetric solid state densities averaged over all crystal directions. (2) For the interstitial part of the electrons (radius 2 rMT) a linear density function is chosen: p(r) = a + b(r - rMT), where b is the gradient of the tabulated values at rMT. The constant a is fitted in such a way, that the volume integral over the inter- stitial part of the crystal unit cell plus the total charge within the muffin tin spheres amount to Zin, where n is the number of atoms in the unit cell. (3) The valence electron density pi within the muffin tin sphere is fitted to the tabulated values by a Slater type function pi = (rx”*-‘))f( > h r , w ere II* is the principal quantum num- ber of the outer electron shell. n* = 3 for the 3s shell of Si. For the function f(r) an ansatz of the form exp(~~~r,air’) is used. The volume integral of pi over the MT-sphere has to be the charge of the valence electrons within the sphere (for Si this amounts to 3.202e [29]). (4) For crystal structures with overlapping MT-spheres (e.g. for Si), the contribution of the p orbitals can also be taken into acount. However, this bounding contribution has only weak influence on the final results in stopping power calculations. This ap- proximative procedure for the electron density calcula- tion yields good results in comparison to first principle calculations from X-ray structure factors [39]. Fig. 1. shows the electron density used in the calculations for Si. The core density pc is the difference between the tabulated values and the calculated valence density. This strongly localized core density is taken into acount for an approximation of the core energy loss with formula (2). At medium energies sufficient accuracy is obtained by this approach. For high energy particles, where the energy loss from the core-electrons in- creases, a more exact approximation as in ref. [40] has to be used for the energy loss contribution from core electrons.

3. Calculations

The motion of an energetic ion within a crystal of thermally vibrating atoms is described by a Monte Carlo calculation. By this simulation the energy, the position and angle of the ion is monitored on its way through the crystal. A measurable lateral and longitu- dinal distribution of ions, their angles and energies are determined by using an appropriate number of ions (more than 5000) in the procedure. The model is not based on an ionic string concept [33], but as the work- ing step in the simulation the distance between two crystal planes perpendicular to the crystal axis or plane under consideration is chosen (z-axis), whereas the impinging ions are distributed homogeneously over the front plane of the elementary cell (x-y axes). In the case of (110) Si the relevant values are: x-y plane D X D/ fi, z distance D/(2&), D = 0.543 nm. In a first part of the model calculations the electron density is tabulated for a volume grid of 60 X 40 X 20 points imposed on the elementary cell. For each of these points in the crystal so many neighbouring atoms are involved in the calculations, that the next not involved atomic shell contributes less than 0.1% to the density. With these density values stopping, straggling, scatter- ing are found as integrals over the direction perpentic- ular to the front plane of the elementary cell (mesh 60 X 40). In the second part of the model, which simu- lates the motion of the ions in the crystal, the forces of all vibrating atoms up to a distance of 3 X D are taken into account through formula (1) and the contributions from the electrons through the above mentioned tabu- lated integrals of stopping, straggling and scattering. The working distance D/(2&?) is added to the actual depth after each step, up to the final depth under

Elsctmnk dersity P ( r )

Radial distance I nm

1. Electronic density for Silicon. 0: spherical symmetric

solid state density. The values are tabulated in ref. [291 up to the muffin tin radius rM ( = 0.147 nm for Si). Drawn curve:

valence electron density pV (integrated charge = 4e for Si).

The differencs pC between the tabulated values and pV is taken as the core density.

J.D. Meyer et al. /Nucl. Instr. and Meth. in Phys. Res. B 89 (1994) 186-190 189

consideration. During their motion the ions always are translated back to the same unit cell in case they leave it (translation invariancy).

4. Results and discussion

To control the capability of our model and theory, the simulated energy distributions are compared with experimental data and other calculations. In this paper the simulations are tested on an experiment where protons of 300 keV are impinging on a 705 nm thick Si crystal in (110) direction. The energy of the transmit- ted particles was monitored by a high-resolution elec- trostatic analyzer [28].

To simulate such a spectrum at least 5000 ions are necessary to cover appropriately the whole impact pa- rameter range. The above mentioned number corre- sponds to a grid of 85 X 60 points. For each point the position of the impinging ion is allowed to be randomly distributed up to half the distance to the next point. In Fig. 2 different strength parameters (Y are used for the simulations. For (Y = 1 only Lindhard’s LDA of Eq. (4) is taken into account, the energy loss of the protons is too low. For (Y = 0.5 local and collective excitations are equal, the peak energy of the transmitted protons is well reproduced, but the total energy distribution is slightly too narrow. The best agreement between ex- periment and simulation is obtained, if the normalized ratio a@, E) between single and collective excitations obtained from Eq. (3) is used. The mean (Y varies between 0.67 for E = 200 keV and 0.65 for E = 300 keV.

Since the simulations are also applied to angular scans from axial to random directions, Fig. 3 shows the simulated random energy loss in Si in comparison with experimental results and calculations of other authors.

I. I * 1, 1 * 1, 1, 1, 11 220 230 240 250 260 270 260 290

Energy I keV

Fig. 2. 0: Energy distribution of 300 keV protons transmitted through 705 nm Si in (110) direction [28]. The dashed dotted line is simulated with (Y = 1, the dashed line with (Y = 0.5 and

the drawn line with LY = &, E) from Eq. (3).

“’ I t

p l.O-

c

5 . 2, :: 0.9 - 0

B b 5 0.8 -

0.7 ' I

2cQ 250 3m

Energy [keV1

Fig. 3. Comparison of the random energy loss in Si between experimental data (OZ ref. [28], A: ref. [38]) with calulations ( n : TRIM91 [29]). The drawn curves are MABIC simulations

(drawn: cr(p, El, dotted: a = 0.5, dashed: (Y = 1).

The triangles [38] and circles [28] are experimental results. The squares are results of TRIM91 [29] simula- tions. The drawn curves are calculated with the same (Y values as in Fig. 2. The results for (Y = 0.5 resp. a! = &, E) are nearly identical, which resembles the fact that the mean value of cy(p, E) is near to 0.5. The agreement ( = 1%) between the experimental and these simulated values is good. It should be noted, that a random TRIM spectrum is = 10% wider than our result since in the TRIM calculations the Bohr strag- gling is used, which is no longer a good approximation in this energy range [25]. In the MABIC model the energy (velocity) and electron density dependent strag- gling is used.

The new model, which takes into account the main interactions and processes influencing the motion of charged particles in crystal lattices proves to be useful and reliable to describe the energy loss (stopping power) and energy distribution of protons, and yields results in good agreement with the experiments.

One of the authors (F. Ditroi) would like to express his thanks to the Alexander von Humboldt Foundation and to Prof. K. Bethge for making possible a research stay at the Institute for Nuclear Physics in Frankfurt.

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