**3. Computer simulation method**

(such as crater formation) and preferentially removing surface asperities leading to flat surfaces. Channeling of low-energy ions in metal and semiconductor single crystals offers the opportunity to create the method of local ion implantation in ultrathin film nanotechnology and surface nano-engineering. Therefore, ranges, energy losses, and profiles of distribution of low-energy ions channeling in crystals have received considerable experimental and theoret‐ ical interest [18–20]. For small crystal depths, the approaches which are used in the analytical theory of orientation effects on the large depths become unacceptable, and a computer simulation method for the channeling process modeling appears to be the most preferable [21, 22]. So, the theoretical investigation of atomic collision processes in crystals caused by particle irradiation and deposition is usually done using computer simulation, because real physical conditions (e.g., complicated interatomic interaction potential, surfaces, interfaces, defects, etc.) can be taken into account much easier than it is possible by using analytical methods [17,

The measurements of differential energy spectra and angular distributions of scattered ions were performed in experimental equipment (energy analyzer of the spherical deflector type)

to analyze the secondary ion masses by means of time-of-flight technique [24, 25]. The experimental setup includes a UHV scattering chamber with the oil-free pumping system and a base pressure in the 10−9 Torr range. During the measurements, the working pressure rises to about 5 × 10−9 Torr. Ions of alkaline metals were obtained in a thermal ion source with a target density of current *J* <sup>0</sup> = 5 ⋅ 10−7 A∙cm−2 under operating conditions. The repeated cycles of an electron bombardment was used for cleaning the target surface. Beams of *Е* 0 = 50–500

Re and Ti, V, Cr polycrystal surfaces under an incidence angle of *ψ* = 55°. At the registration of spectra from contaminated targets, the peak of the straight flight was constantly observed. This peak corresponded to the energy of primary ions *E* o, that is, the reflection from oxide film which behaved as a screen. After cleaning by electron bombardment, this peak disappeared and bell-shaped spectra were observed. The incident and scattering beams were laid in the same plane, perpendicular to the surface of the target in the point of incidence of ions. The size of ion spot on the sample at normal incidence of ions on the surface was 2 mm, and the scatter of angles of incidence of the ion beam did not exceed Δ*ψ* = ±2°. The backscattered ions are collected at a scattering angle θ = 70°. The angular resolution of the device is Δ*θ* = ±2°. Usually, within the method of ion-scattering spectroscopy, the interpretation of both angular and energy distribution of the scattered heavy alkali ions (K, Cs) is based on the differential cross

spectra were processed by computers for averaging of statistical fluctuations of impulse registration with the use of low-frequency digital filter of Spencer. Repeating the deflection of the voltage on the plate of energy analyzer with alteration of cleaning of the target, the impulse

) and energetic (Δ*E*/*E* ≅ 1/125) resolutions and with the capability

being 100% [26]. The measured energy

ions, with a current density *J* <sup>0</sup> = 5 ⋅ 10−7 A∙cm−2, scattered from clean Ta, W,

21–23].

eV of Cs+

**2. Experimental**

362 Radiation Effects in Materials

with high angular (Δ*ψ* ≅ 0.6o

and K+

section of scattering only, the ionization degree *η* <sup>+</sup>

The theoretical investigation of atomic collision processes in crystals caused by ion irradiation is usually done using computer simulation, because real physical conditions (e.g., complicated interatomic interaction potential, surfaces, interfaces, and defects) can be taken into account much easier than it is possible by using analytical methods [4, 5]. The simulation used in our calculations to construct the trajectories of the ions or projectile scattered by target atoms is based on the binary collision approximation [5] with two main assumptions: (1) only binary collisions of ions within target atoms or between two target atoms are considered, and (2) the path in which a projectile goes between collisions is represented by straight-line segments (**Figure 1**). In the binary collision model, particles move along straight-line segments, repre‐ senting asymptotes to their trajectories in laboratory system, and one determines not a particle trajectory but rather the difference between the angles characterizing the initial and final directions of motion. While this approach permits one to cut the required computer time (compared with direct integration of the equations of motion), it also entails a systematic error due to the fact that over short segments of path, the real ion trajectory differs from the asymptotes used to replace the former. This error was estimated in ref. [27] for the Cu–Cu pair, for a number of potentials and three values of energy. It was established that the deviation of an asymptote from the real trajectory is essential only for head-on collisions and high energies.

**Figure 1.** Scheme of the binary collision approximation [27].

For the description of the particle interactions, the repulsive Biersack–Ziegler–Littmark (BZL) potential [28] with regard to the time integral was used. The BZL approximation for the screening function in the Thomas–Fermi potential takes into account the exchange and correlation energies, and the so-called "universal" potential obtained in this way shows good agreement with experiment over a wide range of interatomic separations. Elastic and inelastic energy losses have been summed along trajectories of scattered ions. The inelastic energy losses *ε* (*E* 0, *p*) were regarded as local depending on the impact parameter *p* and included into the scattering kinematics. These losses have been calculated on the basis of Firsov model modified by Kishinevsky [5] and contain direct dependence on the impact parameter:

$$\begin{aligned} \text{2. } \left(E\_0, p\right) &= 0.3 \times 10^{-7} \text{v} \, Z\_1 \left(Z\_1^{12} + Z\_2^{12}\right) \left(Z\_1^{16} + Z\_2^{16}\right) \\ \left[1 - 0.68 V \left(r\_0\right) / E\_r\right] & / \left[1 + 0.67 \, \sqrt[4]{Z\_1 r\_0} / a\_{\text{TF}} \left(Z\_1^{16} + Z\_2^{16}\right)\right] \end{aligned} \tag{1}$$

where *v* and *E <sup>r</sup>* are the velocity and energy of relative atomic motion, *Z* 1 is a greater, and *Z* <sup>2</sup> the smaller of the atomic numbers, and *r* 0 is in units of Å.

The expressions for the ion *E <sup>i</sup>* and recoil *E <sup>r</sup>* energies after binary collision, taking into account inelastic losses, can be written as follows [29]:

$$\begin{aligned} E\_i &= \left(1 + \mu\right)^{-2} E\_0 \left(\cos\theta\_i \pm \sqrt{\left(f\,\mu\right)^2 - \sin^2\theta\_i}\right)^2, \\\\ E\_r &= \mu \left(1 + \mu\right)^{-2} E\_0 \left(\cos\theta\_r \pm \sqrt{\left(f\right)^2 - \sin^2\theta\_r}\right)^2 \end{aligned} \tag{2}$$

where *f* = [1 – (1 + *μ*)/*με*(*E* 0, *p*)/*E* 0]; *θ <sup>i</sup>* and *θ <sup>r</sup>* are the angles of ion and recoil scattering in the laboratory system of coordinate; *E* <sup>0</sup> is initial energy of impinging ion; *p* is impact parameter, and *μ* = *m* 2/*m* 1. In ref. [29] the dependencies of *ε*(*E* 0, *p*) on the basis of the Firsov, Kishinevsky [5], and Oen-Robinson [21] models for Ne+ → Ni pair and low-energy *E* <sup>0</sup> = 1–10 keV have been calculated. Estimating the accuracy of models for various values of the impact parameter in the low-energy range, it is necessary to notice that in a small impact parameter region (*p* < 0.5 Å), it is more preferable to use the Kishinevsky model; however, in the region of large impact parameters, all three models give approximately the same results, and they are useful even when the energy *E* 0 ~ 100 eV. The above-mentioned models were checked experimentally more than once. On the whole, experimental data agree well with these theories; however, in some cases, the calculated values exhibit discrepancies from the measurement which reach 30–40% [21, 27]. In order to consider simultaneous collisions of a particle with the atoms of the adjacent chains, the procedure proposed in ref. [27] was used. The inclusion of the thermal vibrations assumed that the target atoms oscillated independently of one another, and their deflections from the equilibrium position are subject to the normal Gaussian distribution.

Sputtering has been simulated in the primary knock-on regime. Only the primary knock-on recoil (PKR) atoms ejected from first, second, and third layers have been considered. The presence of planar potential energy barrier on the surface was taken into account. The number of incident ions is 4 × 104 . Each new particle is incident on a reset, pure surface. The incident ions and the recoil atoms were followed throughout their slowing down process until their energy falls below a predetermined energy: 25 eV was used for the incident ions, and the surface binding energy was used for the knock-on atoms. The calculations were performed on the crystals comprising up to 120 atomic layers. The simulations were run with the crystal atoms placed stationary at equilibrium lattice sites.

correlation energies, and the so-called "universal" potential obtained in this way shows good agreement with experiment over a wide range of interatomic separations. Elastic and inelastic energy losses have been summed along trajectories of scattered ions. The inelastic energy losses *ε* (*E* 0, *p*) were regarded as local depending on the impact parameter *p* and included into the scattering kinematics. These losses have been calculated on the basis of Firsov model modified

( ) ( )

0 10 1 2

*Vr E Zr a Z Z*

where *v* and *E <sup>r</sup>* are the velocity and energy of relative atomic motion, *Z* 1 is a greater, and *Z* <sup>2</sup>

The expressions for the ion *E <sup>i</sup>* and recoil *E <sup>r</sup>* energies after binary collision, taking into account

( ) ( ( ) ) <sup>2</sup> 2 2 <sup>2</sup> <sup>0</sup> <sup>1</sup> <sup>c</sup> <sup>s</sup> – sin ,o - *EE f i ii* = +

qm

± Ö

( ) ( ( ) ) <sup>2</sup> 2 2 <sup>2</sup>

laboratory system of coordinate; *E* <sup>0</sup> is initial energy of impinging ion; *p* is impact parameter, and *μ* = *m* 2/*m* 1. In ref. [29] the dependencies of *ε*(*E* 0, *p*) on the basis of the Firsov, Kishinevsky

calculated. Estimating the accuracy of models for various values of the impact parameter in the low-energy range, it is necessary to notice that in a small impact parameter region (*p* < 0.5 Å), it is more preferable to use the Kishinevsky model; however, in the region of large impact parameters, all three models give approximately the same results, and they are useful even when the energy *E* 0 ~ 100 eV. The above-mentioned models were checked experimentally more than once. On the whole, experimental data agree well with these theories; however, in some cases, the calculated values exhibit discrepancies from the measurement which reach 30–40% [21, 27]. In order to consider simultaneous collisions of a particle with the atoms of the adjacent chains, the procedure proposed in ref. [27] was used. The inclusion of the thermal vibrations assumed that the target atoms oscillated independently of one another, and their deflections

Sputtering has been simulated in the primary knock-on regime. Only the primary knock-on recoil (PKR) atoms ejected from first, second, and third layers have been considered. The presence of planar potential energy barrier on the surface was taken into account. The number


<sup>0</sup> 1 cos – sin *E Ef r rr*

from the equilibrium position are subject to the normal Gaussian distribution.

 q

<sup>é</sup> ù +Ö + é ù ë û ë û

7 1/2 1/2 1/6 1/6

1/6 1/6

 q

and *θ <sup>r</sup>* are the angles of ion and recoil scattering in the

→ Ni pair and low-energy *E* <sup>0</sup> = 1–10 keV have been

 q

= + ± Ö (2)

. Each new particle is incident on a reset, pure surface. The incident

(1)

by Kishinevsky [5] and contain direct dependence on the impact parameter:


m

mm

, 0.3 10

the smaller of the atomic numbers, and *r* 0 is in units of Å.

inelastic losses, can be written as follows [29]:

where *f* = [1 – (1 + *μ*)/*με*(*E* 0, *p*)/*E* 0]; *θ <sup>i</sup>*

of incident ions is 4 × 104

[5], and Oen-Robinson [21] models for Ne+

e

364 Radiation Effects in Materials

( ) ( )( )

0 11 2 1 2

*E p vZ Z Z Z Z*

1 – 0.68 / / 1 0.67 / *<sup>r</sup> TF*

The channeling simulation program used in the present work is similar by structure to the well-known MARLOWE program and based on the binary collision approximation. But in the case of the solid phase, the binary interaction gets distorted by the influence of neighboring atoms and multiple collisions. It is impossible to calculate inelastic energy losses in this case without exact knowledge of the trajectory of scattering ions. For their calculation, it is necessary to perform computer simulation of ion scattering and channeling in a single crystal. A parallel, uniform, mono-energetic ion beam impinges on an impact area on the surface of a crystal. The angle of incidence of primary ions *ψ* was counted from a target surface. It is assumed that the incident beam is of small density; so, the ions of the beam do not hit twice at the same place. The impact area covers an elementary cell in the transverse plane of channel axis. The number of incident particles is 4 × 104 . The shape of the target area is chosen such that by translating it, one could cover the entire surface of the crystal. Successive multiple scattering of ions from atoms in the rows lying along the principal crystallographic axes is followed in a special search procedure to find the next lattice atom or atoms with which the projectile will interact, with impact parameters for all target atoms forming the walls of a channel calculated for each layer in the crystallite. Around the colliding target atom, the coordinates of the nearest neighbor atoms are consistently set according to the crystal structure of the target. For each set of atoms, the following conditions are checked: (i) it should be at the front part of the ion movement, relatively to a crossing point of asymptotes of the projectile movement directions before and after collision; (ii) the ion impact parameter should be less than *p* lim (*p* lim is the impact parameter corresponding to the scattering angle of 0.05°); (iii) among colliding atoms, it should be the first in turn *p* 1, *p* 2, *p* 3… on the consecutive collisions. After each collision, the scattering angle, energy, and the new movement direction of the channeled ion are determined. It is checked if the projectile is still moving in the given channel. The coordinates of dechanneling are used to obtain the dechanneled and channeled fractions as functions of the depth. The incident ions were followed throughout their slowing-down process, until their energy falls below 25 eV.

In order to consider simultaneous and nearly simultaneous collisions of a particle with the atoms of the adjacent chains, the special procedure proposed in ref. [27] was used. So-called simultaneous collisions which occur if a projectile has a symmetrical position, and which can collide with more than one target atom at the same time, are approximated by successive binary collisions. The inclusion of the thermal vibrations assumed that the target atoms oscillated independently of one another, and their deflections from the equilibrium position are subject to the normal Gaussian distribution. The effect of correlation is equivalent to a reduction of the vibration amplitude of about 5–10%, depending on the effect being looked at [5]. The program allowed the consideration of main peculiarities in channeled particle distribution with depth, such as collision-by-collision details of trajectories, flux-peaking, and difference in specific energy losses for random and channeled trajectories. The number of incident ions is 4 × 104 . Each new particle is incident on a reset, pure surface. The incident ions and the recoil atoms were followed throughout their slowing-down process until their energy fell below a predetermined energy: 25 eV was used for the incident ions, and the surface-binding energy was used for the knock-on atoms. The calculations were performed on the crystals comprising up to 120 atomic layers. The simulations were run with the crystal atoms placed stationary at the equilibrium lattice sites.

The initial energy of incident ions was varied from 0.5 to 10 keV, a grazing angle of incidence *ψ* counted from the target surface was 3–30°, and an azimuth angle of incidence *ξ* realized by rotating the target around its normal and counted from the *<*100> direction was 0–180°. The polar scattering angle *θ* was counted from the primary beam direction, the polar escape angle *δ*––from the target surface, and the azimuthal scattering angle *φ*––from the incidence plane.

In **Figure 2**, the scattering geometry and scheme of a semichannel on the Cu(100) face along the direction <110> and the target area on it are shown. The impact points on the crystal surface filled a rectangle, whose sides were divided into 100 segments in the beam incidence plane (*I*coordinate) and 1000 segments in the perpendicular direction (*J*-coordinate). The sizes of the target area were 1.28 Ả (half-width of the semichannel) on the *J*-coordinate and 2.56 Ả (the interatomic distance along the <110> row) on the *I*-coordinate. A substantial part of the calculations presented in this chapter was made by the computer simulation technique. This was required by the complexity of the scattering, sputtering, and channeling trajectories and by a large number of correlated collision events which prohibit the use of statistical stochastic methods of calculation. Mathematical experiment is similar in some extent to the physical one, while permitting us to extract more information from the latter.

**Figure 2.** Scheme of ion scattering by a surface semichannel on the Cu(100) face and target area located on it. *I* and *J* are the coordinates of the impact points along and transverse to the semichannel axis, respectively, determining the num‐ ber of incidence ions [5].
