Shower approach in the simulation of ion scattering from solids

An efficient approach for the simulation of ion scattering from solids is proposed. For every encountered atom, we take multiple samples of its thermal displacements among those which result in scattering with high probability to finally reach the detector. As a result, the detector is illuminated by intensive “showers,” where each event of detection must be weighted according to the actual probability of the atom displacement. The computational cost of such simulation is orders of magnitude lower than in the direct approach, and a comprehensive analysis of multiple and plural scattering effects becomes possible. We use this method for two purposes. First, the accuracy of the approximate approaches, developed mainly for ion-beam structural analysis, is verified. Second, the possibility to reproduce a wide class of experimental conditions is used to analyze some basic features of ion-solid collisions: the role of double violent collisions in low-energy ion scattering; the origin of the “surface peak” in scattering from amorphous samples; the low-energy tail in the energy spectra of scattered medium-energy ions due to plural scattering; and the degradation of blocking patterns in two-dimensional angular distributions with increasing depth of scattering. As an example of simulation for ions of MeV energies, we verify the time reversibility for channeling and blocking of 1-MeV protons in a W crystal. The possibilities of analysis that our approach offers may be very useful for various applications, in particular, for structural analysis with atomic resolution.

Figure 1

Illustration of the principle of selection of the “hot” region of atomic displacements used in the shower generation. The density of the Gaussian distribution of atomic displacements is represented by the gray cloud, while the “hot” region is indicated as the tube enclosing the region of impact parameters for scattering into the chosen angular cone (left panel). Analogously, the picture at the right panel illustrates the generation of showers of recoiled atoms. $W_i$ and $w_i$ are the weights ascribed to the trajectories of primary and secondary ions, respectively, while $P_i$ designates the integral probability of an atom displacement into the “hot” region [see the explanations following Eq. (2)].

Figure 2

Comparison of simulated angular scans for the scattering of 5-keV Ar${}^{+}$ ions from the surface of a Fe${}_4$N(100) crystal with the experimental results from Ref. [18] (top panel). The yield of recoiled Fe and N atoms is shown in the center and lower panels, respectively. See text for more details.

Figure 3

Energy spectrum of Ar${}^{+}$ ions scattered from the (100) surface of a Fe${}_4$N crystal for the orientation indicated in Fig. 2 by an arrow. The lower part of the figure shows the angular distribution of scattered ions. Results are shown using a gray scale where darker represents higher intensity.

Figure 4

(a) Comparison of our simulation results for 3-keV Ne${}^{+}$ ions scattering from amorphous copper with experimental data [20] and results of simulation with the marlowe code. The simulation results were folded with a Gaussian distribution in order to simulate the experimental energy resolution of 40 eV. The dashed curve corresponds to a calculation in the single-scattering approximation. Panel (b) shows simulations performed by applying additional approximations. The dashed curve corresponds to a simulation using the single-scattering approximation with impact-parameter-dependent energy losses, while the thin solid curve was calculated considering the energy losses as the result of a uniform stopping force.

Figure 5

Results of the simulation of 100-keV He${}^{+}$ ions scattering from a Si crystal. The scattering yield is accumulated from depth ranges of (a) 5–30 nm and (b) 30–55 nm. The displayed angular range lies around the $\langle$112$\rangle$ crystal direction.

Figure 6

Comparison of our simulations (solid curves) of angular scans for 100-keV $p$ scattering from a monolayer-thick YSi${}_2$ film on Si(111) with the result of a vegas simulation (dashed curves) and experimental data [22] (dots). The direction of the incident beam is parallel to (a) the $\langle \overline{1}00\rangle$ and (b) $\langle 0\overline{1}\overline{1}\rangle$ directions of the Si substrate, while the detector lies in the plane $(01\overline{1})$. The structure of surface layers is shown at the top. The yield is normalized to the Rutherford cross section. The dashed-dotted curve in the lower graph shows results for a trial structure with Y atoms located at the top layer.

Figure 7

Backscattering energy spectra of 0.1-MeV $p$ scattered from a 1000-Å Au film (a) and of 0.28-MeV He ion scattering from a 1130-Å Pt foil (b). The simulation results and the calculations in the single-scattering approximation are compared with the rbstrim simulation [25] and with experimental data [26]. Results of simulation in the single-cone approximation are shown in the top panel. The lower graph shows the partial contributions of ions from the internal (MS) and outer (PS) cones.

Figure 8

Channeling and blocking dips calculated for the case of irradiation with 1-MeV protons of a tungsten crystal along the $\langle$100$\rangle$ axial direction with detection at a 135${}^{°}$ scattering angle and in the inverse geometry. The yield is calculated in the energy window corresponding to scattering from a depth range between 2500 and 3500 Å. The blocking dip normalized to the thickness of the layer along the beam direction is also shown.

Figure 9

Schematic illustration of the procedure of flux convolution used in the reversing approach (see the text for explanation of details).