^{1}, M. Murat

^{1,a)}and J. Barak

^{1}

### Abstract

We present results of systematic Monte Carlo calculations of electron transport in silicon for the wide energy range of 0.02–200 keV, obtained in the frame of a single model using verified input data. The results include characteristics of electron transport, such as backscattering coefficients, ranges, transmission, and deposited-energy distributions, which are quantities of importance for electron-beam applications. The calculations of the spatial and temporal evolution of the electron-initiated cascades of secondary electrons yield a better understanding of the electron and ion track structures and related effects in silicon.

I. INTRODUCTION

II. THE MODEL AND CODE

A. Elastic scattering

B. Inelastic scattering

C. A test case: Backscattering from a semi-infinite target

III. ELECTRON RANGES

A. Continuous slowing down range

B. Extrapolated energy deposition range

C. Extrapolated energy transmission range

D. Projected range

IV. DISTRIBUTIONS OF TRANSMISSION AND ENERGY DEPOSITION

V. SPATIAL DISTRIBUTION OF DEPOSITED ENERGY IN ELECTRON TRACKS

VI. TEMPORAL EVENTS

VII. IMPLICATION TO ION TRACKS

VIII. SUMMARY

### Key Topics

- Silicon
- 28.0
- Backscattering
- 17.0
- Monte Carlo methods
- 15.0
- Electrodeposition
- 13.0
- Collisional energy loss
- 12.0

## Figures

Comparison of the differential angular cross section (per atom) for 50 eV electrons in molecule (full circles) calculated using the data of Bettega *et al.* (Ref. 16) to those calculated using the tabulation of ICRU-77 (Ref. 14) in silicon (open triangles) and in carbon and sulfur (open squares). We use .

Comparison of the differential angular cross section (per atom) for 50 eV electrons in molecule (full circles) calculated using the data of Bettega *et al.* (Ref. 16) to those calculated using the tabulation of ICRU-77 (Ref. 14) in silicon (open triangles) and in carbon and sulfur (open squares). We use .

The elastic IMFP in silicon as a function of electron energy.

The elastic IMFP in silicon as a function of electron energy.

The IMFP for inelastic electron scattering in silicon as a function of electron energy . The full and dotted lines show the results of our calculations and those of Ziaja *et al.* (Ref. 29), respectively. The symbols are experimental data of Powell and Jablonski (Ref. 30) and Tanuma *et al.* (Ref. 31).

The IMFP for inelastic electron scattering in silicon as a function of electron energy . The full and dotted lines show the results of our calculations and those of Ziaja *et al.* (Ref. 29), respectively. The symbols are experimental data of Powell and Jablonski (Ref. 30) and Tanuma *et al.* (Ref. 31).

The backscattering coefficients and for silicon (at normal incidence) as a function of . The two dashed curves for are our results for two separation energies and . The full line shows the results of Eq. (1). The circles and the crosses are for the experimental results of Martin *et al.* (Ref. 43) and Hoedl (Ref. 34), respectively. The diamonds and the triangles present the results of PENELOPE and GEANT4 simulations, respectively, taken from Ref. 43. The lower curve with open stars shows our calculated .

The backscattering coefficients and for silicon (at normal incidence) as a function of . The two dashed curves for are our results for two separation energies and . The full line shows the results of Eq. (1). The circles and the crosses are for the experimental results of Martin *et al.* (Ref. 43) and Hoedl (Ref. 34), respectively. The diamonds and the triangles present the results of PENELOPE and GEANT4 simulations, respectively, taken from Ref. 43. The lower curve with open stars shows our calculated .

The angular dependence of the backscattering coefficient for electrons with energies , 10, and 100 keV, incident on a silicon surface at different angles . The lines were computed using Eq. (2). The symbols show the results of our MC simulations.

The angular dependence of the backscattering coefficient for electrons with energies , 10, and 100 keV, incident on a silicon surface at different angles . The lines were computed using Eq. (2). The symbols show the results of our MC simulations.

Illustration of the method to estimate the extrapolated ranges in silicon from the calculated electron transmission curve and the energy deposition curve scaled to its maximum. is the depth into the material. The plots are for 1 keV electrons normally incident on silicon.

Illustration of the method to estimate the extrapolated ranges in silicon from the calculated electron transmission curve and the energy deposition curve scaled to its maximum. is the depth into the material. The plots are for 1 keV electrons normally incident on silicon.

The different electron ranges in silicon described in the text as a function of incident electron energy . The dashed line demonstrates the asymptotic dependence proportional to at high energies.

The different electron ranges in silicon described in the text as a function of incident electron energy . The dashed line demonstrates the asymptotic dependence proportional to at high energies.

The ratio for silicon of calculated by Brigida *et al.* (Ref. 49) and our calculated using Eq. (6) to the experimental of Everhart and Hoff (Ref. 46) extended up to 200 keV.

The ratio for silicon of calculated by Brigida *et al.* (Ref. 49) and our calculated using Eq. (6) to the experimental of Everhart and Hoff (Ref. 46) extended up to 200 keV.

Energy dependence of the projected range in silicon scaled to its value at normal incidence for incidence directions of , 45°, 60°, and 80° from the normal.

Energy dependence of the projected range in silicon scaled to its value at normal incidence for incidence directions of , 45°, 60°, and 80° from the normal.

Transmission curve for 100 keV electrons normally incident on silicon face. Our simulated curve and the fit using Eq. (9) are shown as full and dashed lines, respectively. The curve with open squares is the EMID tabulation (Ref. 50). Note that here is in microns.

Transmission curve for 100 keV electrons normally incident on silicon face. Our simulated curve and the fit using Eq. (9) are shown as full and dashed lines, respectively. The curve with open squares is the EMID tabulation (Ref. 50). Note that here is in microns.

Deposited energy distribution for 100 keV electrons in silicon. The full line and the dashed line show, respectively, our MC results and our fit using Eqs. (6) and (10)–(12). The curve with open squares is the EMID tabulation (Ref. 50).

Deposited energy distribution for 100 keV electrons in silicon. The full line and the dashed line show, respectively, our MC results and our fit using Eqs. (6) and (10)–(12). The curve with open squares is the EMID tabulation (Ref. 50).

The cumulative spherical radial energy distributions in silicon scaled by the incident electron energy . The curves from the top to the bottom are for , 0.2, 0.5, 1, 2, 5, and 10 keV.

The cumulative spherical radial energy distributions in silicon scaled by the incident electron energy . The curves from the top to the bottom are for , 0.2, 0.5, 1, 2, 5, and 10 keV.

Contour plots of in silicon for three values of incident electron energy: 100 eV (top), 1 keV (middle), and 10 keV (bottom).

Contour plots of in silicon for three values of incident electron energy: 100 eV (top), 1 keV (middle), and 10 keV (bottom).

Deposited energy distributions as a function of depth of penetration in an infinite silicon medium for different incident electron energies: 100 eV and 1 keV (bottom and left axes) and for 10 keV (upper and right axes).

Deposited energy distributions as a function of depth of penetration in an infinite silicon medium for different incident electron energies: 100 eV and 1 keV (bottom and left axes) and for 10 keV (upper and right axes).

Temporal evolution of the spectrum (per primary electron) of the first generation of -electrons, generated directly by a primary electron with energy of 5 keV in silicon. The -electron energy is with respect to the bottom of the conduction band. The arrows indicate the positions of the plasmon (denoted as 1), Si-LVV Auger (2), and Si-KLL Auger (3) peaks.

Temporal evolution of the spectrum (per primary electron) of the first generation of -electrons, generated directly by a primary electron with energy of 5 keV in silicon. The -electron energy is with respect to the bottom of the conduction band. The arrows indicate the positions of the plasmon (denoted as 1), Si-LVV Auger (2), and Si-KLL Auger (3) peaks.

The time dependence of the LVV Auger-electron yield (per primary electron) in the cascade process for different electron energies in silicon.

The time dependence of the LVV Auger-electron yield (per primary electron) in the cascade process for different electron energies in silicon.

The time dependence of the average energy and average number (both per incident electron) of cascading hot electrons (i.e., with energy of ) in an infinite silicon medium for different incident electron energies . The curves are (from top to bottom) for , 5, 1, 0.5, 0.2, 0.1, and 0.05 keV.

The time dependence of the average energy and average number (both per incident electron) of cascading hot electrons (i.e., with energy of ) in an infinite silicon medium for different incident electron energies . The curves are (from top to bottom) for , 5, 1, 0.5, 0.2, 0.1, and 0.05 keV.

The total number of cascading electrons , including those that have already stopped, in an infinite silicon medium for different values.

The total number of cascading electrons , including those that have already stopped, in an infinite silicon medium for different values.

Lateral electron density at different times due to a single electron with ejected by an energetic ion propagating in the -direction in an infinite silicon medium.

Lateral electron density at different times due to a single electron with ejected by an energetic ion propagating in the -direction in an infinite silicon medium.

The Debye length in silicon as a function of the radial distance from the track axis for 5 MeV/amu Xe and Ne ions at different times. The symbols are identical to those shown in Fig. 19.

The Debye length in silicon as a function of the radial distance from the track axis for 5 MeV/amu Xe and Ne ions at different times. The symbols are identical to those shown in Fig. 19.

## Tables

The coefficients to be used in Eq. (10) to calculate the twice-scaled depth distribution for the deposited energy.

The coefficients to be used in Eq. (10) to calculate the twice-scaled depth distribution for the deposited energy.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content