About the calculator

The values presented in this app are calculated from simulations performed using CASINO (v3.3), a Monte Carlo-based software developed by Université de Sherbrooke. More information about the software can be found in refs. [1, 2]. A similar database and analysis with all the simulation results can also be found in ref. [3], which has been developed by Delmic, one of the EBEAM partners. The results shown in this app represent only the volume within which the electron looses its energy, on average. Other processes, such as diffusion of the electron-generated carriers, will also have an impact on the interaction volume [4]. These processes are not considered in our calculations.

Simulations

In all simulations for this app, the sample is defined as a 30x30x30 µm3 cube, with density and formula as stated in the calculator. The electron beam is defined with radius 10 nm and acceleration voltage ranging from 1 to 30 kV. For each material and acceleration voltage we simulate 2000 electrons. After running each simulation, we extract a datafile containing the spatial coordinates (x, y, z) and energy of each electron after each elastic scattering event.

Calculation of depth and radius

We assume that the interaction volume of the electrons has cylindrical symmetry, so the spatial coordinate of each interaction can be described in terms of depth z and radius r2 = x2 + y2. For each material and acceleration voltage (E0), we generate a spatial map of energy deposited at each point (r, z) by 2000 electrons. The sum of all these energies corresponds to the total energy deposited on the substrate (Esub). We then calculate the accumulated energy along z, by summing all the values for different radii, and store the depths in which 25, 50, 75 and 99% of Esub has been deposited on the substrate. The radius for deposited energy of 25, 50, 75 and 99% is calculated in a similar manner, by scanning through the accumulated energy along r.

We choose to represent the fraction of energy lost up to 99% instead of 100% to minimize the artifacts due to the stochastic nature of the Monte Carlo simulations. In these simulations, each electron follows a different trajectory, and the depth and radius calculated here only represent average values. In the method described above, the depth (or radius) at 100% deposited energy corresponds to the position of a single electron, the one that travels the furthest, which might be not representative of the collective behavior of the rest of electrons.

Backscattering coefficient

A fraction of the electrons is backscattered, either immediately upon interaction with the sample, or after depositing part of their energy on the substrate. The energy deposited on the sample (Esub) is thus typically lower than the total energy of the electrons. In all cases we check that Esub + Eback = NelE0 (where Nel is the number of simulated electrons, here 2000).  The backscattering coefficient, defined as the number of electrons that end up exiting the sample (being backscattered) divided by the total number of electrons incident on the sample, is also given in the calculator.

Visualization of the interaction volume

The plots shown in the calculator aim to provide a (rough) visualization of the interaction volume of the electron. The plots are generated by using a piriform curve, with parameters a = z99/2 and b = 4/33/2r99, where z99 and r99 are the calculated values of depth and radius for 99% deposited energy.

Usage of this app

This app is aimed at making simulations of the interaction volume of electrons more accessible to users of SEMs and anybody else interested in electron-matter interaction. Please note that the authors of this app do not assume responsibility for any potential inaccuracies or errors in the calculations. In case you want to provide more feedback or report some inconsistency, please contact us at m.sola@amolf.nl.

References

[1] D. Drouin et al. CASINO V2.42 – A fast and easy-to-use modeling tool for Scanning Electron Microscopy and Microanalysis Users, Scanning 29, 92-101 (2007)

[2] H. Demers et al. Three-dimensional electron microscopy simulation with the CASINO Monte Carlo software, Scanning 33, 135-146 (2011)

[3] T. Coenen, Monte Carlo Beam Tracing Simulations for Incoherent Cathodoluminescence Emission (2019)

[4] B. G. Jacobi and D. B. Holt, Cathodoluminescence Microscopy of Inorganic Solids, Springer New York (1990)