SIM0216

ELECTRON MICROSCOPY
Spectra of Electrons Emerging from PMMA
Monte Carlo Simulation of Electron Energy Distributions
Maurizio Dapor
This work describes a Monte Carlo
algorithm which appropriately
takes into account the stochastic
behavior of electron transport in
solids and treats event-by-event all
the elastic and inelastic interactions
between the incident electrons and
the particles of the solid target. The
energy distributions of secondary
and backscattered electrons emerging
from polymethylmethacrylate
(PMMA) irradiated by an electron
beam are simulated and compared
to the available experimental data.
The Spectrum
When an electron beam impinges
on a solid target, many
electrons can be backscattered,
after they interacted
with the atoms and electrons
of the target. A fraction of
them conserves their original
kinetic energy, having suffered
only elastic scattering
collisions with the atoms of the
target. These electrons constitute
the so-called elastic peak,
or zero-loss peak, whose maximum
is located at the energy
of the primary beam. Close to
the elastic peak, another feature
can be observed: it is a
broad peak collecting all the
electrons of the primary beam
which suffered inelastic interactions
with the outer-shell
atomic electrons (plasmons
losses, and inter-band and intra-band
transitions). Another
important feature of the electron
energy spectrum is represented
by the secondaryelectron
emission distribution,
i.e., the energy distribution of
those electrons that, once extracted
from the atoms by inelastic
collisions and having
travelled in the solid, reach
the surface with the energy
sufficient to emerge. The energy
distribution of the secondary
electrons is mainly
confined in the low energy region
of the spectrum, typically
well below 50eV [1].
The Monte Carlo Algorithm
The results presented in this
paper were obtained using
differential and total elastic
scattering cross sections
calculated utilizing Mott theory
[2], i.e. numerically solving
the Dirac equation in a
central field; this procedure
is known as the “relativistic
partial wave expansion
method” and it has been demonstrated
to provide excellent
results when compared to experimental
data. On the side
of the energy losses, the inelastic
mean free paths are
calculated by taking into account
the inelastic interactions
of the incident electrons with
atomic electrons, phonons,
and polarons. The calculation
of the electron-electron inelastic
scattering processes was
performed within the Mermin
theory [3]. Electron–phonon
interactions were described
using the Fröhlich theory [4].
Polaronic effect was modeled
according to the law proposed
by Ganachaud and Mokrani
[5]. Electron trajectories follow
a stochastic process, with
scattering events separated
by straight paths having a distribution
of lengths that follows
a Poisson-type law. Once
the step length is generated,
the elastic or inelastic nature
Fig. 1: Energy distribution of the electrons emerging from PMMA with
energies between 0 and 20eV. Monte Carlo simulated spectrum (red solid
line) is compared to the Joy et al. experimental spectrum [8] (black line).
Data are normalized to a common maximum. The primary energy is 1000eV.
The primary electron beam is normal to the surface. Electrons are accepted
over an angular range from 36° to 48° integrated around the full 360°
azimuth. The zero of the energy scale is located at the vacuum level.
38 • G.I.T. Imaging & Microscopy 2/2016