Fullerenes, Nanotubes, and Carbon Nanostructures, ??: ?-?, 2012
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Simulation of the fast electrons transport in thin metal and fullerite films
Petrenko E.O., Makarets N.V.
Taras Shevchenko Kyiv National University, 01033, Kyiv, Ukraine
Mikoushkin V.M., Gordeev Yu.S.
Ioffe Institute, 194021, St.-Petersburg, Russia
Abstract: A Monte-Carlo simulation of the C60-fullerite film irradiation by
electron beam with energy of several keV incidents normally to the film surface
has been performed. Average transverse coordinates of the primary and secondary
electrons, as well as points of several collision types were constructed in
dependence on the depth. It was shown the main contribution to the
polymerization of the C60-fullerite gives a swarm of secondary electrons with
energies about of tens of eV.
Key words: Fullerite, Ion beam, Polymerization, Simulation.
INTRODUCTION
In [1] it was suggested that the C60-fullerite irradiated by electrons with energies of
several keV, can be used as resistant concerning to radiation. Later, in [2, 3] it was shown that
the Monte-Carlo simulation of electron transport in this material qualitatively describes the
observed features of its polymerization. The aim of this work is to simulate changes of the
C60-fullerite properties under the electron irradiation by using the proposed model.
The trajectories of primary and secondary electrons in the C60-fullerite were modeled
by straight line segments between the points at which they can experienced one of the six
random events: 1) elastic collisions with atoms, 2-4) ionization of one of the three electron
shells of the Carbon atom, 5,6) plasmon or phonon generation. Details of the cross sections
calculations for these processes were described in [2, 3]. Elastic scattering was calculated
according to the Mott-theory by using an optical potential model which takes into account
screening, exchange, correlation and the nearest neighbors according to the muffin-teen model.
Ionization cross sections of electron shells were calculated according to [4], plasmons and
phonons generation – according to [5, 6]. Scattering angles of the primary and ejected
electrons were determined by the energy and momentum conservation laws, and in cases 5,6)
they were vanished. Energy of the electron shells of the Carbon atom into the C60-fullerite
have been calculated by quantum methods, the plasma frequency and electron density – on the
basis of experimental data for the dielectric constant. The analysis of cross sections of these
processes showed that elastic scattering is dominated at low energies, and the plasmons
generation – at the mid and high energies, and slightly exceeds of the valence band ionization.
Excitations of the deepest atom level and phonons generation have negligible probability.
RESULTS AND DISCUSSIONS
Trajectories of several thousand of primary and secondary electrons were traced; a few
dozen-million of their collisions were treated in the calculations, and them the various average
values were calculated for the following samples: all electrons, primary electrons, secondary
electrons of different generations. Energy of primary electrons was equal to 5 keV.
Figure 1 shows the transverse coordinates of the primary electrons with initial energy
of 5 keV and ejected electrons of the first and second generations, depending on the depth
along the direction of the primary electrons. Trajectories were traced to as long as electron
energy decreases to 25 eV and 5 eV. Strictly speaking, the concept of the trajectory of an
electron with energy below about 100 eV inside of substance is not applicable, since its de
Broglie wavelength is comparable with the interatomic distance, therefore interference of the
electron wave function on many atoms need to take into account. But in the case of random
atoms arrangement we can expect a diffuse pattern of electron motion. The figure shows that
secondary electrons are removed from the beam axis is much more far than the primary ones.
Analysis of analogical results has shown when the electrons energy is less than the plasmon
energy (about 25 eV), they still can ionize the fullerite atoms, by knocking out the valence
electrons. But when it becomes less the ionization energy (about 9 eV), the only one energy
loss rests for electrons, namely excitation of fullerite valence electrons and phonons
generation. However, both of them have cross sections an order of magnitude smaller than the
elastic scattering. As a result, electrons have predominantly elastic collisions without energy
loss and only sometimes they undergo the inelastic collisions. Their movement becomes
similar to the diffusion of particles with finite lifetime. Dependence of the elastic scattering
probability on the angle and electrons energy showed that at the energy of the order of ten eV,
it becomes almost isotropic. Therefore the transverse coordinate of the secondary electrons
stops to grow in spite of the experimental fact the mean free path of electrons increases at
these energies. As a result, a region where most of primary electrons lose their energy is
appeared near the beam axis, and second one is created away from the axis, where secondary,
slower electrons have many collisions.
Figure 2 shows the average numbers of three types of collisions along of an electron
path with length of about 15 nm, which is located at different depths. They were plotted for
the primary electrons with energy 5 keV, but the on the first all events were took into account,
when the electron energy was more than 25 eV, while on the second – more than 5 eV. Their
comparison shows that the number of elastic collisions increases by two orders of magnitude
with decreasing of electrons energy and at the end of the path they are experiencing a lot of
elastic collisions. Figure 2 a) shows that the main part of electron energy loses in the
beginning of their path is spent to plasmons generation and fullerite atoms ionization that is
secondary electrons creating. After energy loss below of ionization threshold, they can lose its
energy only on the excitation of valence electrons during its random walk. Some of these
excitations, according to [7], may lead to formation of a double carbon bonds.
Note that plasmons evolution was not considered in the work, but they can accumulate
significant part of the primary electron energy, as it follows from the Figure 2 a) analysis.
Their decay into electron-hole pairs away from the place of generation could significantly
change the resulting picture. On the other hand, this decay can be stimulated by defects,
created by the electrons in the irradiated area. Therefore, the contribution of plasmons into
energy balance of the C60-fullerite irradiated by electrons requires further investigation.
CONCLUSIONS
Numerical simulation of the C60-fullerite irradiation by electrons with energies of
several keV showed that one can distinguish two regions which are characterized by balance
of energy transmitted into the target. In the first one, near the beam axis, the fast electrons
dominate and plasmons are generated together with atoms ionization. In the second one, away
from the axis, the slow secondary electrons are dominated, which have a diffusion character
of movement and energy loss due to excitation of valence electrons.
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