Fullerenes, Nanotubes, and Carbon Nanostructures, ??: ?-?, 2012
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Simulation of fast electron transport in thin fullerite C60 films
Petrenko E.O., Makarets N.V.
Taras Shevchenko Kyiv National University, 01033, Kyiv, Ukraine
Mikoushkin V.M.
Ioffe Institute, 194021, St.-Petersburg, Russia
Abstract: Monte-Carlo simulation of the C60-fullerite film irradiation by 5 keV electron
beam at normal incidence has been performed. Average transverse coordinates of the
primary and secondary electrons as well as the density of collisions of several types
were drawn in dependence on the penetration depth. It was shown that a swarm of low
energy secondary electrons provides efficient excitation of valence electrons and gives
the main contribution to polymerization of C60-fullerite.
Key words: Fullerite; C60; Electron beams; Polymerization, Computer simulation.
INTRODUCTION
Polymerization and other processes of modification induced by electron beams radically
change solubility and evaporability of fullerite C60 [e.g.1]. These processes already have enabled
using the fullerite films as an electron beam resist in electron lithography, though the better
understanding the modification mechanisms is necessary for optimization of the technology. The
Monte-Carlo simulation of electron transport in fullerite C60 performed in Ref. [2, 3] qualitatively
described the main features of the fullerite polymerization. The aim of this work was to develop the
Monte-Carlo description of interaction between irradiating electrons and fullerite C60 to obtain
quantitative characteristics of the interaction and propagation of electrons in this material.
Trajectories of primary and secondary electrons in the C60-fullerite were modeled by straight
lines between the points at which electrons participated in one of the following random events: 1)
elastic collision with atoms, 2-4) ionization of one of the carbon core-level, 5-6) plasmon or phonon
generation. Details of the cross section calculations for these processes were described in [2, 3].
Elastic scattering was calculated according to the Mott-theory by using the optical potential model
which takes into account screening, exchange, correlation and the nearest neighbors according to
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the muffin-teen model. Ionization cross sections of electron shells were calculated according to [4],
generation of plasmons – according to [5] and phonons – according to the Pines-theory. Scattering
angles of the primary and ejected electrons were determined by the energy and momentum
conservation laws, and they were equal zero in the cases 5-6). The plasma frequency and electron
density were calculated on the basis of experimental data for the dielectric constants. The analysis
of all cross sections showed that elastic scattering dominates at low energies, whereas plasmon
generation and valence band ionization are the most effective at the mid and high energies. The
probability of two core-hole excitation and phonon generation were estimated to be negligible.
RESULTS AND DISCUSSIONS
Trajectories of several thousands of primary and secondary electrons and a few dozen
millions of their collisions were considered in the simulation. Then various average values were
calculated for the following samplings: all electrons, primary electrons, secondary electrons of
different generations. The energy of primary electrons was equal to 5 keV.
Fig. 1. Dependence of transverse average coordinates of the primary electrons and electrons
ejected in first and second generations in dependence on the penetration depth: a) for electrons with
energies above 25 eV; b) with energies above 5 eV.
Fig. 1 shows the transverse average coordinates (X,Y) of the primary electrons with initial
energy of 5 keV and ejected secondary electrons of the first and second generations in dependence
on the penetration depth along the direction of the primary electrons (Z). Trajectories were traced
until the electron energy decreases below 25 eV and 5 eV. Strictly speaking, the concept of the
trajectory of an electron with the energy below 100 is not correct since its de Broglie wavelength is
comparable with the interatomic distance. But the possible interference effects were assumed to be
averaged in the case of disordered atomic system and a huge number of electrons considered. The
figure demonstrates that secondary electrons leave the beam axis area much more efficiently than
the primary ones. Analysis of the data showed that electrons can efficiently ionize the valence band
2
when their energy becomes less than the plasmon’s (~ 25 eV). But when the electron energy
diminishes below the ionization potential (~ 9 eV), only excitation of fullerite valence electrons and
phonon generation remain as energy loss channels. However, the cross sections of these channels
are one order of magnitude smaller than of the elastic scattering. Therefore electrons undergo
predominantly elastic collisions without energy losses, and only sometimes inelastic ones. Their
movement becomes similar to the diffusion of particles with finite lifetime. Angular dependence of
the elastic scattering probability is almost isotropic for low energy electrons (dozen eV). Therefore
the transverse coordinate of the secondary electrons stops to grow as it can be seen in Fig. 1. As a
result, primary energetic electrons are concentrated in the area near the beam axis and the low
energy secondary ones are accumulated fare away from the axis and undergo many collisions
finally resulting to valence band excitation.
Fig. 2. Dependence of the average number of collisions per primary electron on the projected
depth for different collision types: a). for electrons with energies above 100 eV; a). for electrons
with energies above 5 eV.
Fig. 2 shows dependence of the average number of collisions per primary electron on the
projected depth for three collision types: elastic, inelastic with plasmon excitation and ionization.
The data in the panel a) and b) were obtained with taken into account electrons with energies above
100 eV and 5 eV correspondingly. Comparison of these panels shows that the number of elastic
collisions increases by two orders of magnitude with decreasing the electron energy and that
electrons undergo a lot of elastic collisions at the end of the path. Fig. 2 a) also shows that the
plasmon generation is the dominating process at the beginning of the primary electron trajectory.
After retarding below ionization threshold, electrons can lose their energy only on excitation of
valence electrons during its random walk. Some of these excitations should lead to formation of
double carbon bonds [6].
It should be noted that evolution of plasmons was not considered in the work, though they
can accumulate essential part of the primary electron energy as it follows from the analysis of Fig. 2
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a). Besides the material heating due to electron-phonon interaction, plasmons can decay into
electron-hole pairs generating additional secondary electrons, which can shape the resulting picture.
Therefore, the role of plasmons in energy balance should be further investigated.
CONCLUSIONS
Numerical simulation of the C60-fullerite irradiation by 5 keV electrons revealed existence
of two areas which are characterized by different kinds of electrons and different balance of the
energy transmission into the target. The first one is located near the beam axis where fast electrons
dominate providing in turn dominating the processes of plasmon generation and ionization. The
second area with domination of slow secondary electrons is distanced from the axis. These electrons
have a diffusion character of motion and lose their energy due to excitation of valence electrons
followed by fullerite dimerization and polymerization.
REFERENCES
[1] Snitov V. V.; Mikoushkin, V. M.; Gordeev Yu. S., Fullerite C60 as electron-beam resist for
“dry” nanolithography, Microelectronic Engineering, 2003, 69, 429-435.
[2] Makarets N.V., Prylutskyy Yu.I., Mikoushkin V.M., Shnitov V.V., Gordeev Yu.S., Computer
Simulation of Fullerite C60 Modification by a Swarm of Secondary Electrons Generated by
Bombarding Electrons in keV Energy Range, Fullerenes, Nanotub. Carbon Nanostruct, 2006, 14(23), 513-518.
[3] Makarets N.V., Prylutskyy Yu.I., Zaloyilo O.V., Gordeev Yu.S., Mikoushkin V.M., Shnitov
V.V., Simulation of fullerite C60 polimerisation under particle beams irradiation, Mol. Cryst. &
Liquid Cryst., 2005, 426, 171-178.
[4] Kim Y.-K., Rudd M.E., Binary-encounter-dipole model for electron-impact ionization, Phys.
Rev, 1994, A50(5), 3954-3967.
[5] Ferrell R.A., Angular Dependence of the Characteristic Energy Loss of Electrons Passing
Through Metal Foils, Phys. Rev., 1956, 101(2), 554-563.
[6] Stafström S., Fagerström J., Electronic structure and stability of fullerene polymers, Appl. Phys.
A:
Mater.
Sci.
Process,
1997,
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307-314.
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Refer. Number,
type of the
presentation
Correspondin
g author,
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Title of the
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Key words
Word count (figure is
equal to 200 words, table
- 200 words, equation 40 words)
3 preferred reviewers from the organizers or the
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Remarks
http://www.ioffe.ru/ACN2011
A206 Poster
Dr
Makarets
N.V.
mmv@univ.k
iev.ua
Simulation of
fast electron
transport in thin
fullerite C60
films
Fullerite; C60;
Electron
beams;
Polymerizatio
n, Computer
simulation
4 figures (x200) = 800
1305 words text = 1305
Total = 2105
1. Adamchuk V.K., St Petersburg State University,
Russia.
Program Committee
2. Sheka E.F., Peoples’ Friendship University of
Russia, 117198 Moscow, Russia
Chairman, Oral, A006
3. Shnitov V.V., Ioffe Institute, Russia
Invited lecture, A379
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