Monte Carlo model for the deposition of electronic energy in solid argon
thin films by keV electrons
R. Vidala),b) and R. A. Baragiola
University of Virginia, Engineering Physics, Thornton Hall, Charlottesville, Virginia 22903
J. Ferrón
INTEC, Universidad Nacional del Litoral and Consejo Nacional de Investigaciones Cientı́ficas y Técnicas,
Güemes 3450, CC 91, 3000 Santa Fe, Argentina
~Received 21 June 1996; accepted for publication 7 August 1996!
The motion of keV electrons in a film of solid argon and the depth distribution of ionizations and
excitations are studied using a Monte Carlo simulation. This method does not only allow for
accurate inclusion of individual cross sections but also for easy inclusion of finite size effects. We
have analyzed the effect of the substrate on electron trajectories and found an important
enhancement of the number of electron–hole pairs and excitons produced near the interface by
electrons reflected from heavy substrates.
© 1996 American Institute of Physics.
@S0021-8979~96!01022-5#
I. INTRODUCTION
The spatial distribution of ionizations, excitations, and
the energy deposited in a material by keV electrons play an
important role in many fields, such as atmospheric,1
solid-state,2–12 and plasma physics.13 The knowledge of
these distributions is an important step in the understanding
of many physical phenomena, such as secondary electron
emission,6,7 cathodoluminescence,8 electron–hole-pair generation in semiconductors,2 low-voltage electron-beam
lithography,9,10 sputtering of frozen gases,11,12 and charging
of insulators.3–5 The experimental measurement of these spatial distributions cannot be undertaken easily, and in only a
few cases are experimental data available. Everhart and
Hoff2 obtained the distribution of deposited electronic energy
by measuring the electron–hole-pair generation in a metal–
oxide–semiconductor ~MOS! device ~aluminum–silicon
dioxide–silicon! irradiated with electrons in the 5–25 keV
energy range. From these measurements they derived an analytical distribution function which they propose should be
valid for targets of atomic number 10,Z,15. The spatial
distribution of the energy deposition has been derived in
some cases in the gas phase14,15 from the distribution of characteristic luminescence, assuming that the luminescence
should be proportional to the deposited energy. From the
theoretical point of view several approaches exist to evaluate
these spatial distributions. The first is the analytic transport
theory which has been used extensively in the field of radiation physics ~see Ref. 16 and references therein!, since
Spencer17 presented a method of solving the transport equation, which converts it into a linked system of differential
equations for the spatial moments of the distribution. In
transport theory it is usual to work in the energy space, and
evaluate what is called the electron degradation spectrum.
Once this spectrum is known, quantities of practical interest,
such as ionization and excitation yields, can be calculated. A
a!
Permanent address: INTEC, Universidad Nacional del Litoral and
CONICET, 3000 Santa Fe, Argentina.
b!
Electronic mail: [email protected]
J. Appl. Phys. 80 (10), 15 November 1996
second approach has been presented recently by Dapor18 that
uses the so-called multiple reflection method, and it has been
applied to evaluate transmitted and backscattered fractions of
electrons, and energy deposited by keV electrons incident on
thin films. Finally, there is the Monte Carlo ~MC! method, in
which histories of individual collisions are simulated for
many particles in a computer, and conclusions are drawn
from the statistics of those histories. The advantage of this
approach is the relative simplicity of the algorithms, its accuracy, and its great usefulness when one wants to study
targets with complicated geometry. The Monte Carlo method
has been applied for a long time to study the electron interaction with matter;19 in a recent example Valkealahti, Schou,
and Nieminen20 used it to calculate the energy deposition of
keV electrons in light elements.
In the present work we use the MC method to calculate
the energy deposition of 0.3–5 keV electrons incident on
solid Ar films. We calculate depth distributions of deposited
energy, ionizations, and excitations to discrete states, and
also the mean energy spent per ionization, the so-called W
value. A feature of our work is the incorporation of finite size
effects to take into account electrons reflected from the substrate backing the Ar films. This situation, not considered in
the work by Valkealahti and co-workers,20 is the usual one in
experiments where the films are grown by vapor deposition
onto cooled substrates. The organization of the article is the
following. In Sec. II we describe our MC code. In Sec. III A
we compare some of its outputs to available experimental
results on different targets and we present depth distributions
of ionizations and excitations, and also W values. In Sec.
III B we discuss the effect of the substrate and the angle of
incidence of the primary electrons.
II. THE MONTE CARLO SIMULATION METHOD
The general characteristics of our MC code have been
already reported.21,22 We consider only two types of interactions between electrons and target atoms: elastic scattering
0021-8979/96/80(10)/5653/6/$10.00
© 1996 American Institute of Physics
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TABLE I. Binding energies for the contributing shells of the different targets ~in eV!. E exc is the lowest
excitation energy in Ar, E val is the effective binding energy of the valence shell for metals. Ar M -shell binding
energies and E exc are from Ref. 27. The other binding energies are from Ref. 28.
Levels
Ar
N2
1
2
3
4
5
L 1 324.0
L 2,3 247.0
M 1 27.2
M 2,3 13.9
E exc 12.0
K
410.0
s 2s b
37.0
s 2s * ,p 2p b 18.0
s 2p b
15.5
Al
K 1559.6
L 1 117.7
L 2,3 73.1
E val 6.8
and inelastic scattering with core and valence electrons. We
used the partial wave expansion method to calculate the differential and total elastic scattering cross sections23 and calculate the phase shifts by solving numerically the radial
Schrödinger equation for an atomic potential of the Thomas–
Fermi–Dirac type.24
Ionizations of core and valence electrons are calculated
separately for each atomic level using the Gryzinski excitation function.25 We used up to five levels to describe the
electronic structure of the target atoms. To take into account
the valence electrons in different metallic substrates, we used
an effective binding energy for the valence level that fits the
total inelastic cross sections evaluated semiempirically by
Tanuma, Powell, and Penn.26 For Ar we have used the
M -shell binding energies appropriate to the solid state.27 In
Table I we present a summary of the different binding energies used.
For Ar we have also included the possibility of excitation processes ~bound–bound transitions! as an alternative
inelastic interaction. For the sake of simplicity in the MC
calculation we considered only the lowest excited states.
These arise when an M -shell electron is excited to a unique
upper-bound level. Excitations from deeper levels are much
smaller than for the M level29 and were thus neglected. For
the calculations, we used the total excitation cross sections
presented by Eggarter.30
The electrons are assumed to follow linear trajectories
between two interactions. If we suppose that the distance
traveled Dx between two collisions follows a Poisson distribution then the step length can be calculated using the relation
Dx52l i ln R,
Ag
Au
M 1,2,3 616.0
M 4,5
369.0
N1
95.0
N 2,3
58.0
E val
15.5
M 4,5 2244
N1–5
471.0
N 6,7O 1
92.0
O 2,3
66.0
E val
16.0
ferential cross sections and assuming an isotropic distribution for the azimuthal angle. If the electron impact produces
an excitation or ionization, the energy loss DE is evaluated
using the Gryzinski excitation function. If DE is greater than
the binding energy E b of the shell involved, it ionizes the
atom. The outgoing secondary electron has kinetic energy
DE2E b and is assumed to be generated isotropically. We
also consider that after an Ar L-shell ionization the atom
decays by emitting, isotropically, an Auger electron of 206
eV average energy.31 The double ionized ion left behind is
counted as two ionizations since it is expected to decay rapidly by charge transfer by the process
Ar211Ar→Ar11Ar11kinetic energy.
The trajectory and collision history of each primary electron and all secondary electrons are traced until they leave
the solid or their kinetic energy goes below a cutoff energy
~14 eV! close to the band gap.
In each simulation more than 60 000 incident electrons
are traced. After each simulation we obtained the depth distribution of ionizations for each shell, the distribution of ex-
~1!
where l i is the electron mean free path in medium i and R is
a uniform random number between 0 and 1. If the electron
crosses the interface from medium i to medium j, the change
in mean free path between different materials must be taken
into account. A way to do this was presented in Ref. 21
where the total traveled distance is corrected as follows:
Dx 8 5Dx i 1 ~ Dx2Dx i ! l j /l i .
~2!
The first term on the right-hand side is the distance traveled
in medium i, and the second term represents the distance
traveled in medium j.
After an elastic interaction the electron only changes its
direction of motion without changing its kinetic energy. The
scattering angle is calculated using the elastic scattering dif5654
J. Appl. Phys., Vol. 80, No. 10, 15 November 1996
FIG. 1. Normalized energy deposition profiles for 1, 2, and 3 keV electrons
incident on nitrogen. The depth is given in units of electron range ~Ref. 15!.
Continuous lines are from MC calculations on solid N2 . Simbols are experimental results from Barrett and Hays ~Ref. 15! for gaseous N2 .
Vidal, Baragiola, and Ferrón
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FIG. 3. Backscattering coefficients as a function of energy for electrons
normally incident on gold, argon, and nitrogen. All electrons emitted from
the sample with energies greater than 50 eV are counted. Open squares are
our MC results, and the dashed lines are drawn just to guide the eye. Solid
circles and solid squares are experimental data for gold ~Ref. 32! and for
nitrogen ~Ref. 33!, respectively. Open triangles are MC results from Ref. 20.
FIG. 2. Energy deposition profiles for 1 and 3 keV electrons incident on a
thick solid argon film. Simbols are MC from this work and the broken lines
are MC results from Valkealahti and co-workers ~Ref. 20!.
differences seen at 3 keV, in the tail of the energy deposition
profile, are probably due to the fact that the Valkealahti and
co-workers curve shown in Fig. 2 is not their true MC profile, but their Gaussian fit to it.
Figure 3 shows backscattering coefficients for normally
incident electrons. This coefficient is defined in the usual
way, as the number of electrons with kinetic energy greater
citations, and the distribution of electronically deposited energy. Unless noted the results presented below are for normal
incidence.
III. RESULTS AND DISCUSSION
A. Depth distributions and W values
We compare some outputs from our MC simulations to
available experimental results as well as to other MC simulations results for thick films. Figure 1 shows our energy
deposition profiles for normally incident electrons on solid
nitrogen and experimental profiles obtained for gaseous molecular nitrogen.15 To make the comparison independent of
sample density and primary electron energy we normalize
our results for the energy deposition as follows:
L ~ z ! 5 ~ R/E d ! D ~ x ! ,
~3!
where L(z) is a dimensionless distribution, z5x/R, D(x) is
the MC energy deposition profile, R is the mean electron
range ~we used the values proposed in Ref. 15!, and E d is the
total deposited energy. As we can see in the figure the agreement is quite satisfactory.
In Fig. 2 we compare our energy deposition profiles for
normally incident electrons on solid Ar with MC results from
Valkealahti and co-workers.20 The agreement between both
MC simulations is quite good at 1 keV incident energy. The
J. Appl. Phys., Vol. 80, No. 10, 15 November 1996
FIG. 4. Depth distributions of ionizations plus excitations to discrete states,
and deposited energy for 0.3 and 3 keV electrons incident onto a thick solid
argon film.
Vidal, Baragiola, and Ferrón
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form one into another by using a mean energy for the creation of either an electron–hole pair or a direct excitation.
This mean energy is 19.7 eV for 3 keV, and the ratio between both distributions is almost constant with depth. Also
the mean energy only decreases slightly as the primary energy decreases, from 19.8 eV for 5 keV to 19.5 eV for 0.3
keV electrons. The W value is nearly independent of projectile energy in the calculated range: 26.2 eV for 0.3 keV to
26.3 eV for 5 keV electrons. This value is close to the experimental values of W526.460.4 eV for 3–5 keV electrons on gaseous Ar,34 and W52761 eV for 30 keV electrons on solid Ar.35
The depth distribution of ionizations and excitations produced by 0.3 and 3 keV electrons incident on a thick solid Ar
film are plotted in Fig. 5. One can see the strong contribution
of secondary electrons to the creation of electron–hole pairs.
The amount of 3p ionizations produced by secondary electrons increases with primary energy from 34% for 0.3 eV
electrons to 61% for 5 keV electrons. Similarly the number
of excitations produced by secondary electrons increased
from 46% to 70% as the electron energy goes from 0.3 to 5
keV. Due to the energy dependence of the ionization and
excitation cross sections, 3p ionizations and excitations to
bound states are clearly favored over processes in other
shells, in the case of secondary electron excitations.
We also calculated G values, the number of a particular
species produced for each 100 eV of electron energy absorbed, a magnitude of common use in radiation research.
They are presented in Table II for the different primary species ~ions and excitons!, as well as for subexcitation electrons ~i.e., electrons that have kinetic energies lower than the
first electronic-excitation threshold, and are therefore incapable of producing excitons or electron–hole pairs!. The excitons are the products of initial excitations and do not include those formed at a later stage from electron–hole
recombination. Table II includes the deposited energy, which
differs from the incident energy due to the energy carried
away by the secondary and backscattered electrons. We have
also included in Table II some results of previous MC simulations for Ar gas.31
FIG. 5. The depth distribution of 3p ionizations and excitations for 0.3 and
3 keV electrons normally incident on a thick solid Ar film. Also shown are
3 p ionizations and excitations produced by secondary electrons.
than 50 eV emitted from the target per incident electron. For
nitrogen and gold, the MC results are compared to experimental backscattering coefficients.32,33 The results for argon
are compared to other MC simulations.20 There is good general agreement in both cases.
Figure 4 depicts the depth distribution of ionizations plus
excitations, as well as the distribution of deposited energy
for 0.3 and 3 keV electrons normally incident on a thick
solid Ar film. It can be seen that the two distributions have
almost the same shape. For that reason it is possible to trans-
TABLE II. Number of initial species produced per 100 eV deposited energy ~G values! for electrons incident
on solid Ar. The values in parentheses are from Ref. 30 for Ar gas.
Electron incident energy ~eV!
5656
300
500
1000
2000
3000
5000
2s ionizations
2p ionizations
3s ionizations
3 p ionizations
Excitations
•••
0.0004
0.289
3.52
1.31
0.0004
0.0029
0.308
3.50
1.29
~1.85!
0.0024
0.0161
0.321
3.46
1.28
~1.75!
0.0048
0.0277
0.323
3.42
1.28
~1.75!
0.0064
0.0323
0.324
3.40
1.27
0.0078
0.0380
0.324
3.38
1.27
Subexcitations
4.03
3.96
~3.93!
3.89
~3.89!
3.85
~3.83!
3.83
3.81
Deposited
energy ~eV!
W ~eV!
245.2
412.2
834.4
1682
2526
4188
26.24
26.22
26.20
26.23
26.27
26.34
J. Appl. Phys., Vol. 80, No. 10, 15 November 1996
Vidal, Baragiola, and Ferrón
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FIG. 8. Increase in the total number of ionizations plus excitations over the
number expected for an Ar-like substrate, for a 400 Å solid Ar film supported on Au, as a function of the incidence angle. The symbols are our MC
results, and the solid line is drawn just to guide the eye.
FIG. 6. Depth distribution of excitations plus ionizations for 1 and 3 keV
electrons normally incident on solid Ar films, supported on a gold substrate,
and for different film thicknesses. Also shown are 3p ionizations plus excitations produced by secondary electrons ~lower curves!.
B. Substrate effects for thin solid Ar films
So far we have discussed depth distributions for ionizations and excitations for electrons incident on thick solid Ar
films, where all the energy of the projectile is either dissi-
FIG. 7. Depth distribution of ionizations plus excitations for 3 keV electrons
incident on a 400 Å solid Ar film supported on Au for different incidence
angles. Results for a thick ~3000 Å! solid Ar film have also been included
for comparison.
J. Appl. Phys., Vol. 80, No. 10, 15 November 1996
pated in the film or is carried away by escaping secondary
and backscattered electrons. One of the key advantage of our
MC code is that we can also study thin films supported on
different substrates. In Fig. 6 we show depth distributions of
ionizations plus excitations for 1 and 3 keV electrons normally incident on Ar films of different thicknesses deposited
on a gold substrate. One can see the strong influence of the
substrate on the generation of electron–hole pairs and excitons near the interface. This is very remarkable when the film
and substrate have very different atomic numbers ~this implies large differences in elastic cross sections for the two
materials!. The enhancement in the number of ionizations
and excitations for heavy substrates is due to electrons reflected from the substrate. The effect tends to decrease when
the thickness of the film approaches the primary electron
penetration depth ~about 340 Å for 1 keV electrons and 1600
Å for 3 keV electrons!. Electron–hole pairs and excitons
generated by secondary electrons show a similar dependence
with depth. For light substrates the contribution of backscattered electrons becomes smaller, and when the substrate is
lighter than Ar ~e.g., Al! the number of ionizations and excitations at the interface is smaller than for a thick argon film
at the same depth from the surface.
We have also studied the change in the depth distributions of ionizations plus excitations with primary electron
incidence angle. That it is shown in Fig. 7 for 3 keV electrons incident on a 400 Å solid Ar film deposited on a gold
substrate. The depth distributions change noticeably with
electron impinging angle. While near the surface the number
of ionizations plus excitations increases monotonously with
incidence angle, the inverse dependence is observed at the
Ar/Au interface. The increase in the energy deposition at the
surface is simply due to the increase of the electron path
length. This increase is nearly proportional to the inverse of
the cosine, and is independent of the film thickness. On the
other hand, the decrease of the number of ionizations plus
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excitations at the interface is produced by roughly the same
effect, i.e., the increase of the path length of the electrons
near the surface reduces the penetration depth, thus the backscattering enhancement due to the substrate turns to be less
important. To quantify the angular dependence of the substrate effect, we need to normalize the total ionization plus
excitation yield with respect to the thick Ar film case, i.e.,
overlayer and substrate are made of Ar. These results are
shown in Fig. 8. There we can see that the substrate effect
diminishes with impinging angle. This diminution is related
to the decrease of the electron penetration depth with angle.
IV. CONCLUSIONS
We have performed MC simulations to evaluate the distributions of ionizations and excitations produced by 0.3–5
keV electron bombardment of argon films supported on different substrates. The simulations predicts an important contribution of secondary electrons to the production of ionizations and excitations. They also show that the depth
distributions of ionizations and excitations can be converted
to a deposited energy distribution by using a mean energy,
which depends only slightly on incident electron energy in
the working range. In addition, the contribution of electrons
backscattered from the substrate to the number of ionizations
and excitations produced near the interface is important for
heavy substrates. An observable consequence of this would
be a noticeable increase in the number of particles emitted
after the decay of electron–hole pairs or excitons ~e.g., electronic sputtering yields, luminescence!, or in the secondary
electron emission coefficient for thin films on heavy substrates.
ACKNOWLEDGMENTS
The authors thank A. Gras-Martı́, M. Inokuti, and J.
Schou for valuable discussions. R.V. would like to acknowledge the support of the CONICET ~Consejo Nacional de
Investigaciones Cientı́ficas y Técnicas de la República Argentina! and the Fundación Antorchas ~A-13019/1!. This
work was supported by NSF ~DMR9510383!, CONICET
~PID 3748/92!, and Fundación Antorchas ~A-13221-019!.
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J. Appl. Phys., Vol. 80, No. 10, 15 November 1996
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