Comprehensive model of electron energy deposition*
Geng Han, Mumit Khan, Yanghua Fang, and Franco Cerrinaa)
Electrical and Computer Engineering and Center for NanoTechnology, University of Wisconsin–Madison,
Wisconsin 53706
共Received 10 June 2002; accepted 14 October 2002兲
We present our effort in developing a complete model of electron energy transfer from fast electrons
共0.1–100 keV兲 to the photoresist. Our model is based on the direct Monte Carlo method, instead of
using continuous slowing down approximation, model and a stopping power curve. We separate the
interaction events into four types: Elastic, ionization, excitation, and plasmon. Our results show that:
First, secondary electrons are major mechanism of energy distribution; and second, plasmons are
very efficient ‘‘friction’’ mechanism but do not create molecular changes; and finally, excitations
lead to molecular changes. © 2002 American Vacuum Society. 关DOI: 10.1116/1.1526633兴
I. INTRODUCTION
In electron-beam lithography, the photoresist is exposed
by the chemical changes associated with the electron energy
loss. In x-ray and extreme UV lithographies, the excitation of
core electrons creates fast photoelectrons that lose energy by
scattering processes, similar to the case of electron-beam
lithography.1 In the energy range of high-energy lithography
共⭓20 eV兲, the main source of energy loss is due to interactions with the bound electrons in the medium. The loss of
energy to the medium through plasmon excitation is akin to a
friction process, removing energy from the beam to the material in the form of heat. Plasmons are relatively insensitive
to the detail of the valence band orbitals because their frequency is determined mainly by the average electron density
具 n(E) 典 and average energy 具E典. To understand the electron
scattering process, the continuous slowing down approximation 共CSDA兲 has been widely used in Monte Carlo
simulations.1–3 However, the CSDA model gives us only an
expected value of energy deposition, and no information on
generations of secondary electrons can be achieved from it.
In this article, we present our effort in developing a complete model of electron energy transfer from fast electrons to
the photoresist. Section II gives the overview and implementation method of our model. Section III focuses on generations of secondary electrons in the photoresist. Section IV
investigates the treatment of plasmons in our model. Section
V discusses modeling at the material interfaces.
II. MODEL DESCRIPTION
To build a complete model of electron energy loss in
high-energy lithographies, two issues have to be addressed:
The first is the electron scattering process in the materials
共propagation兲; the second is how the energy is transferred
from electrons to the photoresist 共deposition兲. Previously, a
theoretical model of inelastic events has been proposed by
Ashley.4 The electron–photoresist interaction model has
been proposed by us based on the method of virtual quanta.5
*No proof corrections received from author prior to publication.
a兲
Author to whom correspondence should be addressed; electronic mail:
[email protected]
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To model the propagation and the electron scattering process,
the direct Monte Carlo 共DMC兲 method6 has been used in our
simulation. In a nutshell, given an electron of energy E, we
compute the cross sections ␴ for the global processes 共elastic, ionization, etc.兲. From ␴ we obtain a mean-free path ␭,
and the interaction location is computed using a standard
exponential mean free path e ⫺x/␭ . Similar approaches are
used in hot electron device modeling.7
Figure 1共a兲 shows a typical trajectory of three 1 keV primary electrons, along with the identification of each type of
events, and Fig. 1共b兲 shows one of the primary electrons
along with its secondaries. The propagation of an electron
can be well described by the CSDA, but this model does not
give any information on local chemical processes. In our
new DMC model, we separate the scattering events into four
types: elastic, ionization, excitation, and plasmon, completely replacing CSDA and stopping power. This model is
stochastic, in the sense that the electrons travel in free flight
between interactions. Cross section data for elastic, ionization, and excitation events are obtained from the EPDL97
database of the Lawrence Livermore National Laboratory.8
Plasmon cross section is computed using the dielectric function and experimental data.9
Figure 2 illustrates the four types of events considered in
this DMC simulation. The elastic events only change the
scattering angle of the electrons without depositing any energy. If an ionization event happens, this model will help us
to first find out the related element and subshell involved in
this ionization event. Then, some of the energy of the primary electron is lost through two ways: One part is passed to
the outgoing secondary electron, the other part is deposited
as bind energy of the relative subshell to the relative atom.
Thus, this atom will be left with a hole in the subshell that
involved in this ionization event. This hole will contribute to
later electron recombination processes, which both Auger
and fluorescence processes are included. These processes
could lead to chemical changes if the valence band is involved 共e.g., LVV Auger兲. The treatment of secondary electrons is computationally challenging. In our approach, each
electron trajectory is followed until the energy falls below a
threshold, typically 10 eV, where the electron is considered
‘‘stopped’’ and that energy is deposited in the material. As
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©2002 American Vacuum Society
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FIG. 3. 25 keV electron-beam simulation setup: 100 nm PMMA as photoresist, Au and Cr as two 50 nm thin film layers, with a Si substrate underneath.
the incident electron to a nearby residual atom. We notice
that the method predicts which atoms are excited and
through what processes. No new electron is generated in this
case, and the lost energy is considered deposited in that material. And finally, if a plasmon event is picked, both direction and energy of the incident electron are changed. The
energy lost to plasmon is considered to be eventually deposited in the material as heat.10 After the scattering event is
decided, the corresponding electron mean-free path is calculated, also based on the cross section data, and the electron
position is updated.
III. GENERATION OF SECONDARY ELECTRONS
FIG. 1. Electron scattering process in the photoresist. 共a兲 The trajectories of
three primary electrons. 共b兲 An example of secondary electrons.
the primary electron generates secondaries, the starting coordinates are put in a queue. When the primary stops, the first
secondary is traced. Any further secondaries are added to the
queue and the process continues until all energy has been
expended. In this way, our implementation considers the contributions from all the possible electrons of every generation.
When an excitation event is picked, the energy is lost from
In order to show the effect of secondary electrons, a simulation of 25 keV electrons passing through a three layers
structure was performed. Figure 3 shows the setup of the
simulation. Figure 4共a兲 shows the generations of electrons by
ionization events in the resist. We use 0 to label the primary
electrons, 1 to label the first secondaries, 2 to label the second secondaries, etc. Figure 4共b兲 shows the relative elements
and subshells involved in the ionization events. Figure 5
gives us the energy deposited in the resist versus the lateral
range of the generations of electrons. The simulation results
have good agreement with published data.11 Our data show
that the secondary electrons have the largest contribution to
the distribution of electron energy deposition in the photoresist.
FIG. 2. Four kinds of electron scattering events considered in the simulation: Elastic, ionization, excitation,
and plasmons.
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FIG. 4. Secondary electrons generated by ionization events in the photoresist, simulation setup as shown in Fig. 3. 共a兲 Generations of secondaries in
the resist. 共b兲 Elements and relative subshells involved in ionization events.
FIG. 6. Loss function and the differential cross section of plasmon events of
PMMA. 共a兲 Calculated loss function at ␪ ⫽0, E 0 ⫽1 keV. 共b兲 Calculated
differential cross section of plasmon events.
IV. PLASMONS
FIG. 5. Energy deposited in the photoresist—proximity effects. Simulation
setup as shown in Fig. 3.
J. Vac. Sci. Technol. B, Vol. 20, No. 6, NovÕDec 2002
The inelastic scattering, according to the energy of the
primary electron, can be with the outer-shell 共conduction or
valence兲 electrons and inner-shell 共core-level兲 electrons. For
inner-shell scattering, as it has been considered in the ionization or excitation events, the electron–atom scattering theory
can be used and is included in the atomic cross section
tables. For outer-shell scattering, the collective effect of plasmon excitation dominates. These plasmon excitations are
longitudinal waves, and thus cannot couple to electronic or
molecular transitions in three-dimensional 共3D兲 systems and
result in nonspecific energy deposition. However, due to the
large cross section of this type of event, they are the largest
contributor to the electron energy loss.
We developed our plasmon model based on the standard
dielectric response of the material to an incident electron,
which gives good results for describing the collective exci-
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Han et al.: Comprehensive model of electron energy deposition
FIG. 7. Energy distribution of plasmon events in the resist. Simulation setup
as shown in Fig. 3.
tations. In this case, the dielectric function can be written as
⑀ ⫽1⫹
␻ 2p
2
␻ 2q ⫹ ␻ avg
⫺ ␻ 2p ⫺ ␻ 共 ␻ ⫹i⌫ exp兲
,
␻ 2q ⫽ ␻ 2p ⫹3 ␯ F2 q 2 /5⫹ប 2 q 4 /4m 2 ,
␻ p⫽
冑
4 ␲ ne 2
,
m
共1兲
共2兲
共3兲
where ␻ p is the plasma frequency, ␻ avg is the expected value
of all the possible oscillators in the material, ⌫ is the standard deviation of the oscillators, ␻ q is the dispersions term
coming from the momentum transfer of the plasmon events,
n is electron density, and ␯ F is the Fermi velocity 共obtained
from the valence band electron density兲. Using this model,
the loss function and the differential cross section of plasmon
excitation for poly共methylmethacrylate兲 共PMMA兲 are calculated, as shown in Fig. 6. This result has good agreement
with the experimental data.9 Figure 7 shows the energy distribution of the plasmon events in the simulation with the
layout as shown in Fig. 3. As we expected, the plasmon
events are centered at about 23 eV, with a wide energy distribution.
The total cross section of plasmon excitation can be calculated by integrating over all the possible energies. Figure 8
shows the total cross section of plasmon events, compared to
the ionization and excitation events. As can be seen, when
the electron energy is greater than 100 eV, plasmon events
will be the major contributor to the electron energy loss process.
To further illustrate this point, we present a simulation of
100 keV electron beam on 200 nm PMMA. Figure 9 shows
the energy loss peak of electrons after transmission through
the PMMA sample. The plasmon peak can be clearly seen in
the energy loss process. Similar results can be found in Ritsko’s experimental work.9
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FIG. 8. Calculated total cross section of plasmon events comparing to ionization and excitation events for PMMA.
V. INTERFACE SIMULATION
There are two issues need to be discussed about the simulation at the material interfaces:
共1兲 calculations of the electron step length when the electron
mean free path crosses the material interfaces; and
共2兲 electron refractions at the interface. This may be important at low energies.
A. Electron step length at the interface
Let us consider a system with three layers of thickness
(z 1 ,z 2 ,z 3 ). An electron is located in material No. 1 will have
a mean-free path ␭ with z * ⫽ P ⫺1 (e ⫺z/␭ 1 ;r) and r苸 关 0,1兴
being a predicted event. If an event z * ⬍z 1 , then the scattering occurs within material No. 1. If z * ⬎z 1 , then the event
will occur in material Nos. 2 or 3: We compute the new
location using a smooth function e ⫺z 1 /␭ 1 •e ⫺(z⫺z 1 ) /␭ 2 suit-
FIG. 9. Energy loss peak of 100 keV electron beam on a 200 nm PMMA
sample.
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case, we follow the treatment by Williams,13 where the angle
␪ between the electron and the outward pointing surface normal can be calculated as
␪ ⫽sin⫺1
冋冉
m * 共 E⫹V 0 兲
mE
冊
1/2
册
sin ␪ ⬘ ,
共5兲
where V 0 is the inner potential, ␪ ⬘ is the angle between the
electron and the inward pointing surface normal of the material, E is the measured kinetic energy of the outgoing electron, m * is the effective mass of the electron, and m is the
free electron mass. Figure 10共b兲 shows the angle difference
between the incident and outgoing electrons at the
PMMA–Si interface.
In the calculation, we have treated m * ⫽m. As can be
seen, at smaller incident angles and high electron energies,
the electron refraction is very small. However, as incident
angle becomes larger and electron energy lower than 30 eV,
the refraction can be large.
VI. SUMMARY AND CONCLUSIONS
FIG. 10. Interface modeling. 共a兲 Electron step length calculation at material
interfaces. 共b兲 Electron refraction angle calculation at PMMA–Si interface
共angles in degrees兲.
ably normalized. Again, if z * is in material No. 2, the event
is accepted; if not the process is repeated. This can be summarized in the formula
ln共 r 兲 ⫽⫺
s1
s2
⫺ ⫺¯⫹
␭1 ␭2
冕
s
sn
⫺
du
,
␭n
共4兲
where, ␭ 1 ,␭ 2 ,...,␭ n , are the mean-free paths in the n different materials, r is a uniform random number, s 1 ,s 1 ,...,s n , is
the distance the electron travels in correspond materials, and
s is the total step length electron travels.
Figure 10共a兲 shows the distribution function P(z) for different lengths of free flight.
We have developed a comprehensive model of the physics
of electron energy loss, and implemented a simulation code
using the DMC method. This code has the ability to simulate
both electron and high-energy photon lithographies. It is also
capable of handling any resist composition, any thin film/
substrate interface, including multilayer system, and arbitrary geometrical layout. In contrast with other work in this
area, we keep track of generations of secondary electrons,
and consider plasmon events in the energy loss process. This
simulation will also be applied to LER studies. DMC is a
preprocessor for computation of the atomic and molecular
excitations. It can also be used as a standard electron propagation code.
The models as presented here are still relatively crude and
need refinement. In particular, one must use a more complete
dielectric function to describe the plasmon process. Future
work will focus on the definition of a more exhaustive dielectric function, and the calculation of cross sections of molecular bonds.
ACKNOWLEDGMENTS
The authors acknowledge extensive discussions with Dr.
L. Ocola 共Argonne National Lab兲 on the physics of the process and on relation of the previous code, LESIS, to our new
model. Discussions with Dr. D. Joy 共University of Tennessee兲 helped us with the definition of the role of plasmons.
This work is based in part by a grant from the Semiconductor
Research Corporation, No. 2002-MJ-985. The Center for
NanoTechnology, University of Wisconsin–Madison, is supported in part by DARPA/ONR Grant No. MDA 972-99-10013, MDA 972-99-1-0018.
B. Electron refraction
One of the important surface effects during low-energy
electron scattering in the material is the electron refraction as
it crosses the potential barrier at the surface. This particular
effect has received some theoretical consideration.12 In our
J. Vac. Sci. Technol. B, Vol. 20, No. 6, NovÕDec 2002
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