Energy transfer between electrons and photoresist: Its relation
to resolution
Geng Han and Franco Cerrinaa)
Electrical and Computer Engineering and Center for NanoTechnology, University of Wisconsin,
Madison, Wisconsin 53706
共Received 14 June 2000; accepted 20 July 2000兲
In the study of high-energy photon and electron lithography the photoresist has been treated as a
uniform and isotropic medium. As dimensions become smaller and exposure fluctuation more
noticeable, we cannot ignore the processes that happen at the molecular scale. Hence, it becomes
important to understand how the energy is coupled from an exciting radiation to the various
molecular components of a photoresist material. In this article, we present the first step in the
development of such a model. We base the analysis on the method of virtual quanta for the incoming
electron, and use the dielectric function response theory to describe the medium. The results
correctly describe the decreasing strength of the interaction on the electron energy, and yield an
estimate of ⬇2 – 3 nm for an average interaction distance. A simple Monte Carlo simulation is
implemented to verify the effect of the fluctuations due to the virtual quanta on a line edge.
© 2000 American Vacuum Society. 关S0734-211X共00兲03706-9兴
I. INTRODUCTION
In high-energy photon and electron lithography the energy is coupled from the original radiation beam to the material along paths that are not completely understood. In
x-ray and extreme ultraviolet 共EUV兲 lithographies the excitation of core electrons creates fast photoelectrons that lose
energy by scattering processes, similar to the case of
electron-beam lithography. Thus we can say that the energy
deposition in the exposure process is dominated by electron
interactions. In high-energy lithography the photoresist is assumed to be described by a dissolution rate function that
depends only on the local absorbed dose. Hence, the resist
has been treated as a uniform and isotropic medium, and no
study of the possible decay paths of the excitation energy
exist because they are not important in this model. As dimensions become smaller and exposure fluctuation more noticeable, we cannot ignore the processes that happen at the molecular scale and it becomes important to understand how the
energy is coupled from an exciting radiation to the various
chemical components of a photoresist material, in particular
the photoacid generator 共PAG兲. Not only the average dose
deposited but also the location of the interaction events become important in a molecular model. The detailed interaction of an electron with the resist matrix is complicated by
the many molecular species 共PAG, resist polymer, plasticizers, remaining solvent, etc.兲. As a start, we consider the case
of PMMA, a relatively simple polymeric resist system.
To model the location of events, the key question to answer is how the energy is transferred from electrons to photoresist, and leads to the bond-breaking transitions. Consider
a beam of 1 keV electrons passing through a PMMA box.
The location of the atoms 共and relative bonds兲 in PMMA
molecules is obtained by molecular models.1 In order to put
things into perspective, we have summarized the main asa兲
Electronic mail: cerrina@nanotech.wisc.edu
3297
J. Vac. Sci. Technol. B 18„6…, NovÕDec 2000
pects of the exposure process in Table I. From this we seek
to determine a more detailed description of the interaction
between electrons and PMMA. The traditional approach to
the exposure process yields only expected 共average兲
values2–4 for the energy lost by the electron. Previous work
has been done to analyze the electron–photoresist
interaction,5 but with incomplete results. For the reasons explained above, there is a vast amount of literature and experimental results on the interaction beam material from the
beam point of view, i.e., one describing in detail the energy
loss process and the energy deposited in the material, but
very little information exists on the effect of the interaction
on the material. The classic model by Greeneich yields an
average value for the number of scissions created by the
energy deposited in the material.6 A model that describes the
energy transfer is complicated by the many possible paths
that the transfer may take. One can define a quantum yield
for the reaction of interest, i.e., the dissociation of the PAG
or the scission of bonds. There are two steps that need to be
explained: first, the creation of an excited state in the molecules of the resist; second, the decay of this excited state
and, in particular, the fraction of energy that ends up in the
PAG activation. In the case of PMMA, where there is no
PAG, the process is simpler.
We seek to develop a model that eventually can be applied to predict on a molecular and bond scale the distribution and location of the interaction between electrons and
materials. Since the physics of the interaction is quite complex, we first review the underlying processes in a simple
model system and then apply the concepts in the development of a simplified stochastic model of PMMA-like exposure. We note that, since the following discussion involves
the optical properties of the resist material, as a first step we
are explicitly limiting ourselves to model systems that are
not necessarily accurate representations of the full interaction
between a resist and a beam of electrons. However, the basic
0734-211XÕ2000Õ18„6…Õ3297Õ6Õ$17.00
©2000 American Vacuum Society
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G. Han and F. Cerrina: Energy transfer between electrons and photoresist
TABLE I. Parameters of PMMA exposure by 1 keV electrons.
Electron energy
Sensitivity
Volume
Stopping power
1
50
403
1.45 共1 keV兲
2.62 共300 eV兲
0.019
⬃700
13
165
g s 共Ref. 2兲
E loss /elec.
Scissions
keV
␮C/cm2
nm3
eV/Å
eV/Å
1/eV
eV
per electron
per PMMA
ideas underlying the model are realistic and transferable: the
inclusion of realistic parameters will be necessary to yield
accurate models.
II. MODEL DESCRIPTION
How does an electron lose energy when passing through a
medium? Following standard treatments,4 we separate the
large angle elastic scattering from the energy loss process,
assumed to be uniform along the electron path 关continuous
slowing down approximation 共CSDA兲兴. The CSDA is described by the loss function,
冉 冊
1
dE
,
共 E 兲 ⬃⫺Im
dS
˜␧ 共 E 兲
共1兲
where ˜␧ is the complex dielectric function. However, Eq. 共1兲
does not include anything about the spatial extent of the
interaction, or its effect on the material.
We introduce the first step in developing such a model.
We base our approach to the analysis of the electrodynamic
interaction between electrons and material on the method of
virtual quanta, as described by Jackson,7 and here we apply
that analysis to the case of resist. Figure 1 shows the physical
basis of this method. When a beam of electrons is passing a
field point S located at a distance b, the electron charge will
induce a time-dependent electric field at S. This electric field
can be decomposed into two terms relative to the electron
path, E储 and E⬜ . Both of these components are time varying,
hence, they can be analyzed as a spectrum in the frequency
domain. The two components of the electric field lead to two
equivalent virtual photon pulses, P 储 and P⬜ . The frequency
components of equivalent pulses are added incoherently, and
FIG. 1. Introduction of method of virtual quanta.
J. Vac. Sci. Technol. B, Vol. 18, No. 6, NovÕDec 2000
3298
the effects of virtual quanta interaction with the struck system are computed. The photons are ‘‘virtual’’ because when
the electron is moving at a constant speed 共in a uniform
medium兲 no radiation is generated. In addition, these ‘‘photons’’ are not the usual propagating plane waves. But, rather,
they describe the electromagnetic interaction between the
passing electron and the material. However, if a resonating
system is located within the field, the energy can be transferred from the electron to the resonating system. The virtual
quanta field is extremely useful in the description of this
interaction. In summary, we can visualize the process as a
Fourier decomposition of the time-varying field of the passing electron. Individual Fourier components represent ‘‘virtual photons.’’ An absorbing element, modeled, for instance,
as a Lorentz oscillator and located within that field, may be
excited and thus remove a real photon from the field. This
energy comes, of course, from the electron kinetic energy
T ⬘ ⫽T⫺ប ␻ 0 .
Thus, we have a new tool for computing the influence of
a pulsed force on a bound oscillator.
In order to describe the interaction we need to determine
the number of transitions per cubic centimeter created by the
electrons that may lead to chain scissions and their spatial
distribution relative to its path. We simplify the analysis by
considering only the virtual photons, ignoring the other scattering processes. In the theory of optical processes, the number of transitions W( ␻ ) per unit time per unit volume induced by a photon field in a medium of dielectric constant
␧⫽␧ 1 ⫹i␧ 2 by a field of strength E( ␻ ) is8
W共 ␻ 兲⫽
再
␧ 2 material
␧ 0␻
␧ 2 E ␻2 共 ␻ 兲 ⇒
2␲ប
E 共 ␻ 兲 external field.
共2兲
The key parameters are the imaginary part of the dielectric function, which represents the material, and the external
electric field, which is caused by the passing electrons. We
will turn now to a discussion of these two terms.
III. ELECTRODYNAMIC INTERACTION: VIRTUAL
QUANTA
Let us begin with the electric field; the discussion will be
abbreviated, and follows closely the work of Jackson. We
can easily compute both components of the electric field as
seen by an oscillator when a electron passes by7 using classical expressions for the E(b,t) field.
In Fig. 2 we plot the two fields, E储 and E⬜ . As we can see
from the curves, if the distance from the electron to the field
point b is fixed, both components of the E field become
narrower when the electron kinetic energy increases, while
the height remains the same for constant b. Therefore, the
total area under the two pulses decreases as the electron energy increases. On the other hand, both components of E
field become larger at smaller b’s.
Since both of the E field components are time varying, the
time dependent signal can be analyzed in the frequency domain. Figure 3 shows the Fourier transform of the two E
field components at fixed distance b. As can be seen, the
bandwidth of both components becomes wider as the elec-
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G. Han and F. Cerrina: Energy transfer between electrons and photoresist
FIG. 2. Two components of the time varying electric fields: top, parallel
component; bottom, perpendicular component.
tron energy increases, but the strength of E( ␻ ) becomes
smaller. This is a direct result of Parseval’s theorem, that
states the area under the curve is conserved under the Fourier
transform. Through Planck’s equation, the frequency spectrum leads to the concept of virtual photons. In Fig. 4 we
show the energy spectrum of both E field components at
fixed distance b. We note that for a given frequency, the
number of virtual photon decreases as the electron kinetic
energy increases. Hence, a fast electron appears to be a localized line source of white radiation, extending from the
infrared to a few hundred electron volts. An increase in electron energy will make this source ‘‘bluer’’ but also ‘‘dimmer.’’ This explains satisfactorily why the interaction between electrons and matters decreases as the electron kinetic
energy increases.
The total energy of the virtual photon field can be calculated by integrating over all the possible b’s and all the fre-
3299
FIG. 4. Solid line: Virtual photon spectrum N(E) in a 0.1% bandwidth at T
⫽ 1, 5, and 10 keV. Dashed line: ␧ 2 for a Lorentzian oscillator.
quencies. This leads to some complex mathematics, omitted
here. We note that since the calculations are based on the
Coulomb potential, the formulas involved in the treatment
tend to diverge for b→0. While these divergences can be
removed using screening, here we keep the simpler formulation of limiting the interaction to a minimum distance b min
共see the later discussion of this point兲. The reason for this is
that the Coulomb potential yields an infinite frequency spectrum as b→0. However, very small values of b are unphysical, because of the following:
共1兲 The bound valence electrons that interact with the passing charge are delocalized in chemical bonds, as shown
in Table II.
共2兲 Because of the smallness of atomic bonds, a molecule is
essentially empty space, and thus it is unlikely that the
passing electron will be very close to any atom or bond
共see below兲.
共3兲 Electrons have a finite spread in energy and momentum.
At 1 keV,
ប
bmin⫽ ⬵0.38 Å.
mv
The largest of the parameters will dominate the interaction and limit the value of b. For now we stop with the
remark that, in general, b⯝0 events are unlikely. The effect
of screening will be the topic of a subsequent paper.
IV. MEDIUM RESPONSE
After solving for the field, let us turn to the problem of the
specific location of energy deposition. It is well known that
photon–matter interaction can be modeled very accurately
TABLE II. Bond parameters.
FIG. 3. Frequency spectrum of the two time varying electric fields components.
JVST B - Microelectronics and Nanometer Structures
Bond type
C–C
C–H
C–O
C–N
Bond length 共Å兲
1.5
1.1
1.43
1.48
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G. Han and F. Cerrina: Energy transfer between electrons and photoresist
with a distribution of Lorentz oscillators, with each oscillator
describing a particular bound electron resonating at a frequency ប ␻ 0 with damping ⌫. These oscillators are the semiclassical description of the optical absorption spectrum of the
medium. It is clear that in our case we will consider the
oscillator to couple to the virtual photon field; as described
before, the spectrum is a function of both electron energy
and distance. In our problem, the key point is that ‘‘oscillators’’ located at different distances from the electron will see
both different spectra and different field strength.
To start, let us consider a single oscillator located at a
distance b from the electron. To put things in perspective, we
can look at the oscillator corresponding to the bond-breaking
transition in PMMA as a guide in determining the resonant
energy ប ␻ 0 to use. Interestingly, the low-energy energy loss
spectrum of electrons in PMMA films was measured a long
time ago.9 For the sake of discussion, let us assume that the
bond breaking is at 5.5 eV 共217 nm兲, one of the transitions
quoted in Ref. 9. We note that, as expected, the experimental
energy loss spectrum of a fast electron is fully consistent
with the dielectric function formulation of Eq. 共1兲.
In general, we can write that the absorbed energy S for jth
oscillator from an electron passing at b j is7
d 2S共 b j , ␻ 兲
⫽ ␻ ␧ 2,j 共 ␻ 兲 兩 E共 b j , ␻ 兲 兩 2 共 J/m3 Hz兲 .
dVd ␻
2⌫ ␻ 2
d 2S共 b j , ␻ 兲 e 2N
⫽
兩 E共 b j , ␻ 兲 兩 2 共 J/m3 Hz兲 ,
dVd ␻
m 共 ␻ 20 ⫺ ␻ 2 兲 2 ⫹⌫ 2 ␻ 2
共4兲
where ⌫⫽ f ␻ ⌫ ⬘ , and f ␻ is the quantum mechanical ‘‘oscillator strength.’’ The number of transitions per unit volume
and frequency for N oscillators per unit volume is given by
the ratio between absorbed energy and photon energy,7 and
thus contains the area of the product of the Lorentzian function with the field strength E( ␻ ) as shown in Fig. 4. The
dotted line is the imaginary part of the dielectric function
that represents absorption from the material,10 and remains
unchanged as the incident electron increases while the virtual
photon spectrum decreases. The product of the two curves
gives us the spectrum of the absorbed energy density, and the
total area under the curves gives us the amount of total energy transferred. Since ␧ 2 is a very narrow function, the energy is coupled around the frequency ␻ 0 . In summary, since
the energy spectrum decreases and the absorption stays constant, the number of transitions will decrease as the electron
energy increases. Of course, this choice of Lorentzian function is purely for the sake of explanation and not necessarily
related to a real transition.
The total number of transitions per unit volume per electron is obtained by integrating over the spectrum (N⫽1).
Considering the fact that the imaginary part of dielectric
function is very narrow and centered at ␻ 0 , the total number
J. Vac. Sci. Technol. B, Vol. 18, No. 6, NovÕDec 2000
FIG. 5. Expected number of transitions per electron per volume vs the electron kinetic energy.
of transitions is independent of ⌫ and proportional to the
value of 兩 E( ␻ 0 ,b j ) 兩 2 . By making the calculations more explicit we obtain
dN共 b j 兲
⫽
dV
共3兲
Substituting the equation for ⑀ 2 for N uniform and identical oscillators per unit volume we obtain the classical formulation
3300
⫽
冕
⬁
0
1 d 2 S 共 ␻ ,b j 兲
d␻
ប ␻ dVd ␻
1 ␲e2
兩 E共 ␻ 0 ,b j 兲 兩 2 共 1/m3兲 ,
ប␻0 m
共5兲
a result containing modified Bessel functions, and the other
symbols have standard meaning.
Let us now consider a specific volume of material, say, in
the form of a box of PMMA. When the electron passes
through this box, it will see a distribution of oscillators at
various minimum approach distance b j , with a distribution
D(b j ). If we use the number of transitions per unit volume
per electron weighted by the distribution of the distance
D(b j ), we can obtain the expected number of transitions per
unit volume per electron to be
冓 冔
兺 nj⫽1 D 共 b j 兲 dN共 b j 兲 /dV
dN
⫽
.
dV
兺 nj⫽1 D 共 b j 兲
共6兲
Figure 5 shows the expected number of transitions per electron per unit volume versus the electron kinetic energy. This
tells us that, as the electron energy increases, the expected
number of transition decreases.
Let us now look at the spatial distribution of the transitions from the electron path. Here it becomes important to
have an estimate of the lower bound on b min . As a test, we
generated a distribution of distance b j by shooting 2400 electrons randomly into the left half side of a 40⫻40⫻40 nm3
PMMA box. If we coupled this distribution with the distribution of the number of transitions per electron versus distance, we obtain the distribution of the number of transitions
versus the impact parameter b, as shown in Fig. 6. This important results shows directly the expected distribution of
scission events around the electron trajectory. We notice the
average is 2.0 nm. In another words, if there is a 1 keV
electron passing through a PMMA box, the range of the in-
3301
G. Han and F. Cerrina: Energy transfer between electrons and photoresist
3301
FIG. 6. Top left: Distribution of b for a PMMA box;
lower left: transition density vs distance b; right panel:
expected transition distribution for the PMMA box.
teraction between that electron and the PMMA molecules is
of the order of 2.0 nm from the electron trajectory.
Even for such a simplified model, it may be interesting to
compare some predictions with the experiment. From our
calculation shown in Fig. 5 at 1 keV the expected number of
transitions per electron per volume is ␩ ⫽5⫻10⫺5 (1/m3).
The energy loss ⌬E of 550 eV absorbed by the medium in
the 40 nm thickness includes these transitions as well as
nonoptical transitions 共plasmons兲, i.e., an implicit efficiency
factor is present in the equation. We obtain
3
dN
⌬V
2 ⫻40
⫽ ␩ ␳ osc⫻
⬵5⫻10⫺5 ⫻9.7⫻
dE
⌬E
550
1
⫽0.028 共 scission/eV兲 .
This number is close to the experiment value of 0.019
scissions/eV.6 The agreement may be coincidental, given the
crude model; however, the final value is in a reasonable
range.
tion; at each layer, we obtain the energy loss from the stopping power.11 Then, from distribution of distance b for a
given energy, combined with use of the experimental value
of 0.019 scission per eV, scissions are created for each electron. Finally, the electron energy is updated and the electron
is moved to the next layer.
Finally, we build a histogram of the event locations. Since
the electrons are generated uniformly in the 0–20 nm interval, any blur is due only to the range of the interaction and
shows directly the granularity of the events. The histogram is
shown in Fig. 7, together with the derivative to put in evidence the edge variations. We obtain a broadening of the
edge of 2.3 nm. This represents the average range of interaction between an ideal electron beam and the substrate. It is
remarkable that such a crude model is close to the experimental observation that the minimum feature size in PMMA
is of the order of 8–10 nm.
V. MONTE CARLO SIMULATION
So far we have determined the external field caused by the
electrons and the distribution of the interactions; our original
goal was to determine the location of these events. Let us
now apply these results to a Monte Carlo simulation. The
main goal of this simulation is to observe the granularity of
the bond breaking. We isolate the virtual quanta contribution
by removing all the other scattering effects.
The PMMA molecules are described by the location of
the atoms, and we consider that the electrons enter the left
half of the box described above. We have used the freely
jointed chain method to compute a realistic distribution of
PMMA molecules.1 The molecular weight of the PMMA is
200 K, with a density ⬵1.0 g/cm3. The box is divided into
10 layers in the z direction, i.e., along the electron propagaJVST B - Microelectronics and Nanometer Structures
FIG. 7. Histogram of the coupling events distribution 共solid line兲 and its
derivate plot 共dashed line兲.
3302
G. Han and F. Cerrina: Energy transfer between electrons and photoresist
VI. SUMMARY AND CONCLUSIONS
We have introduced a simple model based on the use of
the method of virtual quanta and of the dielectric function to
account for the effects observed in lithography. To our
knowledge this is the first time that the virtual quanta method
has been analyzed in detail for application to lithography.
This method, when applied to a fast electron and a material
medium, provides a satisfactory and reasonable explanation
of the physics of the interaction between electrons and photoresists. The general trend agrees with the experiments:
共1兲 The electron energy is coupled to molecule bonds at their
characteristic frequencies.
共2兲 When the electron energy increases, this model correctly
predicts that the energy transferred to the chemical bonds
goes down.
共3兲 The typical range of interaction between an electron and
the medium is of the order of 2 nm for a 1 keV electron.
The model presented here is still crude and needs refinement. In particular, one must use a more complete dielectric
function to describe in a more quantitative way the interactions, and this must explicitly include the higher energy transitions. Also, the inclusion of screening is necessary to remove the unsatisfactory truncation at b min . Future work will
involve the definition of a more exhaustive dielectric function ⑀; other energy coupling mechanisms 共e.g., backbone
excitations to the photoacid generator兲 may be taken into
account phenomenologically by considering suitable yield
values from absorbed energy.
J. Vac. Sci. Technol. B, Vol. 18, No. 6, NovÕDec 2000
3302
ACKNOWLEDGMENTS
This work was based in part by a grant from the Semiconductor Research Corporation, Grant No. 98-LP-452. The
Center for NanoTechnology, University of Wisconsin–
Madison, is supported in part by DARPA/ONR Grant No.
N00014-97-1-0460.
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