Proximity Effect
in E-beam Lithography
By: Araldo van de Kraats
Modified by: Raghunath Murali
Abstract
Electron beam
lithography is able to provide high resolution patterning. However, the effect of
electron scattering in resist and substrate leads to an undesired influence in
the regions adjacent to those exposed by the electron beam. This effect is
called the proximity effect. This paper presents a study on the cause of the
proximity effect and methods to correct for it.
The
electron beam has a wavelength so small that diffraction no longer defines the
lithographic resolution. In electron beam lithography, the resolution is
limited by electron optic aberrations and, more importantly, scattering of
electrons in resist and substrate. These electron scattering effects, often
referred to as the proximity effect, cause exposure of areas surrounding the
area where the electron beam was incident (Fig. 1a). Any pattern written can
suffer significant variation from the intended size because of proximity
effect.
Electron-solid interactions
When
a positive resist is exposed by an electron beam, some molecular chains in the
resist molecules will break, thereby reducing the average molecular weight.
This is accompanied by an increase in solubility and increases the etch rate.
[1-4]. For electron beam lithography it is desirable
to know the three dimensional distribution of energy deposition in the resist
after exposure by the e-beam [5]. Typical electron beam lithography machines
nowadays use electron beams with 10-100 keV energy per electron. Therefore, the
free path of an electron
is 10 um or more, which is at least an order of magnitude more
than the resist thickness. Thus, the electrons can easily penetrate the resist
layer and reach the substrate. As the electrons penetrate the resist and the
substrate, they experience many scattering events. There are two types of
scattering which may take place (Fig. 1b).
incident electron beam
Figure 1a: Proximity effect: exposure at pixel A affects pixel B.
In
forward scattering, an electron can collide with an electron from one of the atoms
in the substrate/resist. The incident electron will change its direction and
transfer part of its energy to the atom. Because of the extra energy, the atom
will become exited (one of its electrons goes to an exited level) or ionized
(one electron leaves the atom, creating a secondary atom in the material). When
the target atom is part of a resist molecule, the molecular chain may break due
to this excitation or ionization. The scattering angle due to inelastic
scattering is, as a rule, small.
In
backscattering, an electron collides with the much heavier nucleus, which
results in an elastic scattering event. The electron retains (most of) its
energy, but changes its direction. The scattering angle may be large in this
case. After large angle scattering events in the substrate, electrons may
return back through the resist at a significant distance from the incident
beam, thereby cause additional resist exposure. This backscattering is what
causes the proximity effect.
Figure 1b. Electron
scattering model (a) incident electron collides with an electron from the
target atom: the scattering angle is small, (b)
incident electron collides with a nucleus: the scattering angle may be large.
As
the primary electrons slow down, much of their energy is dissipated in the form
of secondary electrons with energies in the range 2 to 50 eV. The major part of
the resist exposure is due to these electrons. Since they have low energies,
their range is only a few nanometers. Therefore, they contribute little to the
proximity effect. However, this phenomenon, together with the forward
scattering, effectively causes a widening of the exposure region. This is one
of the main limiting factors in resolution of e-beam lithography machines. The
distance a typical electron travels before losing all its energy depends on
both the energy of the primary electrons and the type of material it is
traveling in. The fraction of electrons that are backscattered, e, is roughly independent of beam
energy. It does, however, have a strong relation to the substrate material.
Substrates with a low atomic number give less backscattering than substrates
with high atomic number.
For
head-on collisions with the nucleus the transfer of energy is given by [6]:
(1)
where
E0 is the incident beam energy and A the atom number of the target.
If E exceeds some displacement energy Ed, which depends on the
atomic weight, bond strength and crystal lattice, the nuclei can be displaced
and the crystal structure may damage. Typical values of Ed,
range from 17 eV for aluminum to 80 eV for diamond. This means, basically no
damage is to
Figure 2. Substrate heating for two
different types of substrates. The upper curve represents a dense
pattern, while the lower is a sparse pattern (spot with an area of 0.25 um2)
(a) bulk quartz photomask, 50 kV beam voltage and 30
A/cm2 current density (b) bulk silicon, 100 kV beam voltage and 50
A/cm2 current density.
be expected
for E0 below 100 keV, unless the target contains hydrogen atoms.
Critical values for some other materials can be found in [6]. Except for damage
due to nuclei displacement, damage may be caused by substrate heating due to high exposure doses. Heating can also modify
resist sensitivity, which can cause unwanted line width variation. A general
analytic heating theory is given in [7], which can quantitatively describe
temperature rise due to beam induced substrate heating. Results for two common
substrates used in electron beam lithography are shown in Fig. 2.
Energy intensity profile
Models of the energy profile
A
proximity effect correction algorithm requires an accurate knowledge of the
energy density profile deposited in the electron resist layer due to a point or
pixel exposure (often called point spread function). In general, this profile
is a function of the system setup. An important property of these profiles is
that the shape is independent of dose as well as position, assuming a planar and
homogeneous substrate[8]. This profile is often
approximated by the sum of two Gaussian distributions [9]:
(2)
representing the forward and the backscattered electrons. C1, C2,
B1 and B2 are constants and r is the distance from the
point of electron incidence. More popular is to write this expression as
follows [10]
(3)
where
η is the ratio of the backscattered energy to the forward-scattered
energy, α is the forward scattering range parameter and β is the
backscattering range parameter. Equation (3) is normalized so that
(4)
Several
researchers have indicated that the double Gaussian function often is
insufficient for expressing the energy density profile. More complex functions
are needed for certain types of substrate, multi-layer substrates and for very
small feature sizes. Improved functions have been proposed, such as triple and
multi Gaussian functions, to express complex phenomena which the double
Gaussian function fails to express. Very accurate results have been obtained by
adding an extra exponential term to the double Gaussian distribution [11].
(5)
The
models discussed above are two dimensional versions of an essential three dimensional
phenomenon. In general, the energy profile depends upon depth as well as
radius. By averaging out the depth dependence, a two dimensional profile can be
obtained out of a three dimensional profile. There are several reasons for
using this simplification. These include a greatly reduced computation time for
the exposure estimation and correction, the fact that it will be difficult to
determine the three dimensional profile accurately, and that the major
difference between profiles at different depths occur in the 0.00 um to 0.01 um
range, which is often well below the minimum feature size[8].
For
certain applications it may be necessary to use a three dimensional profile. In
this case, a
Electron beam lithography process parameters
In
this section, important process parameters and their effect on the proximity
will be discussed. These are electron beam energy, resist type,
resist thickness, exposure time (dose) and development time. Due to the many
small angle scattering events, forward scattering increases the effective beam
diameter. Empirically, it is given by the following formula [5]:
(10)
where
df is the effective beam diameter in
nanometers, Rt is the resist thickness in
nanometers and Vb is the beam voltage in
kilovolts [5]. A slightly different relation for the resist thickness
dependence is given in [13]:
(11)
In
Table I, typical values for the forward and backward scattering are given for a
0.5 um resist on a silicon substrate. Values shown in brackets are
extrapolations. Values for α are calculated
values, while β and η are experimental data.
Table I. Proximity parameters as a function of the beam
energy [13]
|
Beam energy (keV) |
α (um) |
β (um) |
η |
|
5 |
1.33 |
[0.18] |
[0.74] |
|
10 |
0.39 |
[0.60] |
[0.74] |
|
20 |
0.12 |
2.0 |
0.74 |
|
50 |
0.024 |
9.5 |
0.74 |
|
100 |
0.007 |
31.2 |
0.74 |
A convenient method
for quantitatively characterizing the proximity effect is the modulation
transfer function (MTF). This function can be obtained by Fourier transforming
and normalizing the energy density profile:
(12)
where p
is the spatial period. Ideally M
should be 1 for all p. However, in
the presence of electron scattering, M
is less than 1 and is dependent on p.
As put forward in [13], M is allowed
to be smaller than 1, but it is crucial that it is independent of p for the proximity effect to be
corrected. In figure 4, the MTF curve for several electron beam energies are
given (again for a 0.5 um resist layer). For comparison, a spatial passband is
drawn, in which M varies by no more than ±15%.
Figure 4, MTF curve for beam energies of 20, 50 and 100 keV for a silicon substrate with 0.5 um resist thickness.
Analogue to the
MTF for optical lenses, the Y-axis can be thought of as the contrast while the X-axis
presents the spatial period of line pairs. As can be seen from this figure, the
flat part in this curve is much longer for higher beam energies. For example in
100 keV, 0.1 um features have the same contrast as 10 um features (M is independent of p in this range). Contradictory, in 20 keV, the backscattering
component of the 10 um feature will swallow up the 0.1 um feature, due to the
much lower contrast value of the latter. Considering only this, it makes sense
to make the electron beam energy as high as possible. The drawback using high
beam energies is that the sensitivity is reduced and consequently the dose must
be increased. [7] The dose is proportional to the beam voltage, due to the
increased transparency of resist layers at high voltage. [7, 14] Care must be
taken that no damage is caused by nuclei displacement of substrate heating.
From Table I, it
can be concluded that the backscattering effect becomes negligible when
reducing the beam energy below 10 keV. Unfortunately, by decreasing the energy
the forward scattering will increase beyond 1 um. This, however, can be
countered by using smaller resist thickness (see equation 10). In Fig. 5, a MTF
curve is given for a 5 keV beam with a 0.035 um resist thickness and for a 20
keV beam with 0.5 um resist. The passband at 5 keV is [0.5 um, ∞] while
at 20 keV it is [1 um, 5 um]. From this we can conclude that low energy beams
can be useful when using thin resist films. Unfortunately, semiconductor
processes often
Figure 5.
MTF curve for a 5 keV beam with 0.035 um resist thickness and a 20 keV with 20
um resist thickness.
Figure 6.
Distortions in 800 nm resist film with 150 nm line and space width.[14]
require a resist thickness much higher than 0.035
um, so this approach may not be practical in many cases.
Another important
factor in choosing resist thickness is the aperture value, which is defined as
the ratio between the resist thickness and the minimum feature size. Aperture
values around 3 or 4 should be easily obtainable. In [14], aperture values of 5
and higher are reported. When the aperture becomes too high, part of the
pattern may collapse or distort. For example, in figure 6, it can be seen the
outer lines are distorted due to a high aspect ratio.
Multilayer resists
can be used to reduce the proximity effect. Typically the upper layer is used
for patterning, while the lower layer functions to reduce backscattering. This
it does, since the backscatter coefficient n
is lower and the backscatter range β is larger for a polymer than for
silicon. In Fig. 7, an example is given of a MTF curve of a multilayer resist.
The lower layer has a relatively large thickness of 2 um, while the upper layer
is 0.2 um thick. As can be seen, the resolution increases considerably. The
disadvantage is the increased process complexity.
Figure 7.
MTF curve comparing single and multilayer resist.
The correct
exposure time (dose), as well as the development time, is strongly related to
the type of resist and developer used. Negative resist will remain at the exposed
area, while the unexposed parts of the resist will be etched away by the
developer. In positive resist the exposed part will be etched away (Fig. 8). In
negative resist, the edge profile of a line exposed is bell-shaped, caused by
the electron scattering. In positive resist, on the other hand, more control
over the profile is possible. By using a higher dose and/or a short development
time, the edge profile will be dominated by the energy deposition profile, and
will have a shape as in figure 9(a). A low dose and long development time will
yield a shape as in Fig. 9(c). With medium dose and develop, steep vertical
edges can be obtained (Fig. 9(b)) [5, 15-16].
Figure 8.
Negative and positive resist profiles.
Figure 9. Edge profile in positive resist (a)high dose and short development time, (b) medium dose and development time, and (c) low dose and long development time
Proximity effect correction
Exposure estimation
Exposure
estimation is important to simulate the effects of a proximity effect
correction and it is used in many correction schemes. Since the energy
deposition profile gives the response of a single point exposure, the exposure
of a circuit pattern can be mathematically described by a convolution:
with 
(13, 14)
where
E(x,y) is the energy deposited in the resist, f(r)
the point exposure profile and d(x,y) the input dose
as a function of position. The developed image E’(x,y) can be obtained from E(x,y)
by
(15)
where
τ is an experimentally determined development threshold. E’(x,y) = 0 and E’(x,y) = 1 are denoting undeveloped and developed resist
respectively. An overview of this concept is given in Fig. 10 [17].
Figure 10. Exposure
simulation of a circuit pattern.
Small
pixel sizes are necessary to obtain an accurate image. However, convolving a
large circuit using small pixels with the point exposure profile will give
unacceptably long computation times. In [8], an efficient method is developed,
based upon tables with cumulative distribution function of primitive shapes
(e.g. rectangles). Furthermore, the memory requirements are reduced by
separating the total exposure in two components, one due to the forward scattering
component (local exposure), which is sharp and has a short range, and one due
to the backscattering component (global exposure), which is rather flat and has
a long range. The local exposure can be evaluated in a small window around the
critical point, while the global exposure can be evaluated in a coarser grid
without much accuracy degradation.
Exposure correction methods
There
are essentially three methods of proximity effect correction. These are
background correction exposure, shape modification, and dose modification. In
the latter a different dose can be applied to each pixel and is therefore
limited to direct writing electron beam systems.
Dose modification
Many proximity
correction schemes variants have been proposed, which use some form of dose
modification [19-21]. The problem is to determine the required dose for each
pixel with a reasonable accuracy while being computationally practicable.
Numerous
variants of the “self-consistent dose correction” have been developed. In its
simplest form, it is basically the reverse of the exposure estimation discussed
before. Let Qj be the dose applied to
pixel j and let N be the total amount of pixels. The total energy on pixel i will be:
with 
(18, 19)
where
rij is the distance between
pixel centers of i and j. This equation can be
written in matrix notation for all i:
(20)
Solving
this set of equations with matrix operations will provide a proximity effect
corrected pattern. However this “self-consistent” scheme will not be a perfect
correction since only the exposed pixels are considered in this equation [13].
Variants and improvements of this algorithm and their performance are discussed
in [20]. The problem can be simplified, thereby reducing the calculation time,
by splitting the dose modification into a problem of forward scattering and
back scattering correction [20-21]. With dose modification, it is possible to
achieve superior proximity effect correction. The main disadvantage is that
with very large circuits, it may require large computation times.
Shape modification
In
this method, a single dose is used for the entire circuit. The shapes found in
the pattern image are modified in such a way that the developed image will
resemble the intended image as close as possible. A good example of a shape
modification method is the correction scheme in PYRAMID [8, 19]. PYRAMID takes
a pattern with rectangular circuit elements. The circuit is then passed to a
correction hierarchy, which adjusts each element via pre-calculated rule
tables. This rule table is created using exposure estimation as described in
the previous paragraph. The first step is to replace each rectangle with its
inner maximal rectangle, as depicted in Fig. 12. The second step is to correct
the effect of interaction among the different circuit elements. Each edge
facing other circuit elements will be adjusted so that the midpoint of the edge
will be equal to the experimental determined development threshold. Even better
results can be obtained by bending the edge when appropriate. The final step is
to modify the shapes at critical points, that is, junctions between adjacent
rectangles (Fig. 13).
Figure 12: reducing rectangle size for exposure compensation
Figure 13: Shape modification of
critical points.
Problems
can be expected when placing for example very large rectangles very close to small
ones. Experimental results in [19] showed a 0.1 um line 0.3 um next to a 11 x 11 um rectangle was missing completely, due to the
high exposure in the gap. A solution for this is to remove parts of the
interior of the large rectangle, in such a way that the rectangle will still
develop, while backscattering is reduced substantially.
The
major advantage of the shape modification method is that accurate results can
be obtained without being computationlly expensive
and without losing throughput. However, it may not be as flexible as the other
methods and experimental data is needed to obtain the necessary rule-tables.
Background exposure correction
Background exposure correction, often referred to as GHOST, works by
writing a second exposure which is the inverse of the intended image. This is done in such a way
that the background dose is brought to a constant level [17-18]. The main
advantage of this method is that it is one of the easiest proximity effect
correction methods found in the literature and can be used with virtually all
electron beam machines. However, there are several disadvantages with this
method. One problem may be a contrast reduction (although this is true for many
other correction types too). However, the main problem is that it only provides
correction to the backscattering component, where the forward scattering
remains uncorrected [13]. There is also a loss of throughput because of the
double exposure.
Conclusion
Proximity
effect in electron beam lithography was examined. Electron scattering is the
most challenging problem in e-beam lithography for producing very high
resolution images. The physical concepts behind the electron scattering are
explained and the exposure process is quantitatively described using energy
density profile models. The influence of different process
parameters were summarized. Often, the electron beam system’s highest
possible electron beam energy available should be used to obtain the highest
resolution. However, care must be taken that no damage and excessive heating is
caused to the substrate due to the increase dose requirements. When it is
possible to use thin resist layers, low electron beam voltage may be feasible
too. Exposure and development time may be tweaked to obtain vertical edge
profiles, or, when negative or positive slopes, necessary for certain
applications.
Algorithms
that can correct the proximity effect were looked at. There exists a trade-off
in proximity effect correction between speed, complexity and accuracy.
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