Energy dependence of proximity parameters investigated by fitting before measurement tests

J. Vac. Sci. Technol. B, Vol. 15, No. 6, Nov/Dec 1997 (pp. 2298-2302)
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Energy dependence of proximity parameters investigated by fitting before measurement tests

L. I. Aparshina, S. V. Dubonos, S. V. Maksimov, A. A. Svintsov, and S. I. Zaitsev
Institute of Microelectronics Technology, Chernogolovka 142432, Russia.

Some years ago a method for fast and accurate experimental evaluation of the proximity
parameters ,
,
was suggested [S. V. Dubonos et al., Microelectron., Eng. 21,
293 (1993)]. The method, called the fitting before measurement
procedure, is used for regular measurements of
and
in a wide energy range for different bulk substrates (Si, SiO_{2}, mica,
ZrO_{2}, Al_{2}O_{3}, InAs, GaAs) and of a as function of resist thickness and
energy. An empirical relation from the fitting procedure allows one
to extrapolate the
and
values to other substrates and energies. It is demonstrated that a resist
of micron thickness can remarkably reduce the resolution of e-beam lithography.
It is important that the reducing could not be improved by accurate
focusing of the beam but could be overcome only by using a higher accelerating
voltage. A phenomenological relation helps to predict resolution as
function of resist thickness and electron energy.

I. INTRODUCTION

The proximity effect in e-beam lithography is quantitatively described
by the two Gaussian formula (proximity function)

where
is a beam "spot",
is a proximity distance determined by electrons backscattered in the
substrate, and
is the ratio of a dose contribution of backscattered electrons to a
dose contribution of incident electrons. To handle the proximity effect
it is essential to know the "proximity parameters" ,
,
and .
Usually for parameter determination special patterns are exposed, for
which the proximity function can be solved analytically, allowing one
to fit experimental results.^{1-7} Such patterns need measurements
close to heavily overexposed areas, where the result can be influenced
by development processes. This could be one reason for the widespread
use of measured parameters for identical substrates.^{6,7} Using
such methods it is also very difficult to get information about possible
measurement errors. Usually
is less than 0.1 um, a value which is very difficult to measure precisely.
As a consequence, low accuracy of
causes a large error in
due to normalization of the proximity function.

Generally speaking, the beam spot ,
the proximity distance ,
and the dose ratio
are determined by cross sections of elastic and inelastic scattering
of fast electrons in a resist and a substrate and, therefore, depend
on electron energy E, atomic number, atomic weight, density of the substrate
(resist), and resist thickness, h. Exact values of the parameters are
very important for successful correction of the proximity effect in
practical e-beam lithography. The desire to provide experimenters with
these values for different substrates was one of the sources of motivation
for the measurements.

Besides this we have one more goal in mind.
The practical range R_{pr} and the backscattered coefficient
_{BC}
are notions widely used in scanning electron microscopy.^{8}
The practical range R_{pr} determines the spatial resolution
of scanning electron microscopy (SEM) diagnostics whereas the backscattered
coefficient _{BC}
is related to SEM image contrast. There is a close correlation between
,
and R_{pr}, _{BC}.
From a fundamental point of view interaction of fast electrons with
media can be investigated via the dependence of the practical range
on accelerating voltage and material properties.

But experimental measurement of the practical range is difficult and
includes measurement of the transmission coefficient of a set of films
with different thicknesses.^{8} The film thicknesses should
be comparable to and less than the practical range, so two obvious difficulties
with this method are the preparation of films with controllable thickness
in the micron and submicron range and (even) the possibility of preparing
these films. On the other hand, the practical range R_{pr} and
the proximity distance
are related by linear dependence so we suggest using the proximity distance
for characterization of the interaction of fast electrons with matter
instead of the practical range. Such a fast and easy way to measure
the proximity parameters paves a way to investigate elastic and inelastic
cross sections for different materials.

II. ALPHA, BETA, AND ETA TESTS

Some procedures for evaluation of the proximity parameters were suggested.
The common feature of the methods is a consequence of measurement fitting.
As mentioned earlier, such methods need to measure distances in nanometers,
which leads to unacceptably low accuracy of
and .
Several years ago a method^{9} for fast and accurate experimental
evaluation of the parameters ,
,
and
was suggested and some results for Si and GaAs were obtained. The method
can be characterized as the fitting before measurement method. Here
we used the method for measurement of
and
in the wide energy diapason for different bulk substrates and for different
resist thicknesses.

FIG. 1. Test pattern used for
determination.

FIG. 3. Test pattern design for
determination.

a)

b)

c)

FIG. 2. (a) Part of an experimental image with the
pattern designed for a beam spot equal to 100 nm. It is seen that
real beam spot is smaller. (b) Part of an experimental image with
the
pattern designed for beam spot equal to 80 nm. The real beam spot
is very close to the one used in the calculation. (c) Part of an
experimental image with the
pattern designed for a beam spot equal to 60 nm. There is no a straight
line and the real beam spot is higher.

FIG. 4. Experimental image of the test pattern designed for =3.9
um (Si) demonstrates the coincidence of expected and real values.

FIG. 5. Experimental image of the test pattern allows the conclusion that
parameter
is equal to 0.7

A. Fitting before measurement method

Special patterns have been designed where the doses of all the features
are numerically calculated by PROXY^{10,11} (a PC software package
for proximity correction) using a given set of parameters. If the used
parameters are "true", the pattern developed will show a special
boundary as a straight horizontal line. If this line is bent, the parameters
used are not correct. Along this line all features have just the right
dose which avoids any effects caused by the development process due
to overexposure.

Fortunately such tests can be done specifically for
and also for ,
allowing measurement of these parameters independently by comparing
several exposed patterns (preferable after "lift off") calculated
with different parameter values.
can be measured by patterns which were calculated by using a true value
for .

Looking just for straight horizontal lines in different patterns also
gives a clear impression of the accuracy of the measured parameter values.
The test patterns were generated by special procedures implemented in
PROXY. Lithographic data of the patterns were exposed with JSM840 under
PROXY control as well.

B. Test pattern for

This pattern consists of many isolated vertical lines with increasing
widths ranging from values less than
up to approximately 5.
Each line is split into small vertical elements separated by a gap of
0.1 um (see Fig. 1).

Along all line elements in one vertical level the required dose for
a full exposure is numerically calculated by PROXY using a given
value. Values used for and
are not critical, but must
be large compared to ,
which is normally the case. In the vertical direction all doses are
scaled down, e.g., by a factor of 0.95, from step to step. Assuming
the bottom ends of all the lines are overexposed and the top ends are
underexposed there must always be a clear boundary in between. The gaps
between the vertical elements are used for making the "lift off"
process easier.

Where the value used for
is correct, the boundary between fully exposed and underexposed vertical
elements will form a straight horizontal line! If the true
is smaller than the one used for calculation, the narrow lines will
be longer than the wide ones and visa versa.

The parameters and
will just move this line up and down, if is
very large compared to ^{'},
if not, the left side of the pattern will still indicate whether a is
too high or too low.

A PMMA based resist of 0.5 um thickness was used in the experiments.
Figures 2(a)-2(c) show only three patterns from several exposed on a
substrate after development and "lift off", calculated with
a values between 60 and 100 nm. The straightest and most horizontal
boundary can be as- signed to be just between the two patterns of Figs.
2(b) and 2(c) and this gives a true
value of about 75 nm. The comparison of the patterns makes it clear
that this method allows an a determination within approximately 10%.

C. Test pattern for

While the calculation of the test pattern for
mainly considers the loss of dose in each line due to forward scattering,
the test pattern for
is also based on the gain of dose in a probe line by backscattered electrons
from the overexposed pattern.

Figure 3 shows this test pattern where a small probe line is positioned
in the middle between two wide and overexposed lines. Numerical calculation
is done in such a way that half of the probe line dose is exposed by
the incident beam while the other half has contributions by backscattered
electrons from the areas as well. The total test pattern contains many
such vertical line groups. In the horizontal direction the gaps between
the overexposed lines are increasing while in the vertical direction
the lines are again separated into small elements where the doses are
scaled down in the same way as before.

Where the
value used for calculation is correct, all probe lines will end up the
same height, again forming a straight boundary (note: this is just a
boundary for the inner probe line; the outer overexposed lines have
to be ignored). Changes in
or
will move this straight horizontal boundary only up or down but will
not influence the straightness and the angle, therefore this test pattern
depends on
only.
If the boundary decreases to the right, it shows that the real value
must be smaller than the assumed one and vice versa.

Figure 4 shows such a pattern for Si again after "lift off",
calculated for a value
equal to 3.9 um, which turned to be the most horizontal boundary, formed
by the upper ends of all the inner probe lines. So it was concluded
that 3.9 um was the value
for the Si substrate at 30 keV. Also, here it was assumed that the accuracy
achievable is approximately 10%.

D. Test pattern for

This test pattern is very similar to the test pattern for
(see Fig. 1), but the linewidth ranges from less than b to approximately 5.

With this pattern the result depends strongly on h and on and
the two effects cannot be clearly separated. Therefore is important
that the dose distribution in this pattern is calculated with a true
value,
which was measured before. The left side of this pattern (Fig. 5) depends
mainly on b whereas the right side is more related to .
Using a true
value the height of the horizontal boundary is already given. The true
value will now correspond to that pattern, where the boundary on the
right side is the same height as that on the left side. A higher boundary
on the right side indicates that the true value is higher than the
one used for the calculation and visa versa.

III. ENERGY DEPENDENCE OF BETA

Due to the small thickness of the resist and its low density the contribution
by the resist in the scattering of fast electrons and in proximity distance
is considered negligible. Beta values were measured for different substrates,
most of which are of particular interest for microelectronics. The results
are shown in Table I. In Table I results of fitting to the power law
=K*E^{p}
are presented. The energy of the electrons is measured in keV and values
K(K_{max}, K_{min}) are given in microns. Exponent p
is dimensionless. The standard fitting procedure gives the confidence
interval (p_{min}, p_{max}) for p and (K_{max},
K_{min}) for prefactor K at a confidence level of 95%. The power
law was expected due to the similarity of the proximity distance to
the practical range (diffusion depth) known from SEM diagnostics.^{8}
The exponent values are close to those measured for the practical range
but much easier to measure. The data are shown in graph form in Fig. 6.

FIG. 6. Dependence of proximity distance
on the energy (accelerated voltage) of electrons for several bulk substrates.

A backscattering coefficient _{BS}
is a ratio of the backscattered electrons to the number of incident
electrons. It is a well known fact that the coefficient is independent of electron
energy in the range 5-100 keV. The proximity parameter
is a ratio of absorbed dose induced by backscattered electrons to an
absorbed dose related to incident electrons. One could expect a close
similarity between these two coef- ficients. Indeed, measurement of
by the method showed energy independence of parameter
in the range 10-40 keV. The
values are shown in the Table I.

IV. ENERGY AND RESIST THICKNESS DEPENDENCIES OF ALPHA

Now, after developing methods of the proximity effect, the correction
spatial resolution is frequently determined by spot diameter of an electron
beam, .

Table I. Proximity parameters (as function of electron energy E) and h for different substrates.
The fitting procedure based on the formula =K*(E/1
keV)^{p} gives the mean values of p and K with the confidence interval (p_{min}, p_{max}) and (K_{min}
,K_{max)} at a confidence probability of 95%.

Fig. 7. Dependence of electron beam spreading ^{2}
as function of resist thickness h^{3 }measured for three different electron energies shows the power dependence expected
from the theory of small-angle scattering.

The spot diameter is determined in turn by an initial beam spot, _{0
} , and by beam spreading due to small-angle scattering in resist,
d:

where h is the resist thickness and L_{eff} is proportional
to a so-called transport length l_{tr} of fast electrons. The
transport length ltr is determined by elastic (low-angle) scattering
of the electrons in the resist and it is known that it is related to
electron energy by parabolic dependence l_{tr}
E^{2}. It is seen that even for an infinitely narrow beam there
is a physical reason for finite beam size. There are some theoretical
estimates of beam spreading in resist but it is very interesting to
measure the value experimentally.

The above mentioned alpha test was used for systematic investigation
of spreading as a function of the resist thick- ness and electron energy.
A remarkable feature of the method is that it allows the measurement
to be carried out in the submicron and nanometer range using an ordinary
SEM (and even an optical microscope) with relative accuracy (not less
than 10%) due to the special form of the test pattern. A PMMA based
resist of varying thicknesses (0.07-1.35 um) was used at three different
electron energies (15, 25, and 35 keV). All the data sets (Fig. 7) demonstrate
the expected power dependence on resist thickness h and energy E. A
fitting procedure based on the formula

gives values of the initial beam spot _{0
} and a phenomenological constant A listed in Table II. Extrapolation
of the dependence to zero thickness gives the value of the initial beam
_{0 } as about 55-60 nm.

Table II. Parameters _{0
} and A extracted from the formula for spreading of the electron beam due to small-angle scattering in resist,
^{2}=_{0}^{2}+A*h^{3}/E^{2}.

Additional conclusions about the correctness can be made from the
independence of fitting parameters _{0 }
and A on electron energy.

From a practical point of view it is important (to point out) that
a resist of micron thickness can remarkably reduce the resolution of
e-beam lithography. It is important that the reducing could not be improved
by accurate focusing of the beam but could be overcome only by using
higher accelerating voltage. The measured spreading provides a quantitative
tool for estimation of the resist thickness and energy influence on
electron lithography accuracy.

Such experiments are of fundamental interest because they pave the way
for measurement of the elastic scattering cross section of fast electrons
with matter and could provide unique and important information.

ACKNOWLEDGMENTS

The authors are obliged to the anonymous referees who reviewed this
paper for their comments which led to the im- provements of both the
style and the presentation of this article. This work was supported
in part by the Russian Foundation of Fundamental Research (Grant Nos. 96-02-19798 and 97-02-17318).

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