New Model for Simulation of Exposure Process in Complex Nonplanar Resist-Substrate Structures

Introduction

Predictive and efficient lithography simulation is an
important component of the semiconductor industry efforts to develop the
next generation of deep submicron technologies. Emerging technologies
are based on elements with very small feature sizes and extremely complex
and nonplanar topographies. Therefore lithography processing has to provide
high resolution with large depth of focus. Simultaneously such effects
as nonplanar reflections and notching as well as refractive index dependence
on local absorbed dose are very critical for printing small mask elements
using short wavelength radiation. This work presents a new approach for
simulating the exposure process, which takes into account these effects
in complex nonplanar resist-substrate structures. The method is based
on numerical solution of the Helmholtz equation for the electric field
in the media with complex refractive index n(x,y,z,I), where I is the
dose previously absorbed at the point. Propagation of initial electromagnetic
field and fields reflected from the material interface elements are calculated
using the Beam Propagation Method (BMP) [1]. Because the method is very
general it can be used for different types of radiation (UV, EUV, X-ray)
as well as for multiexposure processes and multilayer and nonlinear resists.
The method is implemented as a separate module, which is interfaced with
imaging and developing modules of * Optolith*. Complete exposure
simulations for a typical 2D structure take 2-20 sec on a Sun Ultra-10
workstation.

Model

The exposure model previously implemented into Optolith is based on the Ray Tracing Method (RTM). RTM cannot accurately calculate diffraction on small features and does not allow to account for optical nonlinearities of the photoresist. The new exposure model has been developed to achieve several goals:

- More accurate simulation of the exposure process by
taking into account diffraction effects.

- Include a capability to simulate non-linear effect
of the intensity distribution on the local optical properties of the
resist material.

- Improve the simulator performance.

The Beam Propagation Method (BPM) is used to solve the Helmholtz equation for electromagnetic field inside the structure. During the simulation the field distribution is formed as the superposition of incident light, all the reflections from all elements of the resist-substrate interface and secondary reflection(s) from the upper resist surface.

The formal descriptions of the BPM can be found in [1]. In [2-4] some applications using BPM are described too.

In this model the Helmholtz equation for the electric
field E into the media with complex refractive index *n(x, y, z)*

where *k* is the wave number, is solved in two main
stages:

- First, the diffraction over a small spatial step along
the propagation is calculated thus obtaining the new field amplitude
distribution without absorption taken into account;

- Then the actual field distribution is computed as a product of this amplitude distribution and the distribution of the complex absorption over the step.

Let the wave is propagated along z-axis. We find the
solution as a quasi plane wave *E=A(x, y, z)exp(inkz) *with slowly
varied amplitude *A* ( *A* is modified with * z* slower
than phase term *inkz*). In this case, the Fourier image of current
distribution *A* in the plane *z=z0 *is
defined as

After propagating over a small step each component of obtains additional phase shift corresponded to the value of . Thus, the amplitude distribution at (without accounting for absorption) can be written as

Due to difference of actual optical properties from ones for vacuum, the field at the new plane is computed simply:

The algorithm is repeated recursively step-by-step over all simulation domain. The same calculations are applied to reflections from all segments of the resist boundaries. And the whole procedure is repeated NUM.REFL times, where NUM.REFL is specified in the EXPOSURE statement.

This approach allows the dependence of the refraction
index n on intensity I to be taken into account. The following formula
for *n(x,y,z,I)* is implemented:

Here *=
-CI(x, y, z, t)MPAC* , where *I(x, y, z, t)
*is the current intensity distribution, *C* is the Mack's C-parameter.
This effect is calculated by accumulating the total dose in several steps
so the intensity distribution is formed as the sum of distributions obtained
for each step. In this case, the previously accumulated intensities are
used to compute current values of the refractive index for each point
inside the resist area.

Changes in Optolith Syntax

Several additional parameters have been included in the OPTOLITH syntax to control the new model. The BPM parameter in the EXPOSURE statement specifies that Beam Propagation Method is to be used during exposure simulation. This parameter is now default. If the RTM parameter is specified instead the old Ray Tracing Method will be used.

To simulate the dose effect on the resist optical properties the difference of the complex refraction index for exposed and unexposed resist has to be specified. It is assumed that standard value of the refraction index corresponds to the case of unexposed resist. The following OPTICAL statement:

optical name.resist=RESIST1 i.line delta.real=0.1 delta.imag=-0.03

specifies that the real part of the refraction index increases by 0.1 for completely exposed resist while the imaginary part of the refraction index (absorption) is reduced by 0.03 for completely exposed resist.

Example

The figures illustrate influence of the "dose effect"
on the developed profile. Artificial conditions were simulated to outline
the importance of the dose effect. In Figure 1 and 2 the intensity distributions
in the resist over a non-planar substrate are shown for the cases of constant
refraction index *n0 = 1.4 + i . 0.02 *(Figure
1) and the index varied with dose from n0 for unexposed resist to *n1
= 1.4 + i . 0.04* for completely exposed one (Figure 2). The exposure
level near the center of the substrate deepening differs substantially
for these two cases. In the both cases there is the local maximum of the
intensity due to reflections from the slope walls. However, in the second
case, this maximum is visibly lower due to modification of the resist
properties during the exposure. These quantitative differences in the
intensity distributions result in completely different resist development
profiles (see Figure 3, 4). In the first case the feature is unresolved
(Figure3) while in the second case (Figure4) the whole lithography process
was successful.

Figure 1. The distribution of intensity (dose) in the resist

over a non-planar substrate. The resist refraction

index is not modified during the exposure.

Figure 2. The distribution of intensity (dose) in the resist over a non-planar

substrate. The resist refraction index is modified during the exposure

from for unexposed resist to for completely exposed one.

Figure 3. The resist development profile corresponding

to the dose distribution in Figure1.

Figure 4. The resist development profile corresponding

to the dose distribution in Figure2.

New capabilities of the * Optolith* simulator
allow to determine the characteristics of actual resists as well as to
optimize parameters of the technological process.

**References**

- J. Van Roey, J.van der Donk, P.E. Lagasse.
Beam-propagation method: analysis and assessment. J. Opt.Soc. Am.,

Vol.71, No. 7, July 1981, p. 803. - J. Z. Y. Guo, F. Cerrina. Modeling X-ray proximity
lithography. IBM J. Res. Develop.,

Vol. 37, No. 3, May 1993, p. 331. - A. Erdmann, C. L. Henderson, C. G. Wilson, W. Henke.
Influence of optical nonlinearities of the photoresist on the photolithographic
process: basics. SPIE Proc., Vol. 3051, 1997, p. 529. [4] A. Erdmann,
C. L. Henderson, C. G. Wilson, R. R. Dammel. Some aspects of thick film
resist performance and modeling. SPIE Proc.,

Vol. 3333, 1998, p. 1201.