This thesis describes the latest embodiment of a three-dimensional electromagnetic simulation program called TEMPEST which is implemented on the connection machines CM-2 and CM-5, and is used to predict and study technology trade-offs of interest in photolithography. Highlights of the new algorithm include generalization to three-dimensional calculation, analysis of dispersive materials, an efficient absorbing boundary condition, oblique incidence, and image synthesis based on Hopkins' formulation.
The finite propagation speed of electromagnetic waves makes the time-domain finite-difference approach a natural choice for implementation on parallel computer architectures. This thesis addresses algorithmic issues including the accuracy and stability of the numerical scheme, and numerical boundary conditions. The conventional time-domain finite-difference scheme is second order accurate and requires 15 simulation nodes per wavelength to achieve a 2% accuracy. Stability of the scheme depends on the ratio between the spatial and temporal discretizations. Analogous to previous work in plasma physics, instability of the algorithm due to highly dispersive materials has been alleviated by calculating explicitly the time-domain convolution relation between the electric field and the electric displacement. A novel boundary condition derived from the harmonic nature of electromagnetic waves is used to bound the simulation domain with minimal artificial reflection.
Implementation of a software package which caters to user convenience in data processing and remote use of the connection machine is also included. A link between TEMPEST and the optical image simulation program SPLAT allows the study of the interaction among mask topography effects, partial coherence effects, and lens aberrations.
Validation of TEMPEST via experimental comparison and the usefulness of the program in predicting and assessing complex technological issues in photolithography are presented. In particular, TEMPEST is used to predict important effects such as glass edge scattering in phase-shifting masks and resonance in dielectric ridges. These predictions have been subsequently confirmed experimentally. Transmission loss and polarization effects in small contact holes are characterized as a function of the feature size. TEMPEST is also shown to be well-suited for analyzing three-dimensional effects of reflection from underlying topography during photoresist exposure which can cause variations in the critical dimensions of the features being formed.

Rigorous three-dimensional time-domain finite-difference electromagnetic simulation

A two-dimensional fast Fourier transform technique is proposed for accelerating
the computation of scattering characteristics of features on surfaces by using
the discrete-dipole approximation. The two-dimensional fast Fourier transform
reduces the CPU execution time dependence on the number of dipoles N from O(N2) to O(N log N). The capabilities and flexibility of a discrete-dipole
code implementing the technique are demonstrated with scattering results from
circuit features on surfaces.

/pdf/discrete-dipole-approximation-for-scattering-by-features-on-4efn4m7l48.pdf

Discrete-dipole approximation for scattering by features on surfaces by means of a two-dimensional fast Fourier transform technique

Light scattering from particles or structures on a surface is calculated by using a modification of the coupled-dipole method. The interaction matrix is modified for the presence of the surface by inclusion of the dipole reflection terms, calculated with the Sommerfeld integral formulation. The incident plane wave is replaced by the incident plus the surface reflected waves. In some cases the dipole reflections may be approximated by a Fresnel image dipole. Light scattering for structures up to a wavelength in size can be calculated in a reasonable amount of time.

Calculation of light scattering from particles and structures on a surface by the coupled-dipole method

In conjunction with the development of a bidirectional scattering metrology project, a large number of papers pertaining to the theory and measurement of bidirectional scattering from optical surfaces were collected and categorized. This collection includes papers that deal with various aspects of the bidirectional scattering distribution function (BSDF), its measurement, interpretation, use, and implications. Each paper is classified in one or more subject categories on the basis of its technical content. The subject categories are included just to serve as a key to the most salient characteristics of each paper cited. Because of the interest in this field, this bibliography is being published as a service to the public.

/pdf/bidirectional-scattering-distribution-function-bsdf-a-418hurcg4u.pdf

Bidirectional Scattering Distribution Function (BSDF): A Systematized Bibliography.

Polarization and interface reflection effects are examined for technology issues in mask making of chrome edge shape, overcoating, double phase-shifting removal of defects, and reflective masks. This study is based on the massively parallel rigorous electromagnetic simulator TEMPEST1, with extensions to include the TM polarization and boundary conditions for outgoing waves in optical materials. The analysis was carried out on a CM-2 connection machine with remote electronic access. A local workstation was used to write and preview the geometry as well as process images from files of diffraction efficiencies downloaded to the workstation from the network. Through this remote analysis procedure, the four proto-typical mask case studies described by Doi et al.2 were examined for polarization and boundary condition effects.
For chrome masks of one wavelength wide opening in a IX system at 0.248 |im, moderate differences between the TE and the TM polarizations were observed in the peak intensities (10%). The TM polarization in general showed higher transmission and lower sidelobes. The two polarizations showed a similar increase in linewidth and peak intensity with decreasing chrome edge slope. The difference was not as pronounced for a 5X system at 0.365 |Lim. Overcoating chrome masks with anti-reflection layers improved resolution for both polarizations. For 5X chromeless phase-shifting masks at 0.365 pm, neither polarization effect nor phase-shifter edge slope was important. The peak intensity at the phase-shifting section changed by only 1% and the linewidth varied by less than 2% when the phase-shifter edge slope changed from 90° to 45°, keeping the mid-points of the phase-shifter edge fixed. To remove defects on such masks by double phase-shifting may drop the intensity level to 70% of the clear field value for a 0.1 X/NA sized defect. For IX reflective masks at one wavelength, both polarizations displayed similar behaviors. These reflective masks with built-in material-based phase-shifting improved the image slope, and the TM polarization was found to have slightly less ringing than the TE polarization.

Polarization effects in mask transmission

This paper demonstrates numerical solutions of Maxwell's equations for laser diffraction and scattering from submicron objects on silicon wafers. Discrete numerical solvers using time domain, explicit finite element (FE) and finite difference (FD) methods have been implemented and vectorized for supercomputers. Both 2-D and 3-D FE and FD simulations have been performed for fibers and spheres (.5 to 1.6 microns) in free-space and on bare silicon, illuminated at normal incidence by a He-Ne laser beam. The wave solvers were verified against exact solutions for cylinders and spheres in free-space. Laboratory scattering experiments have been conducted with latex spheres on bare silicon, and the results compared to the corresponding numerical simulations. Simulations of glass and silicon fibers on a layer of photoresist over silicon examined differences in exposure patterns as well as the implementation of bleaching in nonlinear calculations. The results indicate that there is no inherent limitation to rigorously simulating many of the optical problems associated with IC fabrication, inspection, and design. The idea here is not to obtain just "exact" solutions for unsolved problems, but rather to provide the analyst with a broader capability for numerical experiments and simulations. In this way computers and mathematical algorithms can augment ad hoc approximations and laboratory experiments. The ultimate goal of this research is to simulate imaging systems and resist exposure patterns, and incorporate some of these results in process simulation codes like SAMPLE.

Lee K. Galbraith

Papers

Calculation Of Light Scatter From Structures On Silicon Surfaces