OVERVIEW
APPLICATIONS
INTERACTIVE APPLETS
HISTORY OF THE METHODS/FLOW CHART
PUBLICATIONS
EDUCATIONAL MATERIAL
ACKNOWLEDGEMENTS
ABOUT THE AUTHOR/CV
Copyright:
1996, 1999, 2006
J.A. Sethian
Applications to Semiconductor Profile Modeling
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The central idea is to employ the Narrow Band level set method to track the interface. Fast Marching Methods are used to couple the physics through the construction of extension velocities. The flux functions which ultimately determine the growth of the profile depends on such factors as material dependent etch rates, visibilities, non-convex flux laws, surface diffusion, as well as re-deposition and re-emission factors.
General References
A large collection of papers have been written on this topic, and are described below. We also refer the reader to two special web pages on this topic:
Annotated References:
-
Ref. 1 is the first paper in this area, and shows how to track
semiconductor profiles in two space dimensions.
-
Ref. 2 extends this work to three dimensional profiles.
-
Ref. 3 extends the work to additional effects,
including consider re-emission, re-deposition, and surface diffusion.
-
Ref. 4 introduces Fast Marching Methods for photolithography development.
-
Ref. 5 is a review article combining all the techniques.
-
Ref. 6 studies the computational performance of the algorithms.
-
Ref. 7 models high density plasma problems.
-
Ref. 8 studies the growth of voids.
-
Refs. 9,10, and 11 are reviews.
-
Ref. 12 moves all of the methodology to unstructured triangulated meshes.
-
Ref. 13 studies metal sintering, surface diffusion, and reflow of materials,
New Book and Resource on Level Set and Fast Marching Methods
Example:
The figure on the left is a numerical simulations of plasma
enhanced chemical vapor deposition. The figure on the right is
an experimental profile of the same effect.
Details may be found in
Level Set Methods and Fast Marching Methods, J.A. Sethian, 1999
References:
-
A Level Set Approach to a Unified Model for
Etching, Deposition, and Lithography I: Algorithms and Two-Dimensional
Simulations
: Adalsteinsson, D,. and Sethian, J.A.,
Journal of Computational Physics, 120, 1, pp. 128--144, 1995.
Abstract
We apply a level set formulation to the problem of surface advancement
in a two-dimensional topography simulation of deposition, etching,
and lithography processes in integrated circuit fabrication.
The level set formulation is based on solving
a Hamilton-Jacobi type equation for a propagating level set function,
using techniques borrowed from hyperbolic conservation laws. Topological
changes, corner and cusp development, and accurate determination of
geometric properties such as curvature and normal direction are
naturally obtained in this setting. The equations of motion of a unified
model, including the effects of isotropic and unidirectional deposition
and etching, visibility, surface diffusion, reflection, and material
dependent etch/deposition rates are presented and adapted to a level set
formulation. The development of this model and algorithm naturally extends
to three dimensions in a straightforward manner, and is
described in Part II of this paper.
Download publications
-
A Level Set Approach to a Unified Model for Etching, Deposition,
and Lithography II: Three-Dimensional Simulations
: Adalsteinsson, D., and Sethian, J.A.,
Journal of Computational Physics,
122, 2, pp. 348--366, 1995.
Abstract
We apply a level set formulation to the problem of surface advancement
in three-dimensional topography simulation of deposition, etching,
and lithography processes in integrated circuit fabrication.
The level set formulation is based on solving
a Hamilton-Jacobi type equation for a propagating level set function,
using techniques borrowed from hyperbolic conservation laws. Topological
changes, corner and cusp development, and accurate determination of
geometric properties such as curvature and normal direction are
naturally obtained in this setting. The equations of motion of a unified
model, including the effects of isotropic and unidirectional deposition
and etching, visibility, surface diffusion, reflection, and material
dependent etch/deposition rates are presented and adapted to a level set
formulation. In Part I of this paper, the basic equations and algorithms
for two dimensional simulations were developed. In this paper, the
extension to three dimensions is presented. We show a large collection
of simulations, including three-dimensional etching and deposition into
cavities under the effects of visibility, directional and source flux
functions, evolution of lithographic profiles, discontinuous etch rates
In Part III of this paper, effects of reflection, re-emission,
surface diffusion, and multiple materials will be presented.
Download publications
-
A Level Set Approach to a Unified Model for Etching,
Deposition,
and Lithography III:
Re-Deposition, Re-Emission, Surface Diffusion, and Complex
Simulations
:
Adalsteinsson, D., and Sethian, J.A.,
Journal of Computational Physics, 138, 1, pp. 193-223, 1997.
Abstract
Previously, Adalsteinsson and Sethian have applied the level set formulation
to the problem of
surface advancement in two and three-dimensional topography simulation of
deposition, etching, and lithography processes in integrated circuit
fabrication. The level set formulation is based on solving
a Hamilton-Jacobi type equation for a propagating level set function,
using techniques borrowed from hyperbolic conservation laws. Topological
changes, corner and cusp development, and accurate determination of
geometric properties such as curvature and normal direction are
naturally obtained in this setting.
Part I presented the basic equations and algorithms
for two dimensional simulations, including the effects of
isotropic and uni-directional deposition
and etching, visibility, reflection, and material
dependent etch/deposition rates.
Part II focused on the
extension to three dimensions.
This paper completes the series, and add the effects of
re-deposition, re-emission, and surface diffusion.
This requires the solution of the
transport equations for arbitrary geometries, and leads to simulations that
contain multiple simultaneous competing effects of
visibility, directional and source flux
coefficients for the re-emission and re-deposition
functions, multi-layered fronts and thin film layers.
Download publications
-
A Level Set Approach to a Unified Model for Etching,
Deposition,
and Lithography III:
Re-Deposition, Re-Emission, Surface Diffusion, and Complex
Simulations
:
Adalsteinsson, D., and Sethian, J.A.,
Journal of Computational Physics, 138, 1, pp. 193-223, 1997.
Abstract
Previously, Adalsteinsson and Sethian have applied the level set formulation
to the problem of
surface advancement in two and three-dimensional topography simulation of
deposition, etching, and lithography processes in integrated circuit
fabrication. The level set formulation is based on solving
a Hamilton-Jacobi type equation for a propagating level set function,
using techniques borrowed from hyperbolic conservation laws. Topological
changes, corner and cusp development, and accurate determination of
geometric properties such as curvature and normal direction are
naturally obtained in this setting.
Part I presented the basic equations and algorithms
for two dimensional simulations, including the effects of
isotropic and uni-directional deposition
and etching, visibility, reflection, and material
dependent etch/deposition rates.
Part II focused on the
extension to three dimensions.
This paper completes the series, and add the effects of
re-deposition, re-emission, and surface diffusion.
This requires the solution of the
transport equations for arbitrary geometries, and leads to simulations that
contain multiple simultaneous competing effects of
visibility, directional and source flux
coefficients for the re-emission and re-deposition
functions, multi-layered fronts and thin film layers.
Download publications
-
Fast Marching Level Set Methods for
Three Dimensional Photolithography
: Sethian, J.A.,
Proceedings, SPIE 1996 International Symposium on Microlithography, Santa
Clara, California, March, 1996.
Abstract
We present detailed timings of a
fast marching level set method
introduced by Sethian for surface advancement in photoresist development.
The method merges
fast narrow band level set methods
Eikonal equation, and fast heap sort algorithms.
The resulting method can perform the development stage of the
three-dimensional photoresist process in 80 seconds on a
Sparc10 for a 200x200x200 grid.
Download publications
-
An Overview of Level Set Methods for Etching,
Deposition, and Lithography
: Sethian, J.A., and Adalsteinsson, D.,
IEEE Transactions on Semiconductor Devices, 1996.
10, 1, pp.167-184, 1997.
Abstract
The range of surface evolution problems in etching, deposition, and
lithography development offers significant challenge for numerical
methods in front tracking. Level set methods for evolving interfaces
are specifically designed for profiles which can develop sharp corners,
change topology, and undergo orders of magnitude changes in speed.
They are based on solving a Hamilton-Jacobi type equation for a level set
function, using techniques borrowed from hyperbolic conservation laws.
Over the past few years, a body of level set methods have been
developed with application to microfabrication problems.
In this paper, we give an overview of
these techniques, describe the implementation in etching, deposition, and
lithography simulations, and present a collection of fast level set methods,
each aimed at a particular application. In the case of photoresist development
and isotropic etching/deposition,
the fast marching level set method
introduced
by Sethian
can track the three-dimensional photoresist process through a
$200 \times 200 \times 200$ rate function grid in under 55 seconds on a
Sparc10.
In the case of more complex etching and deposition,
the Narrow Band level set method, introduced in by Adalsteinsson and
Sethian, can be used to handle problems in which the speed of the interface
delicately depends on the orientation of the interface vs. an incoming
beam, the effects of visibility, surface tension, reflection and
re-emission, and complex three-dimensional effects.
Our applications include photoresist development, etching/deposition
problems under the effects of masking, visibility, complex flux integrations
over sources, non-convex sputter deposition problems, and simultaneous
deposition and etch phenomena.
Download publications
-
Computational Performance of Level Set Methods for
Etching, Deposition, and Lithography Development
: Sethian, J.A.,
Abstract
The application of level set techniques to problems
two and three dimensional
surface evolution in etching, deposition, and lithography development
have been described in a series of papers, see [1,6].
The techniques are robust, accurate, unbreakable, and extremely fast, and
can be
applied to highly complex two and three dimensional surface
topography evolutions in [1,6]),
including sensitive flux/visibility integration laws,
simultaneous etching and deposition, effects of
non-convex sputter laws demonstrating faceting, as well as ion-sputtered
re-deposition and re-emission with low sticking coefficients, and
surface diffusion.
Download publications
-
High Density Plasma Deposition Modeling Using Level
Set Methods
: Adalsteinsson, D., Sethian, J.A., and Rey, J.C.,
Proceedings Second International Dielectrics for
VLSI/ULCI Multilevel Interconnection Conference, Santa Clara, California,
pp. 116-123, Feb,. 1996.
Abstract
Several silicon dioxide chemical vapor deposition processes using high
density plasma sources have been recently proposed in the literature
for deposition of self-planarizing inter-level dielectric
deposition. All these processes exhibit the competitive effect of
simultaneous deposition and etching mechanisms. This paper describes the
use of a robust simulation technique that can include all physical
mechanisms involved in these processes.
We demonstrate results applied to two and three-dimensional problems analyzing
ion milling, simultaneous etching and deposition, and multiple effects of
re-emission and redeposition.
Download publications
-
Voids in Plasma Enhanced CVD models
: Adalsteinsson, D., Sethian, J.A., and Rey, J.C.,
Proceedings Third International Dielectrics for
VLSI/ULCI Multilevel Interconnection Conference, Santa Clara, California,
Feb,. 1997.
Abstract
In this paper we describe the implementation of Plasma Enhanced CVD
(PECVD) models. We show numerical results for a fully three dimensional
structure using level set method techniques. The terms being simulated
contain both an isotropic and a source deposition term,
along with the effects of reflection and re-emission.
Download publications
-
Level Set Methods for 3D Re-Deposition and Re-emission Simulations
: Adalsteinsson, D., and Sethian, J.A.,
Abstract
Over the past few years [2,3], level set methods have shown to be valuable
tools in simulating the effects of etching and deposition on surface
topography issues. Level Set methods are computational techniques
which approximate the equations of motion
for a propagating front by transforming them into an initial value partial
differential equation, whose unique solution gives the position of the
front. Corners and cusps are naturally handled, and topological change
occurs in a straightforward and rigorous manner with no special user
intervention.
The techniques are robust, accurate, unbreakable, and extremely fast, and
can be applied to highly complex surface evolutions.
In this paper, we discuss the extension of these techniques to problems
including re-emission and redeposition, both with linear and non-linear
flux functions, as well as to problems including thin films and
emerging triple points. Our focus is on three-dimensional simulations, which
require particular attention to fast solvers for computing visibility,
rapid techniques for building the interaction matrix to approximate the
integral equation for the total flux at each point of the interface,
fast summation techniques for evaluating the associated integral
equation, and techniques for tracking multiple interfaces.
We discuss these issues and depth, and present a series of realistic
computational examples, including timing numbers, for building accurate
re-emission/re-deposition and thin film/sidewall activation simulations.
Download publications
-
Etching, Deposition, and Lithography Development:
Theory, Algorithms, and Computational Requirements
: Adalsteinsson, D., and Sethian, J.A.,
Abstract
The range of surface evolution problems in etching, deposition, and
lithography development offers significant computational challenges.
In a series of papers, (Ref. 1, 2, 3, and 5 above),
level set methods for front tracking have been used to simulate a wide
range of semi-conductor simulation, including lithography development and
etching and deposition simulation, including the affects of
visibility, masking, non-convex sputter laws, re-deposition, re-emission,
and surface diffusion.
A careful design of appropriate algorithms yields a wide range of
computational requirements, from personal computers for lithography
development to full supercomputers for solving the integral equation in
re-emission problems with smaller sticking coefficients.
In this paper, we briefly review the level set approach to these problems
and discuss some aspects of the computational requirements.
Download publications
-
Level Set Methods for Surface Advancement in Lithography, Etching and Deposition
: Sethian, J.A.,
Abstract
We describe set of numerical techniques, known as level set methods,
for computing the complex motion of two and three dimensional
surface evolution in etching, deposition, and lithography development.
The techniques are robust, accurate, unbreakable, and extremely fast, and
can be
applied to highly complex surface evolutions. For example, calculation of
the three-dimensional profile advancement for lithography development
takes under 3 seconds on a $80 \times 80 \times 80$ grid on a Sparc 10.
We show the application of these techniques to
a variety of process manufacturing problems in two and
three dimensions, including sensitive flux/visibility integration laws,
simultaneous etching and deposition, effects of
non-convex sputter laws demonstrating faceting, as well as ion-sputtered
re-deposition and re-emission with low sticking coefficients, and
surface diffusion.
Download publications
-
Numerical Schemes for the Hamilton-Jacobi and Level
Set Equations on Triangulated Domains
: Barth, T.J., and Sethian, J.A.,
Journal of Computational Phys., 145, 1, pp. 1-40, 1998.
Abstract
Borrowing from techniques developed for conservation law
equations, numerical schemes which
discretize the Hamilton-Jacobi (H-J), level set, and Eikonal equations
on triangulated domains are presented. The first scheme
is a provably monotone discretization for the H-J
equations. Unfortunately, the basic scheme lacks proper
Lipschitz continuity of the numerical Hamiltonian. By employing
a ``virtual'' edge flipping technique, Lipschitz continuity of
the numerical flux is restored on acute triangulations.
Next, schemes are introduced and developed based on the weaker concept
of positive coefficient approximations for homogeneous Hamiltonians.
These schemes possess a discrete maximum principle on arbitrary
triangulations and naturally exhibit Lipschitz continuity
of the numerical Hamiltonian under mild assumptions on the data and
Hamiltonian. Finally, a class of Petrov-Galerkin
approximations are considered. These schemes are stabilized
via a least-squares bilinear form. The Petrov-Galerkin schemes do
not possess a discrete maximum principle but generalize to high order
accuracy. Discretization of the level set
equation also requires the numerical approximation of a mean
curvature term. A simple mass-lumped Galerkin approximation is presented
and analyzed using maximum principle analysis.
The use of unstructured meshes permits several forms of mesh adaptation
which have been incorporated into numerical examples. These
numerical examples include discretizations of convex and nonconvex
forms of the H-J equation, the Eikonal equation, and the level set equation.
Download publications
-
Motion by
Intrinsic Laplacian of Curvature
: Chopp, D.L., and Sethian, J.A.,
accepted for publication, to appear, Interfaces and Free Boundaries,
March 1999.
Abstract
In this paper, we discuss numerical schemes to model the
motion of curves and surfaces under the intrinsic
Laplacian of curvature. This is an intrinsically
difficult problem, due to the lack of a maximum
principle and the delicate nature of computing
an equation of motion which includes a fourth
derivative term. We design and analyze a host
of algorithms to try and follow motion under this
flow, and discuss the virtues and pitfalls of each.
Synthesizing the results of these various
algorithms, we provide a technique which is
stable and handles very delicate motion in two and
three dimensions. We apply this algorithm to problems
of surface diffusion flow, which is of value for
problems in surface diffusion, metal reflow in
semiconductor manufacturing, sintering, and elastic
membrane simulations. In addition, we provide
examples of the extension of this technique to
anisototropic diffusivity and surface enery which
results in an anisotropic form of the equation of motion.
Download publications
New Book and Resource on Level Set and Fast Marching Methods
|
|
|
Abstract
We apply a level set formulation to the problem of surface advancement
in a two-dimensional topography simulation of deposition, etching,
and lithography processes in integrated circuit fabrication.
The level set formulation is based on solving
a Hamilton-Jacobi type equation for a propagating level set function,
using techniques borrowed from hyperbolic conservation laws. Topological
changes, corner and cusp development, and accurate determination of
geometric properties such as curvature and normal direction are
naturally obtained in this setting. The equations of motion of a unified
model, including the effects of isotropic and unidirectional deposition
and etching, visibility, surface diffusion, reflection, and material
dependent etch/deposition rates are presented and adapted to a level set
formulation. The development of this model and algorithm naturally extends
to three dimensions in a straightforward manner, and is
described in Part II of this paper.
Download publications
We apply a level set formulation to the problem of surface advancement in a two-dimensional topography simulation of deposition, etching, and lithography processes in integrated circuit fabrication. The level set formulation is based on solving a Hamilton-Jacobi type equation for a propagating level set function, using techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. The equations of motion of a unified model, including the effects of isotropic and unidirectional deposition and etching, visibility, surface diffusion, reflection, and material dependent etch/deposition rates are presented and adapted to a level set formulation. The development of this model and algorithm naturally extends to three dimensions in a straightforward manner, and is described in Part II of this paper.
Abstract
We apply a level set formulation to the problem of surface advancement
in three-dimensional topography simulation of deposition, etching,
and lithography processes in integrated circuit fabrication.
The level set formulation is based on solving
a Hamilton-Jacobi type equation for a propagating level set function,
using techniques borrowed from hyperbolic conservation laws. Topological
changes, corner and cusp development, and accurate determination of
geometric properties such as curvature and normal direction are
naturally obtained in this setting. The equations of motion of a unified
model, including the effects of isotropic and unidirectional deposition
and etching, visibility, surface diffusion, reflection, and material
dependent etch/deposition rates are presented and adapted to a level set
formulation. In Part I of this paper, the basic equations and algorithms
for two dimensional simulations were developed. In this paper, the
extension to three dimensions is presented. We show a large collection
of simulations, including three-dimensional etching and deposition into
cavities under the effects of visibility, directional and source flux
functions, evolution of lithographic profiles, discontinuous etch rates
In Part III of this paper, effects of reflection, re-emission,
surface diffusion, and multiple materials will be presented.
Download publications
We apply a level set formulation to the problem of surface advancement in three-dimensional topography simulation of deposition, etching, and lithography processes in integrated circuit fabrication. The level set formulation is based on solving a Hamilton-Jacobi type equation for a propagating level set function, using techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. The equations of motion of a unified model, including the effects of isotropic and unidirectional deposition and etching, visibility, surface diffusion, reflection, and material dependent etch/deposition rates are presented and adapted to a level set formulation. In Part I of this paper, the basic equations and algorithms for two dimensional simulations were developed. In this paper, the extension to three dimensions is presented. We show a large collection of simulations, including three-dimensional etching and deposition into cavities under the effects of visibility, directional and source flux functions, evolution of lithographic profiles, discontinuous etch rates In Part III of this paper, effects of reflection, re-emission, surface diffusion, and multiple materials will be presented.
Abstract
Previously, Adalsteinsson and Sethian have applied the level set formulation
to the problem of
surface advancement in two and three-dimensional topography simulation of
deposition, etching, and lithography processes in integrated circuit
fabrication. The level set formulation is based on solving
a Hamilton-Jacobi type equation for a propagating level set function,
using techniques borrowed from hyperbolic conservation laws. Topological
changes, corner and cusp development, and accurate determination of
geometric properties such as curvature and normal direction are
naturally obtained in this setting.
Part I presented the basic equations and algorithms
for two dimensional simulations, including the effects of
isotropic and uni-directional deposition
and etching, visibility, reflection, and material
dependent etch/deposition rates.
Part II focused on the
extension to three dimensions.
This paper completes the series, and add the effects of
re-deposition, re-emission, and surface diffusion.
This requires the solution of the
transport equations for arbitrary geometries, and leads to simulations that
contain multiple simultaneous competing effects of
visibility, directional and source flux
coefficients for the re-emission and re-deposition
functions, multi-layered fronts and thin film layers.
Download publications
Previously, Adalsteinsson and Sethian have applied the level set formulation to the problem of surface advancement in two and three-dimensional topography simulation of deposition, etching, and lithography processes in integrated circuit fabrication. The level set formulation is based on solving a Hamilton-Jacobi type equation for a propagating level set function, using techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. Part I presented the basic equations and algorithms for two dimensional simulations, including the effects of isotropic and uni-directional deposition and etching, visibility, reflection, and material dependent etch/deposition rates. Part II focused on the extension to three dimensions. This paper completes the series, and add the effects of re-deposition, re-emission, and surface diffusion. This requires the solution of the transport equations for arbitrary geometries, and leads to simulations that contain multiple simultaneous competing effects of visibility, directional and source flux coefficients for the re-emission and re-deposition functions, multi-layered fronts and thin film layers.
Abstract
Previously, Adalsteinsson and Sethian have applied the level set formulation
to the problem of
surface advancement in two and three-dimensional topography simulation of
deposition, etching, and lithography processes in integrated circuit
fabrication. The level set formulation is based on solving
a Hamilton-Jacobi type equation for a propagating level set function,
using techniques borrowed from hyperbolic conservation laws. Topological
changes, corner and cusp development, and accurate determination of
geometric properties such as curvature and normal direction are
naturally obtained in this setting.
Part I presented the basic equations and algorithms
for two dimensional simulations, including the effects of
isotropic and uni-directional deposition
and etching, visibility, reflection, and material
dependent etch/deposition rates.
Part II focused on the
extension to three dimensions.
This paper completes the series, and add the effects of
re-deposition, re-emission, and surface diffusion.
This requires the solution of the
transport equations for arbitrary geometries, and leads to simulations that
contain multiple simultaneous competing effects of
visibility, directional and source flux
coefficients for the re-emission and re-deposition
functions, multi-layered fronts and thin film layers.
Download publications
Previously, Adalsteinsson and Sethian have applied the level set formulation to the problem of surface advancement in two and three-dimensional topography simulation of deposition, etching, and lithography processes in integrated circuit fabrication. The level set formulation is based on solving a Hamilton-Jacobi type equation for a propagating level set function, using techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. Part I presented the basic equations and algorithms for two dimensional simulations, including the effects of isotropic and uni-directional deposition and etching, visibility, reflection, and material dependent etch/deposition rates. Part II focused on the extension to three dimensions. This paper completes the series, and add the effects of re-deposition, re-emission, and surface diffusion. This requires the solution of the transport equations for arbitrary geometries, and leads to simulations that contain multiple simultaneous competing effects of visibility, directional and source flux coefficients for the re-emission and re-deposition functions, multi-layered fronts and thin film layers.
Abstract
We present detailed timings of a
fast marching level set method
introduced by Sethian for surface advancement in photoresist development.
The method merges
fast narrow band level set methods
Eikonal equation, and fast heap sort algorithms.
The resulting method can perform the development stage of the
three-dimensional photoresist process in 80 seconds on a
Sparc10 for a 200x200x200 grid.
Download publications
We present detailed timings of a fast marching level set method introduced by Sethian for surface advancement in photoresist development. The method merges fast narrow band level set methods Eikonal equation, and fast heap sort algorithms. The resulting method can perform the development stage of the three-dimensional photoresist process in 80 seconds on a Sparc10 for a 200x200x200 grid.
Abstract
The range of surface evolution problems in etching, deposition, and
lithography development offers significant challenge for numerical
methods in front tracking. Level set methods for evolving interfaces
are specifically designed for profiles which can develop sharp corners,
change topology, and undergo orders of magnitude changes in speed.
They are based on solving a Hamilton-Jacobi type equation for a level set
function, using techniques borrowed from hyperbolic conservation laws.
Over the past few years, a body of level set methods have been
developed with application to microfabrication problems.
In this paper, we give an overview of
these techniques, describe the implementation in etching, deposition, and
lithography simulations, and present a collection of fast level set methods,
each aimed at a particular application. In the case of photoresist development
and isotropic etching/deposition,
the fast marching level set method
introduced
by Sethian
can track the three-dimensional photoresist process through a
$200 \times 200 \times 200$ rate function grid in under 55 seconds on a
Sparc10.
In the case of more complex etching and deposition,
the Narrow Band level set method, introduced in by Adalsteinsson and
Sethian, can be used to handle problems in which the speed of the interface
delicately depends on the orientation of the interface vs. an incoming
beam, the effects of visibility, surface tension, reflection and
re-emission, and complex three-dimensional effects.
Our applications include photoresist development, etching/deposition
problems under the effects of masking, visibility, complex flux integrations
over sources, non-convex sputter deposition problems, and simultaneous
deposition and etch phenomena.
Download publications
The range of surface evolution problems in etching, deposition, and
lithography development offers significant challenge for numerical
methods in front tracking. Level set methods for evolving interfaces
are specifically designed for profiles which can develop sharp corners,
change topology, and undergo orders of magnitude changes in speed.
They are based on solving a Hamilton-Jacobi type equation for a level set
function, using techniques borrowed from hyperbolic conservation laws.
Over the past few years, a body of level set methods have been
developed with application to microfabrication problems.
In this paper, we give an overview of
these techniques, describe the implementation in etching, deposition, and
lithography simulations, and present a collection of fast level set methods,
each aimed at a particular application. In the case of photoresist development
and isotropic etching/deposition,
the fast marching level set method
introduced
by Sethian
can track the three-dimensional photoresist process through a
$200 \times 200 \times 200$ rate function grid in under 55 seconds on a
Sparc10.
In the case of more complex etching and deposition,
the Narrow Band level set method, introduced in by Adalsteinsson and
Sethian, can be used to handle problems in which the speed of the interface
delicately depends on the orientation of the interface vs. an incoming
beam, the effects of visibility, surface tension, reflection and
re-emission, and complex three-dimensional effects.
Our applications include photoresist development, etching/deposition
problems under the effects of masking, visibility, complex flux integrations
over sources, non-convex sputter deposition problems, and simultaneous
deposition and etch phenomena.
Abstract
The application of level set techniques to problems
two and three dimensional
surface evolution in etching, deposition, and lithography development
have been described in a series of papers, see [1,6].
The techniques are robust, accurate, unbreakable, and extremely fast, and
can be
applied to highly complex two and three dimensional surface
topography evolutions in [1,6]),
including sensitive flux/visibility integration laws,
simultaneous etching and deposition, effects of
non-convex sputter laws demonstrating faceting, as well as ion-sputtered
re-deposition and re-emission with low sticking coefficients, and
surface diffusion.
Download publications
The application of level set techniques to problems two and three dimensional surface evolution in etching, deposition, and lithography development have been described in a series of papers, see [1,6]. The techniques are robust, accurate, unbreakable, and extremely fast, and can be applied to highly complex two and three dimensional surface topography evolutions in [1,6]), including sensitive flux/visibility integration laws, simultaneous etching and deposition, effects of non-convex sputter laws demonstrating faceting, as well as ion-sputtered re-deposition and re-emission with low sticking coefficients, and surface diffusion.
Abstract
Several silicon dioxide chemical vapor deposition processes using high
density plasma sources have been recently proposed in the literature
for deposition of self-planarizing inter-level dielectric
deposition. All these processes exhibit the competitive effect of
simultaneous deposition and etching mechanisms. This paper describes the
use of a robust simulation technique that can include all physical
mechanisms involved in these processes.
We demonstrate results applied to two and three-dimensional problems analyzing
ion milling, simultaneous etching and deposition, and multiple effects of
re-emission and redeposition.
Download publications
Several silicon dioxide chemical vapor deposition processes using high density plasma sources have been recently proposed in the literature for deposition of self-planarizing inter-level dielectric deposition. All these processes exhibit the competitive effect of simultaneous deposition and etching mechanisms. This paper describes the use of a robust simulation technique that can include all physical mechanisms involved in these processes. We demonstrate results applied to two and three-dimensional problems analyzing ion milling, simultaneous etching and deposition, and multiple effects of re-emission and redeposition.
Abstract
In this paper we describe the implementation of Plasma Enhanced CVD
(PECVD) models. We show numerical results for a fully three dimensional
structure using level set method techniques. The terms being simulated
contain both an isotropic and a source deposition term,
along with the effects of reflection and re-emission.
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In this paper we describe the implementation of Plasma Enhanced CVD (PECVD) models. We show numerical results for a fully three dimensional structure using level set method techniques. The terms being simulated contain both an isotropic and a source deposition term, along with the effects of reflection and re-emission.
Abstract
Over the past few years [2,3], level set methods have shown to be valuable
tools in simulating the effects of etching and deposition on surface
topography issues. Level Set methods are computational techniques
which approximate the equations of motion
for a propagating front by transforming them into an initial value partial
differential equation, whose unique solution gives the position of the
front. Corners and cusps are naturally handled, and topological change
occurs in a straightforward and rigorous manner with no special user
intervention.
The techniques are robust, accurate, unbreakable, and extremely fast, and
can be applied to highly complex surface evolutions.
In this paper, we discuss the extension of these techniques to problems
including re-emission and redeposition, both with linear and non-linear
flux functions, as well as to problems including thin films and
emerging triple points. Our focus is on three-dimensional simulations, which
require particular attention to fast solvers for computing visibility,
rapid techniques for building the interaction matrix to approximate the
integral equation for the total flux at each point of the interface,
fast summation techniques for evaluating the associated integral
equation, and techniques for tracking multiple interfaces.
We discuss these issues and depth, and present a series of realistic
computational examples, including timing numbers, for building accurate
re-emission/re-deposition and thin film/sidewall activation simulations.
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Over the past few years [2,3], level set methods have shown to be valuable
tools in simulating the effects of etching and deposition on surface
topography issues. Level Set methods are computational techniques
which approximate the equations of motion
for a propagating front by transforming them into an initial value partial
differential equation, whose unique solution gives the position of the
front. Corners and cusps are naturally handled, and topological change
occurs in a straightforward and rigorous manner with no special user
intervention.
The techniques are robust, accurate, unbreakable, and extremely fast, and
can be applied to highly complex surface evolutions.
In this paper, we discuss the extension of these techniques to problems
including re-emission and redeposition, both with linear and non-linear
flux functions, as well as to problems including thin films and
emerging triple points. Our focus is on three-dimensional simulations, which
require particular attention to fast solvers for computing visibility,
rapid techniques for building the interaction matrix to approximate the
integral equation for the total flux at each point of the interface,
fast summation techniques for evaluating the associated integral
equation, and techniques for tracking multiple interfaces.
We discuss these issues and depth, and present a series of realistic
computational examples, including timing numbers, for building accurate
re-emission/re-deposition and thin film/sidewall activation simulations.
Abstract
The range of surface evolution problems in etching, deposition, and
lithography development offers significant computational challenges.
In a series of papers, (Ref. 1, 2, 3, and 5 above),
level set methods for front tracking have been used to simulate a wide
range of semi-conductor simulation, including lithography development and
etching and deposition simulation, including the affects of
visibility, masking, non-convex sputter laws, re-deposition, re-emission,
and surface diffusion.
A careful design of appropriate algorithms yields a wide range of
computational requirements, from personal computers for lithography
development to full supercomputers for solving the integral equation in
re-emission problems with smaller sticking coefficients.
In this paper, we briefly review the level set approach to these problems
and discuss some aspects of the computational requirements.
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The range of surface evolution problems in etching, deposition, and lithography development offers significant computational challenges. In a series of papers, (Ref. 1, 2, 3, and 5 above), level set methods for front tracking have been used to simulate a wide range of semi-conductor simulation, including lithography development and etching and deposition simulation, including the affects of visibility, masking, non-convex sputter laws, re-deposition, re-emission, and surface diffusion. A careful design of appropriate algorithms yields a wide range of computational requirements, from personal computers for lithography development to full supercomputers for solving the integral equation in re-emission problems with smaller sticking coefficients. In this paper, we briefly review the level set approach to these problems and discuss some aspects of the computational requirements.
Abstract
We describe set of numerical techniques, known as level set methods,
for computing the complex motion of two and three dimensional
surface evolution in etching, deposition, and lithography development.
The techniques are robust, accurate, unbreakable, and extremely fast, and
can be
applied to highly complex surface evolutions. For example, calculation of
the three-dimensional profile advancement for lithography development
takes under 3 seconds on a $80 \times 80 \times 80$ grid on a Sparc 10.
We show the application of these techniques to
a variety of process manufacturing problems in two and
three dimensions, including sensitive flux/visibility integration laws,
simultaneous etching and deposition, effects of
non-convex sputter laws demonstrating faceting, as well as ion-sputtered
re-deposition and re-emission with low sticking coefficients, and
surface diffusion.
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We describe set of numerical techniques, known as level set methods, for computing the complex motion of two and three dimensional surface evolution in etching, deposition, and lithography development. The techniques are robust, accurate, unbreakable, and extremely fast, and can be applied to highly complex surface evolutions. For example, calculation of the three-dimensional profile advancement for lithography development takes under 3 seconds on a $80 \times 80 \times 80$ grid on a Sparc 10. We show the application of these techniques to a variety of process manufacturing problems in two and three dimensions, including sensitive flux/visibility integration laws, simultaneous etching and deposition, effects of non-convex sputter laws demonstrating faceting, as well as ion-sputtered re-deposition and re-emission with low sticking coefficients, and surface diffusion.
Abstract
Borrowing from techniques developed for conservation law
equations, numerical schemes which
discretize the Hamilton-Jacobi (H-J), level set, and Eikonal equations
on triangulated domains are presented. The first scheme
is a provably monotone discretization for the H-J
equations. Unfortunately, the basic scheme lacks proper
Lipschitz continuity of the numerical Hamiltonian. By employing
a ``virtual'' edge flipping technique, Lipschitz continuity of
the numerical flux is restored on acute triangulations.
Next, schemes are introduced and developed based on the weaker concept
of positive coefficient approximations for homogeneous Hamiltonians.
These schemes possess a discrete maximum principle on arbitrary
triangulations and naturally exhibit Lipschitz continuity
of the numerical Hamiltonian under mild assumptions on the data and
Hamiltonian. Finally, a class of Petrov-Galerkin
approximations are considered. These schemes are stabilized
via a least-squares bilinear form. The Petrov-Galerkin schemes do
not possess a discrete maximum principle but generalize to high order
accuracy. Discretization of the level set
equation also requires the numerical approximation of a mean
curvature term. A simple mass-lumped Galerkin approximation is presented
and analyzed using maximum principle analysis.
The use of unstructured meshes permits several forms of mesh adaptation
which have been incorporated into numerical examples. These
numerical examples include discretizations of convex and nonconvex
forms of the H-J equation, the Eikonal equation, and the level set equation.
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Borrowing from techniques developed for conservation law equations, numerical schemes which discretize the Hamilton-Jacobi (H-J), level set, and Eikonal equations on triangulated domains are presented. The first scheme is a provably monotone discretization for the H-J equations. Unfortunately, the basic scheme lacks proper Lipschitz continuity of the numerical Hamiltonian. By employing a ``virtual'' edge flipping technique, Lipschitz continuity of the numerical flux is restored on acute triangulations. Next, schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous Hamiltonians. These schemes possess a discrete maximum principle on arbitrary triangulations and naturally exhibit Lipschitz continuity of the numerical Hamiltonian under mild assumptions on the data and Hamiltonian. Finally, a class of Petrov-Galerkin approximations are considered. These schemes are stabilized via a least-squares bilinear form. The Petrov-Galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy. Discretization of the level set equation also requires the numerical approximation of a mean curvature term. A simple mass-lumped Galerkin approximation is presented and analyzed using maximum principle analysis. The use of unstructured meshes permits several forms of mesh adaptation which have been incorporated into numerical examples. These numerical examples include discretizations of convex and nonconvex forms of the H-J equation, the Eikonal equation, and the level set equation.
Abstract
In this paper, we discuss numerical schemes to model the
motion of curves and surfaces under the intrinsic
Laplacian of curvature. This is an intrinsically
difficult problem, due to the lack of a maximum
principle and the delicate nature of computing
an equation of motion which includes a fourth
derivative term. We design and analyze a host
of algorithms to try and follow motion under this
flow, and discuss the virtues and pitfalls of each.
Synthesizing the results of these various
algorithms, we provide a technique which is
stable and handles very delicate motion in two and
three dimensions. We apply this algorithm to problems
of surface diffusion flow, which is of value for
problems in surface diffusion, metal reflow in
semiconductor manufacturing, sintering, and elastic
membrane simulations. In addition, we provide
examples of the extension of this technique to
anisototropic diffusivity and surface enery which
results in an anisotropic form of the equation of motion.
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In this paper, we discuss numerical schemes to model the motion of curves and surfaces under the intrinsic Laplacian of curvature. This is an intrinsically difficult problem, due to the lack of a maximum principle and the delicate nature of computing an equation of motion which includes a fourth derivative term. We design and analyze a host of algorithms to try and follow motion under this flow, and discuss the virtues and pitfalls of each. Synthesizing the results of these various algorithms, we provide a technique which is stable and handles very delicate motion in two and three dimensions. We apply this algorithm to problems of surface diffusion flow, which is of value for problems in surface diffusion, metal reflow in semiconductor manufacturing, sintering, and elastic membrane simulations. In addition, we provide examples of the extension of this technique to anisototropic diffusivity and surface enery which results in an anisotropic form of the equation of motion.