Table Of ContentProgress in Numerical Simulation for Microelectronics
Vol. 3
Edited by
K. Merten, Siemens AG, Munich
A. Gilg, Siemens AG, Munich
Alfred Kersch
William J. Morokoff
Transport Simulation
in Microelectronics
1995 Birkhauser Verlag
Basel· Boston· Berlin
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Introduction
Computer simulation of semiconductor processing equipment and devices
requires the use of a wide variety of numerical methods. Of these methods,
the Monte Carlo approach is perhaps most fundamentally suited to mod
eling physical events occurring on microscopic scales which are intricately
connected to the particle structure of nature. Here physical phenomena can
be simulated by following simulation particles (such as electrons, molecules,
photons, etc.) through a statistical sampling of scattering events. Monte
Carlo is, however, generally looked on as a last resort due to the extremely
slow convergence of these methods. It is of interest, then, to examine when
in microelectronics it is necessary to use Monte Carlo methods, how such
methods may be improved, and what are the alternatives. This book ad
dresses three general areas of simulation which frequently arise in semicon
ductor modeling where Monte Carlo methods playa significant role. In the
first chapter the basic mathematical theory of the Boltzmann equation for
particle transport is presented. The following chapters are devoted to the
modeling of the transport processes and the associated Monte Carlo meth
ods. Specific examples of industrial applications illustrate the effectiveness
and importance of these methods.
Two of these areas concern simulation of physical particles which
may be assigned a time dependent position and velocity. This includes
the molecules of a dilute gas used in such processing equipment as chemi
cal vapor decomposition reactors and sputtering reactors. We also consider
charged particles moving within a semiconductor lattice. In both cases vari
ous forms of the Boltzmann equation describe the time evolution of a prob
ability distribution function for the position and velocity of a particle. It is
this equation that the Monte Carlo particle simulations attempt to solve.
Analysis of the Boltzmann equation indicates when other, faster numerical
methods are suitable and suggests possible hybrid approaches which make
limited use of Monte Carlo.
The third topic addressed here is radiative transfer which arises in
Rapid Thermal Processing reactors, as well as many other areas. Here the
simulated particles are photons which may be scattered or absorbed and
thereby transfer energy from a source to a background or surface. Here the
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6
Boltzmann equation may also be used as the mathematical model; however,
now the speed of the particles is fixed at the speed of light, and only the
velocity direction varies. We present here a less widely known treatment of
the problem using Quasi-Monte Carlo methods whereby random sequences
are replaced by more uniformly distributed deterministic sequences. This
approach is particularly well suited to the linear, time-independent problems
which often arise in radiative transport.
All of the key aspects of Boltzmann transport (non-linear scattering,
linear scattering off a background and surface scattering) are addressed in
the examples given. Numerous analogies may draw to other applications.
For example, the simulation of small scale features of semiconductor device
structures is closely related to radiation transport simulation. Ion imple
mentation problems may be solved with methods similar to those used in
the simulation of the sputtering reactor. And plasma simulation may be
treated with a combination of methods used for gas dynamics and charge
transport.
This book is the result of two different directions of research - ap
plied mathematics and physical modeling - being applied to the area of
numerical simulation of transport phenomena on the level of the Boltz
mann equation. From the mathematical side, William Morokoff, working
in a theoretically oriented, academic research environment, has attempted
to unite these seemingly diverse areas through the similarities in their un
derlying mathematical models and to present a rigorous foundation for the
numerical methods used. From the physical modeling side, Alfred Kersch,
working in the context of an industrial research lab, has focused on present
ing problems and examples of true industrial relevance and connecting these
with the theoretically validated simulation methods. The authors have tried,
through a close collaboration, to close the gap which often exists between
the two approaches.
We greatfully acknowledge the opportunity given by the editors Knut
Merten and Albert Gilg to include this book in their series "Progress of
Numerical Simulation in Microelectronics (PNSM)". We would also like
to acknowledge the cooperation of the people of the Siemens equipment
simulation group, especially Chr.Werner and R.P. Brinkmann. Furthermore
we thank Th. Vogelsang for the many things we learned from him and his
PhD thesis, and Hermann Jacobs for his encouragement and support.
The collaboration between the authors had its origins in a joint pro
gram between Siemens and the Institute for Mathematics and Its Appli
cations (IMA) at the University of Minnesota. We therefore give special
thanks to the IMA and its director A vner Friedman for making this project
possible. Thanks also to Russel Caflisch of UCLA for his support and advice,
as well as to the National Science Foundation. Finally we wish to acknowl
edge the Universities of Minnesota, Arizona, California (Los Angeles) and
L'Aquila (Italy) for the use of their resources in completing this project.
Contents
1 The Boltzm.ann Equation 11
1.1 The Liouville Equation. 11
1.2 Rarefied Gases . . . . . 13
1.2.1 Equilibrium ... 19
1.2.2 Knudsen Number and Fluid Equations. 21
1.2.3 Multiple Species ... 24
1.2.4 Boundary Conditions ...... . 25
1.3 Radiation Transport . . . . . . . . . . . . 28
1.3.1 The Radiation Transport Equation 29
1.3.2 Boundary Conditions 31
1.4 Electron Transport . . . . . . . . . . . 34
1.4.1 Classical Vlasov-Poisson .... 35
1.4.2 Semi-Classical Vlasov-Poisson . 37
1.4.3 Semi-Classical Boltzmann Equation 41
1.4.4 Equilibrium........ 43
1.4.5 Hydrodynamic Equations 44
1.4.6 Electrons and Holes . 47
1.4.7 Boundary Conditions 47
1.5 Summary . . . . . . . . . . . 48
2 Modeling of Gas Flow 49
2.1 Typical Reactors .. 51
2.2 Hydrodynamic Equations 52
2.2.1 Balance Equations 53
2.2.2 Generalized Forces 55
2.3 Near Hydrodynamic Flows. 58
2.3.1 Estimates of Transport Coefficients. 59
2.3.2 Slip Boundary Conditions 61
2.4 Transition Regime Flows ..... 62
2.4.1 Scattering Angle Models. 63
2.4.2 Boundary Conditions 71
2.5 Free Molecular Flow ... 73
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8 CONTENTS
3 NUlnerical Methods for Rarefied Gas Dymunics 77
3.1 Direct Simulation Monte Carlo 79
3.1.1 Spatial Discretization 80
3.1.2 Time Step .... 81
3.1.3 Collision Step . . 82
3.1.4 Convection Step 88
3.1.5 Boundaries 89
3.2 Computing Results ... 91
3.3 Extensions. . . . . . . . 93
3.3.1 Multiple Species 93
3.3.2 Axisymmetric Flow. 94
3.3.3 Gas Phase Chemical Reactions 96
3.4 Simplified Flows ....... 97
3.4.1 Test Particle Method . 97
3.4.2 Free Molecular Flow 99
4 Gas Transport Simulations 101
4.1 Near Hydrodynamic Effects ........ · 102
4.1.1 Determination of Slip Coefficients .102
4.1.2 The Diffusion Coefficient · 111
4.2 UHV- CVD Reactor . . . . . · 114
4.3 Sputtering Reactors ..... · 122
4.3.1 The Reactor Model. . .124
4.3.2 Velocity Distributions · 127
4.3.3 Collimated Sputtering · 136
5 Modeling of Radiative Heat Transfer 145
5.1 Rapid Thermal Processing . . . . . . . · 145
5.2 Semi-transparent Materials ...... .146
5.2.1 The McMahon Approximation .147
5.2.2 Example: Single Coating Layer · 151
5.3 Optical Properties of Surfaces . . · 155
5.4 Solution of the Integral Equation · 157
5.4.1 The Series Solution. · 157
5.4.2 Specular Reflection . · 163
5.4.3 Diffuse Reflection . · 165
5.5 The Rendering Equation . . · 167
6 Monte Carlo for Radiation Transport 175
6.1 Monte Carlo Solution Procedure .175
6.2 Quasi-Monte Carlo Methods ... · 180
6.2.1 Discrepancy . . . . . . . . · 182
6.2.2 Quasi-Random Sequences · 182
6.2.3 Integration and Discrepancy . .184
CONTENTS 9
6.3 Coupling Radiation and Gas Transport . 187
7 Radiation Transport Shnulations 189
7.1 Case Study of an RTP Reactor . . . . . . . . . . . . . . 189
7.1.1 A Simplified Radiative Heat Transfer Reactor. . 190
7.1.2 Numerical Experiment . 191
7.1.3 Numerical Results . . . . 193
7.1.4 Conclusions....... .200
7.2 Scalar Control RTCVD-reactor . 200
7.3 Multi-variable Control RTCVD-reactor . . 204
8 Modeling of Charge Transport 209
8.1 Numerical Methods for Linear Transport. . 211
8.1.1 Event Based Monte Carlo . . 211
8.1.2 History Based Monte Carlo . 213
8.2 Further Comparisons. . . . . . . . . 213
8.2.1 Distribution Functions. . . . 213
8.2.2 Mass and Charge Currents . 216
A Monte Carlo Methods 219
References 225
Subject Index 232
Chapter 1
The Boltzmann Equation
Systems consisting of a large number of particles are often most easily de
scribed by a probability density function f(x,p, t), which gives the proba
bility of finding a particle (molecule, photon, electron, etc.) at a position
x and with momentum p. Macroscopic quantities such as mass density and
temperature are found as averages over momentum space of the function f.
It is then of interest to determine an equation for the time and space evolu
tion of such distribution functions. When the evolution includes effects due
to particle collisions, such an equation is called a Boltzmann equation. In
this chapter we discuss the Boltzmann equation for rarefied gases, radia
tion and electrons, including a sketch of the derivation and the underlying
assumptions.
1.1 The Liouville Equation
The starting point for the derivation of the Boltzmann equation is the Li
ouville equation, which describes the evolution of the probability density
function for finding a system of particles at a given point in phase space.
Therefore we give here a physical derivation of the Liouville equation start
ing from the equations of individual particle dymanics. Depending on the
nature of the particle interactions and forces acting on the particles, one
obtains different equations describing the evolution of a dilute gas or of
electrons in a solid. We start by considering a system of N particles with
positions and momenta (Xi,Pi) E D x R3 for i = 1 .. . N, where D is a
subset of R3 describing position space. Given the form of the force acting
on each. particle, in principle it would be possible to determine the state of
the system at any time after initial conditions are specified. This is excep
tionally impractical, however, as the number of particles N is on the order
of 1020. Moreover, we are often only interested in macroscopic properties of
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