of the material surface is described by a Hamilton-Jacobi equation. Nonlocal e ects such as ...... a curve stores pointers to its boundary nodes. A curve also has ...
A Three-Dimensional Front Tracking Algorithm for Etching and Deposition Processes A Dissertation Presented by Glenn William VanDerWoude to The Graduate School in Partial Ful llment of the Requirements for the Degree of Doctor of Philosophy in Applied Mathematics and Statistics State University of New York at Stony Brook May, 2000
State University of New York at Stony Brook The Graduate School Glenn William VanDerWoude We, the dissertation committee for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of this dissertation. Prof. James G. Glimm, Dissertation Director Department of Applied Mathematics and Statistics Prof. Folkert M. Tangerman, Chairman of Thesis Committee Department of Applied Mathematics and Statistics Prof. W. Brent Lindquist, Member Department of Applied Mathematics and Statistics Dr. Steven M. Rossnagel, Outside Member IBM Thomas J. Watson Research Center, Yorktown Heights, New York This dissertation is accepted by the Graduate School.
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Abstract of the Dissertation A Three-Dimensional Front Tracking Algorithm for Etching and Deposition Processes by Glenn William VanDerWoude Doctor of Philosophy in Applied Mathematics and Statistics State University of New York at Stony Brook 2000
A three-dimensional front tracking algorithm is presented for the study of etching and deposition in the context of semiconductor manufacturing. The evolution of the material surface is described by a Hamilton-Jacobi equation. Nonlocal e ects such as resputtering and visibility require that the model be extended to include the dependency of the Hamiltonian on the global surface shape. The solution algorithm achieves sharp resolution in the evolution of surface edges and corners. The method is applied to a variety of etching and deposition processes. Examples showing the e ects of resputtering and redeposition are also iii
given. The numerical results agree with other physical and numerical experiments, including the development of three-dimensional features. The front tracking algorithm has been modi ed for use on parallel architectures. The communication of the front has been extended to include interfaces with tracked edges and corners in the surface. Routines have been introduced to resolve topological changes to the interface that occur during communication.
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in memory of my father with thanks to my mother to the glory of God
Table of Contents List of Illustrations
ix
Acknowledgments
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1 Introduction
1
2 Modeling Surface Evolution in Etching and Deposition
8
2.1 2.2 2.3 2.4
The Hamilton-Jacobi Equation . . . . . . . . . . . . . . . . . . . Viscosity Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . Local Flux Functions . . . . . . . . . . . . . . . . . . . . . . . . . Global E ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Resputtering and Redeposition Models . . . . . . . . . . . 2.4.2 View factors . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Sticking probability . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Extension of the view factor operator to edges and corners
3 A Front Tracking Algorithm for Hamilton-Jacobi Equations
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3.1 The Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 vi
3.1.1 Storage of topological information . 3.1.2 Interface retriangulation . . . . . . 3.1.3 Untangling the interface . . . . . . 3.2 Propagation of the Interface . . . . . . . . 3.2.1 Surface propagation . . . . . . . . . 3.2.2 Curve propagation . . . . . . . . . 3.2.3 Node propagation . . . . . . . . . . 3.2.4 Visibility . . . . . . . . . . . . . . .
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4 Numerical Simulations
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4.1 Isotropic Deposition . . . . . . . . . . . . . . . . . . . . . 4.2 Ion Etching . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Ion etching of oxide . . . . . . . . . . . . . . . . . . 4.2.2 Comparison to other triangulated-mesh schemes . . 4.3 Physical Vapor Deposition . . . . . . . . . . . . . . . . . . 4.3.1 Simulation of physical vapor deposition . . . . . . . 4.3.2 Collimated physical vapor deposition . . . . . . . . 4.3.3 Ionized physical vapor deposition . . . . . . . . . . 4.4 Resputtering . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Ionized physical vapor deposition with resputtering 4.4.2 Sticking probability . . . . . . . . . . . . . . . . . .
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5 Parallelization of the Interface for Three-dimensional Front Tracking 64 vii
5.1 Domain Decomposition . . . . . . . . . . . . 5.2 Communication of the Interface . . . . . . . 5.2.1 Removal and creation of bu er zones 5.2.2 Attaching the bu er to the interface 5.3 State Information at the Interface . . . . . .
References
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List of Figures 2.1 The local representation of a concave edge at a point X is shown in (a), with one of the superderivatives at X and the associated ambient hyperplane. The local representation of a convex edge is shown in (b), with a subderivative and its material hyperplane. . . . . . . . . . . . . 2.2 The local representation of a concave triple-point corner. Three planar regions with normals N ; N ; N intersect in a corner X . Each vector N = a N + a N + a N ; ai 0 is parallel to a superderivative. 2.3 Isotropic etching: (a) the etching rate C () and (b) the associated Hamiltonian H^ (P ); here P = jr ^(X^ )j. . . . . . . . . . . . . . . . . 2.4 Low-temperature etching of silicon oxide by CHF3 ions: cross-sections through the X ? Z plane for (a) the etching rate C () and (b) the associated Hamiltonian H^ (P ); P = jr ^j [72, 38]. Figure (c) shows the 3D view of H^ (r ^). . . . . . . . . . . . . . . . . . . . . . . . . . 0
0
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4.1 The Hamiltonian H (r (X )) = 1 + r (X ) for isotropic deposition with a deposition rate of 1. . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 Isotropic deposition into a trench. . . . . . . . . . . . . . . . . . . . . 43 ix
4.3 The Hamiltonian H (r ) for the low-temperature etching of silicon oxide by CHF3 ions, as given by (4.1). The nonconvex nature of the Hamiltonian can lead to faceting of the surface. . . . . . . . . . . . . 4.4 Ion etching of a rectangular trench with length:width:depth ratio of 6:3:5 according to the nonconvex Hamiltonian in (4.1). The process simulated is low-temperature etching of silicon oxide by CHF3 ions. . 4.5 The Hamiltonian H = cos? ()(4 cos () ? 5 cos()) for the ion etching experiment discussed in Section 4.2.2. . . . . . . . . . . . . . . . . 4.6 Ion etching of a structure according to the normal velocity in (4.2). The initial interface is a tower structure with a trench in its top surface. As the structure is etched away, the walls develop smooth sloping surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Physical vapor deposition into a trench according to (4.3): (a) the upper regions of the trench receive more material, creating overhangs, while (b) the corners lag behind the overhangs, which grow into each other to form creases at the corners. . . . . . . . . . . . . . . . . . . 4.8 The e ect of aspect ratio on physical vapor deposition. Three trenches of di erent widths are propagated for the same amount of time. The higher the aspect ratio of the trench, the less coating the walls and
oor will receive. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Collimated physical vapor deposition. Cross-sections of PVD into a trench with collimation angles of 52 ; 14, and 1 . The lower collimation angles produce more deposition on the oor and thinner overhangs. 1
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4.10 I-PVD with increasing ion:neutral ux ratios. (a) Cross-sections of an initial trench and the results of I-PVD with ion:neutral ratios of 1:2, 1:1, 2:1, 3:1, and 4:1; the higher ratios lead to smaller overhangs and greater lling of the trench. (b) Overhead view of the trench after I-PVD with a 4:1 ion:neutral ratio. . . . . . . . . . . . . . . . . . . . 4.11 The etching rate for Al [53], used to determine the sputtering yield during ion deposition [39]. The function is normalized so that Y (0) = 1. 4.12 Ion deposition of Al with resputtering (Y (0) = 0:8) into a trench with hidden regions. Resputtering of material inside the structure causes redeposition underneath the overhangs. The process is a mixture of both etching and deposition. . . . . . . . . . . . . . . . . . . . . . . . 4.13 Ionized PVD with resputtering: (a) a trench is subjected to I-PVD with conditions as in Figure 4.10, except that resputtering and redeposition are now considered. (b) The redeposition results in a more complete coating along the walls and lower edges of the trench. . . . . 4.14 Ionized PVD with resputtering and variable sticking probability. The sticking probability for the three runs is 1.0, 0.75, and 0.5. Lower sticking probability provides less deposition in the lower regions of the trench. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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56 59
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5.1 A global domain is divided into nine computational subdomains. The center computational subdomain is shaded dark. The corresponding virtual subdomain, containing a bu er region extending into the adjacent subdomains, is shaded light. . . . . . . . . . . . . . . . . . . . 5.2 (a) The interface as restricted to a given computational subdomain, with the four neighboring bu ers to be attached to complete the virtual subdomain. (b) The local interface for the given processor after the bu ers are attached. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Removal of the sections of the curve outside the computational subdomain (dotted lines) will result in four separate, disconnected curves (solid lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Removing the bu er zone from the interface can lead to topological changes: an interface contains a closed curve in (a) which after removal of bu er triangles and bonds becomes an open curve with boundary nodes (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 The interface in (a) contains a curve that crosses the subdomain boundary multiple times. The removal of bu er triangles and bonds splits the curve into disjoint sections. Each section becomes a separate curve with boundary nodes. . . . . . . . . . . . . . . . . . . . . . . . . 5.6 The inverse of the operation in Figure 5.4: two overlapping curves are merged together to form one closed curve. . . . . . . . . . . . . . 5.7 The inverse of the operation in Figure 5.5: several overlapping curves are merged together to form one single curve. . . . . . . . . . . . . . . xii
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72 74 75
5.8 The bu er information in (a) is to be attached to the local interface. The two surfaces forming the upper plateaus in the local interface will both be attached to the corresponding surface in the bu er. The three surfaces are merged together to form one surface in the new interface (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.9 A high-density polytropic gas jet injected into an ambient environment: the parallelization of this experiment requires curve splitting and merging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
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Acknowledgments Along the road to the completion of this work, there have been so many who have aided this traveler on the way. I am grateful for them all. First, I would like to thank Folkert Tangerman. His advice and guidance, given with insight and patience, have been invaluable. His con dence in me has meant even more. I am proud to have worked with him. I wish to thank James Glimm for his support on this journey. He has given good counsel, scienti c insight, and direction, plus the appropriate push. Steve Rossnagel, Satoshi Hamaguchi, and David Ross supplied not only most of the physical models used in this work, but they provided helpful discussions, as well. On the programming side, Dechun Tan did the hard work of introducing the Hamilton-Jacobi physics into FronTier, as well as much more. I am so grateful for my parents and their love and encouragement throughout my life. There is so much so say, but in particular, I hold dearly in my heart one discussion with each of them where these qualities shone brightly to help me nd my way to this point. Arun Venkatarangan has been my close friend, with whom I have walked stride
for stride down this path. If I am nishing slightly before him, it is only because on that last step I have a bigger foot. Next, the acronyms: TVC, CCC, IVG, AMS. Your friendship and partnership in so many areas, in so many ways, have been a great joy during this trip. You have opened up your hearts, homes, and lives. I hope you all know how special you are to me. Finally, I thank the God of grace who has brought me here and who will lead me on. I, too, have been evolving during this process, hopefully deeper than just the surface. I trust that these studies have taught me something of that grace, and I look forward to knowing more.
Chapter 1 Introduction Understanding the evolution of a surface undergoing an etching or deposition process is important for ecient manufacturing of semiconductor chips. Advances in semiconductor technology have led not only to marked decreases in the size of features, but also to changes in the shape of these features. New and more complex processes have been introduced to the industry to further improve production. Flexible, accurate simulations provide an important means of analyzing the e ect of such innovations on surface morphology. The purpose of this work is to present a three-dimensional front tracking algorithm for surface evolution of etching and deposition problems. The system is modeled by a Hamilton-Jacobi equation, with the Hamiltonian chosen to simulate di erent physical processes. This algorithm builds on three pillars: the mathematical theory of viscosity solutions of Hamilton-Jacobi equations, robust front tracking algorithms for other applications in two and three dimensions, and the characteri1
zation and modeling of di erent production methods and materials used in etching and deposition processes. Viscosity solutions of Hamilton-Jacobi equations were introduced by Crandall and Lions [12, 11, 57]. In general, solutions of Hamilton-Jacobi equations have singularities and are nonunique. A viscosity solution satis es speci ed inequalities, known as entropy or admissibility conditions, at singularities in order to select the correct physical solution. These admissibility conditions are the generalized form of the entropy conditions for scalar conservation laws presented by Olenik [61, 62]. Entropy conditions for Hamilton-Jacobi equations have previously been discussed by Hopf [10, 43, 44]. The term viscosity solution comes from the method of \vanishing viscosity" used to nd solutions in these earlier works. Much work has been done to develop nite di erence schemes that converge to viscosity solutions. See, for instance, the work of Osher for scalar conservation laws [59, 63] and HamiltonJacobi equations [64], as well as the work of Crandall and Lions and Souganidis [13, 83, 58]. Chapter 2 will discuss the Hamilton-Jacobi model and admissibility conditions that are used in the etching and deposition simulations in this work, including the extension of the model to include global e ects such as redeposition. Many approaches have been taken for the three-dimensional simulation of surface evolution for etching and deposition. Lagrangian methods represent the boundary between distinct volumes by a discrete mesh of points. The numerical algorithms then trace the motion of the boundary points. An example of a Lagrangian method for etching and deposition is the ray-trace algorithm [87]. The ray-trace method 2
advances each point of the initial grid independently. This method is easy to implement, but the lack of connectivity between the rays makes reconstruction of the nal interface dicult. More importantly, the mesh cannot be re ned during the simulation, so some regions may never be reached by the initial choice of rays. Marker-string methods address these issues by representing the material interface with a mobile grid, with mesh points connected to each other by segments, typically quadrilateral blocks or a triangulated surface. Initial three-dimensional marker-string approaches modeled the evolving surface using a Hamilton-Jacobi equation and solving it using the method of characteristics [80]. Propagation of grid points is based on characteristic lines. When characteristics intersect, unphysical regions of negative volume form, which must be removed, forming edges in the surface. Removing these loops requires robust mesh maintenance algorithms. In addition, correct formation and propagation of edges requires the application of entropy conditions to the Hamiltonian. Neureuther has presented a marker-string algorithm that enforces the correct entropy conditions, with application to a number of physical problems [86, 87, 76, 41, 50]. Katardjiev's variation on the method of characteristics creates the convex envelope of the possible characteristics emerging from each point [51, 52]. This method is able to accurately handle edge formation such as faceting of surfaces during ion etching, without requiring delooping. Lagrangian-type methods have also been used by Cale and Raupp for line-of-sight models for etching and deposition [56]. Eulerian-type algorithms, such as cell-based methods and level set methods, which use a xed coordinate system for the computational grid, have also been 3
applied to etching and deposition [45]. Cell-based methods discretize the domain into regular blocks which are assigned a parameter representing the volume fraction of material in the cell [49, 41]. Cell methods avoid delooping problems and thus bene t from ease of implementation. They have been found, however, to introduce directional bias to the solution [41]. Calculation of the surface gradient can also be inaccurate [78], which hinders the use of cell methods for ux functions dependent on surface orientation. Level set methods for interface evolution, proposed by Osher and Sethian [65], have also been applied to etching and deposition problems [77, 78, 4, 41, 45]. An upwind nite-di erence scheme is used to solve a Hamilton-Jacobi equation over the whole domain. The material surface is then extracted as the zero level set of that function. In order to satisfy entropy conditions, conservation law techniques are included in the di erence approximations. Level set methods have shown particular bene t for modeling curvature-driven ows. Topological changes in three-dimensions have been demonstrated for simulations of etching [1] and deposition [2]. Ross studied ion etching by applying a shock capturing algorithm to a twodimensional conservation law derived from the Hamilton-Jacobi equation [72, 71, 73]. Hamaguchi et al. presented a shock tracking algorithm to solve two-dimensional Hamilton-Jacobi equations in this context [38, 36, 37]. The surface was represented by a discretized curve which was propagated by solving Riemann problems with proper admissibility conditions at each point. Hamaguchi and Rossnagel extended this algorithm to study a variety of etching and deposition problems, including multiple deposition sources and redeposition of etched material, with favorable compar4
ison to physical experiments [39, 40]. This work applies the front tracking code FronTier, developed at New York University and the University of Stony Brook, to etching and deposition problems [9, 21, 25, 26, 30]. Front tracking is a mixed Lagrangian and Eulerian numerical method that represents singularities in a piecewise smooth solution by a lowerdimensional adaptive grid, called the interface [35]. The interface is composed of smooth manifolds associated with physical waves in the system. For instance, in R , material interfaces of codimension 1 are represented by smooth surfaces, while a shock wave or edge of codimension 2 is represented by a smooth curve. The interface elements are then used to track the evolution of the singularities. Instead of applying general shock resolution algorithms to every point, the propagation algorithms make use of the codimension information to apply routines of appropriate complexity. This approach allows for coarse grids while providing accurate solutions to problems with important singularities. Front tracking methods have been used for a variety of elds, including gas dynamics [9, 35, 6], porous-media uid ow [28, 82], and elastic-plastic deformation in solid dynamics [88, 89]. The ability to represent material interfaces and sharp features make front tracking an appropriate tool to study geometric development in etching and deposition processes. For three-dimensional front tracking, the discontinuity is represented by a collection of triangulated surfaces, curves, and nodes called the interface. Front tracking studies have been performed for three-dimensional studies in compressible gas dynamics with smooth interfaces [18, 22, 19]. The introduction of sharp features (edges and corners) into the interface brings new geometric and mathematical 3
5
complexities. The inherent nature of surface evolution as modeled by a HamiltonJacobi equation provides a suitable platform for developing front tracking interface techniques and was an additional motivating factor in extending the front tracking method to this problem. The behavior of corners in an surface evolving according to a Hamilton-Jacobi equation leads to two-dimensional Riemann problems. A general theory for higherdimensional Riemann problems in Hamilton-Jacobi equations has been presented, including constructions for many two-dimensional Riemann problems [27]. These solutions provide the basis for the front tracking algorithms implemented in this method [33]. Chapter 3 will present a three-dimensional front tracking algorithm, modifying the FronTier package to study etching and deposition processes. The etching and deposition algorithms for FronTier are applied to a series of problems chosen to test and demonstrate di erent aspects of the model. Reference is made to physical experiments and other numerical experiments, both two- and threedimensional, to provide comparison for the algorithm and establish validity. Ion etching [53, 72, 38] is considered for both theoretical and experimental ux functions. Physical vapor deposition (PVD) [74] is studied, including the e ects of aspect ratio [39] and collimation angle [75]. PVD has a broad angular distribution [74], which leads to crease formation in the corners, a particular three-dimensional e ect [3]. Also simulated is ionized PVD [74, 39], which provides a mixed deposition stream of ionized and neutral particles. Examples demonstrating the global scattering e ects of resputtering and redeposition [33, 39, 60, 2] are also given for di erent problems. 6
Chapter 4 will present and discuss the results of these numerical experiments. To improve computational capability and performance, parallel computing methods have been implemented in three-dimensional front tracking algorithms [22]. Each processor is assigned a subset of the domain. For hyperbolic problems, information from the neighboring subdomains must be accessed in order to correctly propagate the local information [17]. Thus, a bu er strip from each neighboring subdomain is attached to the local interface. The updating of the information in the bu er zones is accomplished using a scatter-gather process [46]. The operations involved in this update may result in topological changes to the local interface. Parallel routines for scatter-gather in FronTier are presented here, which accomplish the bu er update for more general interfaces, including edges and nodes, and associated topological changes. These new parallel algorithms, developed in the context of solving Hamilton-Jacobi equations, are not problem speci c, and open up new areas for front tracking simulation. Chapter 5 will present the parallel algorithms for front tracking interfaces.
7
Chapter 2 Modeling Surface Evolution in Etching and Deposition We study the evolution of a material surface during an etching and/or deposition process, making the following assumptions:
we look at the system at a macroscopic level, treating the material surface as a continuum;
the material surface is piecewise smooth; all dynamics occur at the material surface; surface di usion and ow are negligible. Let C (X ); X 2 R , denote the velocity in the normal direction at a point X on the surface. C represents the rate function, either the deposition rate or the etching 3
8
rate. To avoid having to specify deposition or etching, C will often be referred to as the normal velocity. We choose the orientation of the normal as pointing from the material into the ambient environment. Thus, positive C (X ) indicates deposition, while negative C (X ) indicates etching. We allow for the following dependencies in the normal velocity:
Model 1 (general model). The normal velocity may depend on position, (for example, photoresist etching [15, 41]); local surface gradient , (ion etching [72]); and global surface shape (resputtering/visibility [39]).
2.1 The Hamilton-Jacobi Equation Let the material surface S at time t be represented by the hypersurface
(X; t) = 0; : R n R ! R : +
If the surface velocity has the dependencies given by Model 1, then the equation of motion for S is
t (X; t) + H (X; ; r (X )) = 0; with the initial condition
(X; 0) = (X ): 0
9
(2.1)
The ux function, or Hamiltonian, H is related to the normal velocity C by r (X ) )jr (X )j: (2.2) H (X; (X ); r (X )) = C (X; (X ); jr (X )j Equation (2.1) provides a framework for studying surface evolution driven by both local and global factors. Global e ects such as redeposition of etched material will be addressed in section 2.4. A local model for a variety of etching and deposition processes requires an additional restriction on the form of the normal velocity.
Model 2 (local model). The normal velocity at a point X depends solely on the position of X and the unit surface normal at X , i.e., C = C (X; jrr ((XX ))j ).
With this simpli cation, the equation of motion (2.1) takes the form of a Hamilton-Jacobi equation:
t (X; t) + H (X; r (X; t)) = 0:
(2.3)
2.2 Viscosity Solutions A large body of work exists for solutions to Hamilton-Jacobi equations. The method of characteristics may be used to solve (2.3) for C initial data [48]. This method will provide a unique solution up until the formation of singularities at the intersection of characteristics [54, 81], at which time the solution is no longer unique. Therefore, one must impose additional conditions to select the physically relevant solution. Crandall and Lions introduced the notion of viscosity solutions for HamiltonJacobi equations [11, 12, 57]. Key aspects of viscosity solutions are listed below: 1
10
A viscosity solution solves (2.3) at points of di erentiability. A viscosity solution satis es the inequalities found below in De nition 2 at points of non-di erentiability.
For general conditions (e.g., H continuous and (X; 0) uniformly continuous), there exists a unique viscosity solution (among potentially many non-viscosity solutions).
The solution provided by the \vanishing viscosity" method, that is, adding a di usion term (X; t) to (2.3) and nding the limit of the solutions as goes to 0, is a viscosity solution [11]. A viscosity solution must satisfy appropriate admissibility conditions at all points of non-di erentiability. These admissibility conditions are based on the notion of subderivatives and superderivatives [11, 27].
De nition 1. Consider a continuous function (X; t). A vector P = (Px; Pt) is a superderivative of at a point (X ; t ) if lim sup (X; t) ? (X ; t ) ? Px (X ? X ) ? Pt(t ? t ) 0 jj(X; t) ? (X ; t )jj X;t ! X ;t and a subderivative if (X; t) ? (X ; t ) ? Px (X ? X ) ? Pt (t ? t ) 0: lim inf X;t ! X ;t jj(X; t) ? (X ; t )jj 0
0
(
)
(
)
(
0
0
0 0)
0
(
0
0
0
0
0
0
0
0 0)
0
0
The set of subderivatives and the set of superderivatives at a point are closed and convex. The set of subderivatives and the set of superderivatives are both 11
nonempty if and only if is di erentiable at X , in which case each set will contain only the gradient vector. Otherwise, either the set of subderivatives or the set of superderivatives, or both sets, will be empty [27].
De nition 2 (Admissibility Conditions). 1. A continuous function (X; t) is a viscosity solution of the Hamilton-Jacobi equation (2.3) if for all points (X0 ; t0) it satis es the following (\admissibility") conditions:
for every superderivative (Px; Pt) of at (X ; t ) 0
0
Pt + H (Px) 0
(2.4)
and for every subderivative (Px; Pt) of at (X ; t ) 0
0
Pt + H (Px) 0:
(2.5)
2. If a continuous function (X; t) satis es conditions (2.4) and (2.5) at a point (X0; t0 ), we say that is admissible at (X0 ; t0).
If is di erentiable at (X ; t ), then is admissible at (X ; t ). If has no subderivatives or superderivatives at (X ; t ), the solution is admissible at that point. A more sophisticated discussion of the admissibility conditions for the Hamilton-Jacobi equation (2.3) in the context of etching and deposition is found in [27]. A simpler description is given here to demonstrate the application of the admissibility conditions in three-dimensional space to nondi erentiable features in a material 0
0
0
0
12
0
0
surface: edges and corners. It will be useful to view the hypersurface = 0 as a graph with respect to some direction z. In this case, the system may be described by a reduced Hamilton-Jacobi equation, with one less spatial variable. Let X^ be the orthogonal lower-dimensional coordinate. Then the hypersurface = 0 is a graph of a function ^ with ^ t) ? z: (X; t) = ^(X;
(2.6)
The transformation (2.6) was discussed in [27]. In particular, it was shown that ^ t) is a viscosity solution of ^(X; ^ r ^(X^ )) = 0 ^t + H^ (X;
(2.7)
^ r ^(X^ )) = H (X; r (X )) H^ (X;
(2.8)
with
if and only if (X; t) de ned by (2.6) is a viscosity solution of (2.3). Equation (2.7) is also a Hamilton-Jacobi equation. We can illustrate the subderivatives and superderivatives using a local model for the material surface S that represents the limit behavior of S at a point X in a piecewise linear fashion. Any edge in the surface entering a point X is represented by a line tangent to that edge at X . The smooth portions of the surface adjacent to X are then planar regions intersecting at X along the edges. (This is not a general model | for instance, a cone with a vertex at X cannot be represented in this manner). Each subderivative and superderivative correspond to a hyperplane that intersects S at X . 13
Subderivative Material Hyperplane
Superderivative
X
Ambient Hyperplane
X
Material Surface
Material Surface
(a) Concave edge
(b) Convex edge
Figure 2.1: The local representation of a concave edge at a point X is shown in (a), with one of the superderivatives at X and the associated ambient hyperplane. The local representation of a convex edge is shown in (b), with a subderivative and its material hyperplane.
De nition 3. An ambient (material) hyperplane for a point X on a surface S is a hyperplane passing through X that de nes an open half-space that within a neighborhood of X is entirely ambient (material) space.
Each ambient hyperplane has a normal which is a superderivative of the surface at X , and each material hyperplane has a normal which is a subderivative. Figure 2.1(a) shows one of the ambient hyperplanes and its superderivative for a point on an edge, while Figure 2.1(b) shows a material hyperplane and subderivative for a di erent edge. Distinguishing which admissibility condition must be tested requires a categorization of surface singularities based on the convexity of the material surface. 14
Recall that a subset of R n is said to be convex if for any two of its points, it also contains the line segment connecting these. A real-valued function de ned on a convex subset of R n is said to be convex if the region above its graph is convex, while it is said to be concave if the region below its graph is convex. We rst consider the case of a point on an edge in the material surface. By the local piecewise linear model, the interface at a point on an edge may be represented as two half-spaces intersecting in a line tangent to the edge at that point. We choose a coordinate system such that the surface is a graph within a neighborhood of the point, so that the convexity of the edge can be de ned.
De nition 4. An edge is convex at a point if locally the ambient space at the point is convex. An edge is concave if locally the material at the point is convex.
For example, the edges of a \material" cube are concave. The convex/concave categorization of singularity points at edges is complete for the local representation of an edge. (If the two half-spaces are parallel, the surface is di erentiable at the point.) Note that the set of ambient hyperplanes (superderivatives) is nonempty for a point on a concave edge, while the set of material hyperplanes (subderivatives) is nonempty for a point on a convex edge. Figure 2.1 gives an example for both a convex and a concave edge. Let X be a point on an edge. Let N and N be the unit normal vectors of the two half-spaces in the local representation of the interface at X . De ne N (X ) as the set of all vectors of the form a N + a N ; a ; a 0. Then if the edge is convex at X , any nonzero vector N 2 N (X ) is the positive scalar multiple of one of the 0
0
0
1
15
1
1
0
1
subderivatives at X . If the edge is concave, then for any nonzero vector N 2 N (X ), N is the positive scalar multiple of one of the superderivatives at X . The admissibility conditions for a point on an edge are also known as the Olenik conditions [61, 62], which can be expressed as follows:
a solution is admissible at a point X on a convex edge if 8ai 0 H (X; a N + a N ) a H (X; N ) + a H (X; N ); 0
0
1
1
0
0
1
1
(2.9)
a solution is admissible at a point X on concave edge if 8ai 0 H (X; a N + a N ) a H (X; N ) + a H (X; N ): 0
0
1
1
0
0
1
1
(2.10)
If an initial condition contains an inadmissible edge, that edge in the viscosity solution will bifurcate into a combination of admissible edges and smooth surfaces. The application of the admissibility conditions to corners will be given here for \triple-point" corners | those that are formed by the intersection of three surfaces. The two-dimensional Riemann problems that occur at more complex corners are discussed in [27]. A corner formed by intersection of three smooth surfaces is represented as three planar regions intersecting at the corner, with pairwise intersections between the planar regions occurring along lines tangent at the corner to the edges running into the corner. Again, choose a coordinate system such that the surface at the corner is a graph. Figure 2.2 gives an example of a triple-point corner.
De nition 5. A corner is convex if locally the ambient space is convex at the corner. A corner is concave if locally the material is convex at the corner.
16
N N0 X N2
N1
Figure 2.2: The local representation of a concave triple-point corner. Three planar regions with normals N ; N ; N intersect in a corner X . Each vector N = a N + a N + a N ; ai 0 is parallel to a superderivative. 0
1
1
2
1
2
0
0
2
Convexity, respectively concavity, of a corner, requires that all edges running into the corner are convex, respectively concave. For example, the corners of a \material" cube are concave. Corners which are neither convex nor concave are possible, such as a surface with saddle behavior at a corner. The corner formed at the origin of the surface z = jxj ? jyj is neither convex nor concave. Let X be a convex or concave corner at which three smooth regions intersect. Let N ; N ; N be the unit vector normals of the three planar regions in the local model. De ne N (X ) as the set of vectors of the form a N + a N + a N ; ai 0. For a convex corner, any nonzero vector N 2 N (X ) is the positive scalar multiple of one of the subderivatives at X . If the corner is concave, then for any nonzero vector N 2 N (X ), N is the positive scalar multiple of one of the superderivatives at X . An example for a concave triple-point corner is shown in Figure 2.2. The admissibility conditions for convex and concave corners are the higher0
1
2
0
17
0
1
1
2
2
dimensional generalization of the Olenik conditions [27]:
the admissibility test for a corner rst requires that the edges intersecting at the corner are themselves admissible; then
a solution is admissible at a convex corner X if 8ai 0 H (X; a N + a N + a N ) a H (X; N ) + a H (X; N ) + a H (X; N ); 0
0
1
1
2
2
0
0
1
1
2
2
(2.11)
a solution is admissible at a concave corner X if 8ai 0 H (X; a N + a N + a N ) a H (X; N ) + a H (X; N ) + a H (X; N ): 0
0
1
1
2
2
0
0
1
1
2
2
(2.12) Note that if N (X ) is empty, the solution at X is admissible. If an initial condition contains an inadmissible corner, that corner in the viscosity solution will bifurcate into a combination of admissible corners, admissible edges, and smooth surfaces.
2.3 Local Flux Functions Di erent physical processes are simulated by choosing the appropriate Hamiltonian, or ux function in the model. Chemical etching baths are modeled by a position-dependent Hamiltonian H (X ) [78, 41]. Many processes that have a directional source ray, such as ion etching or ion-based deposition, can be modeled as a gradient-dependent Hamiltonian H (r ). The rest of this section will discuss the application of the admissibility conditions to Hamiltonians of this gradient-dependent form. 18
A model is commonly de ned as a two-dimensional ux function F (r ^(X^ )) which assumes the surface is a graph. The Hamiltonian H^ (r ^(X^ )) is obtained by rotating F around the vertical axis, as in Figure 2.4(c). We rst demonstrate the application of admissibility conditions (2.9) and (2.10) to edges in the surface. Good discussions on the application of admissibility conditions can be found in [62, 81, 54]. We consider a convex edge, with the two halfplanes intersecting at the edge having gradient vectors P and P . Equation (2.9) states that if the chord between H^ (P ) and H^ (P ) is greater than H^ (P ); P 2 P P , then ^ is admissible. If the chord is less than H^ (P ) at any vector P 2 P P , then ^ is inadmissible. In this case, the admissible solution can be found by creating the upper convex hull of H^ (P ) between P and P . A straight line segment raised above the graph of H^ in the convex hull will be an edge in the surface (a shock). The gradients of the two surfaces at the edge is de ned by the values of P at the endpoints of the straight segment. The portions of the convex hull that trace H^ will result in smooth rounded regions of the surface (rarefaction waves). A greater number of in ection points in H^ leads to a greater number of alternating shocks and rarefaction waves. Similarly, equation (2.10) is applied to concave edges by creating the lower convex hull. The application of the admissibility conditions (2.11) and (2.12) to corners in the surface requires the solution of two-dimensional Riemann problems as presented in [27]. Let X be a convex (concave) triple-point corner: three planar surfaces of gradients P ; P ; P intersecting at X , with the three edges between the surfaces all convex (concave). The application of the admissibility condition to the corner states 0
0
1
0
1
0
0
0
1
2
19
1
1
1
that the triangle between the points H^ (P ); H^ (P ), and H^ (P ) must be above (below) the graph of H^ (P ) for all P in the triangle P P P . Otherwise, the admissible solution will be a combination of corners, edges, and smooth surfaces. Examples of solutions to two-dimensional Riemann problems for Hamilton-Jacobi equations were discussed in [67, 68, 66]. Flux functions in the literature are often given in terms of a cross-section of H^ (r ^) on the lower-dimensional coordinate system X^ de ned by (2.6) [39, 72, 77]. In such cases, the Hamiltonian H^ for (2.7) is a rotationally symmetric function of the given cross-section such as seen in Figure 2.4(c). The ux function in many etching and deposition processes is dependent upon the orientation of the surface with respect to the source. In isotropic processes, however, there is no dependence on direction. The normal velocity in this case is simply a constant. An isotropic etching rate and corresponding Hamiltonian H^ are shown in Figure 2.3. Some chemical etching processes, as well as some low-pressure chemical vapor deposition systems, exhibit isotropic behavior [8, 45, 55, 70]. Unidirectional deposition also leads to a simple model. We assume that the deposition streams come from a planar source with directions parallel to each other. We also assume that the particles will stick where they initially strike the surface. The normal velocity C is then proportional to cos(), where is the angle between the incoming ux direction and the normal of the surface at the point of impact. For ion etching, the etching rate C can be signi cantly more complex. For many materials, C is nonconvex, as the erosion of material is very sensitive to the angle of impact. As the number of in ection points in the resulting Hamiltonian increases, 0
1
2
0
20
1
2
0.00
0.00
-0.50
H
etching rate C
-2.00
-1.00
-4.00 -1.50
-6.00 -2.00 -50
0
50
-5
Degrees
0
5
P
(a) Etching rate
(b) Hamiltonian
Figure 2.3: Isotropic etching: (a) the etching rate C () and (b) the associated Hamiltonian H^ (P ); here P = jr ^(X^ )j. so does the amount of faceting (formation of edges) that will occur during etching. The creation of edges is described by the application of the admissibility conditions to the Hamiltonian. The etching rate function varies according to the nature of the material: crystalline materials may have etching rate functions with many peaks and valleys [73]. Amorphous materials often have rates with four in ection points (a \double-hump" function) [73], such as the one seen in Figure 2.4, which shows the etching rate C and Hamiltonian H^ for the low-temperature etching of silicon oxide by CHF3 ions [72, 38].
21
1.0 1.2
C
H
1.0
0.5
0.8
0.6
-5
0
5
-5
0
Degrees
5
P
(a) Etching rate
(b) Hamiltonian
–0.8 H –1 –1.2 –3
–2
–1
y0
1
2
3
3
2
1
0x
–1
–2
–3
(c) 3D Hamiltonian
Figure 2.4: Low-temperature etching of silicon oxide by CHF3 ions: cross-sections through the X ? Z plane for (a) the etching rate C () and (b) the associated Hamiltonian H^ (P ); P = jr ^j [72, 38]. Figure (c) shows the 3D view of H^ (r ^).
22
2.4 Global E ects In order to simulate global e ects such as visibility or redeposition, we must extend beyond the limits of Model 2. The ux function must be allowed to depend upon the shape of the surface, as well. We recall the general equation of motion (2.1)
@ (X; t) + H (X; (X ); r (X )) = 0; @t which has been referred to as a Hamilton-Jacobi \type" equation [2].
2.4.1 Resputtering and Redeposition Models In some deposition and etching processes, incoming material may in uence the evolution of the surface in areas other than the point of initial impact. Resputtering occurs in a deposition process when particles that have been sputtered from a source target are deposited onto the substrate with enough energy to cause ejection of material [39]. The ejected, or resputtered, material may be redeposited elsewhere on the surface. Similarly, material removed during an etching process may be redeposited on the surface [79]. Redeposition can also occur when deposited particles do not stick at the point of impact but bounce o the surface to land elsewhere. Redeposition becomes a new source of material for a point on the surface that depends on how that point is situated with respect to the rest of the surface. A more complete model must account for such nonlocal scattering of material. We incorporate the e ects of surface-to-surface deposition by including a visibility 23
model in the ux function. We begin by de ning this model for smooth regions and will later address visibility for edges and corners.
Model 3. We assume that the material ux C at a point X is of the following form [39]:
C (X ) = Cs(X ) + Cr (X ):
(2.13)
Here Cs(X ) is the material ux at X due to reception of material directly from the source, while Cr (X ) is the ux at X due to the redeposited material that has been scattered to X from elsewhere on the surface. The source ux Cs is determined by experimental or analytic means. Cs may be dependent on the position of X and the angle of impact, and depends on the global surface shape only in determining whether the source is visible from the point X . The function Cr , however, relates the ux at X to material coming from the rest of the surface. Thus, Cr has an innate dependence on the global shape of the surface.
2.4.2 View factors For many materials, sputtering is characterized by a uniform velocity distribution [39, 74]. This is often known as the \cosine" distribution, due to the cosine dependence in the resulting emission rate: the emission rate along a given vector v is equal to the emission rate in the normal direction times the cosine of the angle between v and the normal, for jj =2. We assume that all sputtering | whether the initial sputtering of a source target or the resputtering caused by the deposition 24
stream | results in a cosine distribution [39, 40]. View factors, used in radiative heat transfer, provide a means of analysis for the scattering of sputtered material. The view factor operator V [84] is de ned for an oriented interface M in the following manner: Let f be a surface density; for a point X in M where M is smooth, de ne
V f (X ) =
Z M
K (X; Y ) f (Y ) dA(Y ):
(2.14)
Here the integration dA(Y ) is with respect to surface area and the integral kernel K (X; Y ) is de ned as ) cos(Y ) ; K (X; Y ) = cos( dX(X; (2.15) Y) provided X and Y \are visible to each other", while K (X; Y ) = 0 otherwise. In this formula, d(X; Y ) denotes the distance between X and Y , and X (resp. Y ) denotes the angle of the line segment XY with the normal at X (resp. Y ). V has a geometric interpretation: Let X be a smooth point in M and M 0 a subsurface of M , visible from the point X . Then 2
Z M0
Z 1 K (X; Y )dA(Y ) =
p(M 0 )
cos()d ;
where p is the projection onto the hemisphere of directions at X pointing into the ambient space and denotes the solid angle on this hemisphere. We rst clarify the notion \visible to each other" through a separation into local and global surface properties. For points X and Y in the surface M , de ne a function (X; Y ) as 1, provided the line segment XY does not intersect M at an intermediary point, and 0 otherwise. Then for points X and Y to be \visible to 25
each other" we require that (X; Y ) = 1 and that N (X ), the surface normal at X , and N (Y ), the surface normal at Y , satisfy
N (X ) (Y ? X ) 0; N (Y ) (X ? Y ) 0: When the projection equals the entire hemisphere the d integral equals R exactly. Therefore M K (X; Y )dA(Y ) is less than or equal to 1. The view factor operator V has the following general properties: 0
1. V is positive: f 0 ) V f 0. 2. V 1 1, and V 1 = 1 if and only if M is closed. 3. V is selfadjoint on the space of L functions on M . 2
4. V f (X ) is continuous at points where the interface is smooth. We thus assume that the redeposited ux Cr is of the form
Cr (X ) = (V Ce)(X ):
(2.16)
Ce(Y ) denotes the ejected material ux from a point Y on the surface. (V Ce)(X ) then denotes the material deposited at X which it receives as ejected material from the rest of the surface. Given our de nition, Ce must be non-negative to be physically signi cant. The equation for the ux with scattering then takes the form
C (X ) = Cs(X ) + V Ce(X ): 26
(2.17)
2.4.3 Sticking probability We mention a particular class of problems using equation (2.17) that involve sticking probabilities. The sticking probability S is a phenomenological parameter: particles stick with probability S [37, 79]. The assumption that those particles which do not stick (1) scatter again with a cosine distribution and (2) stick at the next point they hit leads to the following model. Denote by Ct (X ) the total amount of material ux originating from the source and landing at the point X . Then Cs(X ) = S Ct(X ) and Ce(X ) = (1 ? S ) Ct(X ). Therefore, in terms of Ct,
Model 4. C (X ) = S Ct(X ) + (1 ? S ) V Ct (X ); An enhancement of this model is to allow arbitrarily many rebounds before sticking is guaranteed, which leads to a consistency relation between C and Ct :
Model 5. C (X ) = S Ct (X ) + (1 ? S )V C (X );
(2.18)
This model is equivalent to (I ? (1 ? S )V )C = SCt: This equation can then be solved directly or iteratively for C .
27
(2.19)
2.4.4 Extension of the view factor operator to edges and corners Since edges and corners have zero measure in the interface, they do not contribute to the integral involved in the view factor operator. However, the de nition of the view factor operator V must be extended so that we can evaluate V sensibly for points on edges and corners, where normals are not uniquely de ned. Let a denote maxfa; 0g for a real number a. We rewrite the integral kernel K in the de nition of V as +
K (X; Y ) = K (X; N (X ); Y ); N (Y ) (X ? Y )) K (X; N; Y ) = (X; Y ) (N (Y ? X))d((X; Y) = (N (X; Y ) (Y ? X )(Nd((YX;) Y ()X ? Y )) ) : +
(2.20)
+
4
+
4
+
(2.21)
In this formulation we assume that the interface is smooth at the point Y , so that N (Y ) is uniquely de ned as a function of Y and does not need to be introduced as an additional argument of K . De ne the view factor operator for arbitrary points X in the interface and arbitrary choice of the normal vector N as follows:
V f (X; N ) =
Z M
K (X; N; Y ) f (Y ) dA(Y ):
(2.22)
We will suppress the explicit N dependence when the interface is smooth near X and the normal is therefore uniquely de ned.
28
Chapter 3 A Front Tracking Algorithm for Hamilton-Jacobi Equations Front tracking is a numerical method designed to focus computational resources on areas of particular interest and complexity [9]. Two separate grids are used to represent the solution to a system of partial di erential equations. The rst grid is typically a xed nite di erence grid [35], though nite element meshes have been utilized, as well [28]. The second grid is a mobile lower-dimensional grid that tracks discontinuities and other singularities in the interface, such as shocks, material interfaces, and contact discontinuities. This work uses the FronTier front tracking package developed at New York University and the University of Stony Brook [9, 21, 25, 26, 35, 42, 6, 22, 20]. FronTier has been applied to problems in a variety of elds, including gas dynamics [9, 35, 6], porous-media uid ow [29, 28, 82], and solid dynamics [88, 89]. The front tracking algorithm for FronTier 29
was originally based on a method proposed by Richtmeyer and Morton [69] and has gone through many updates and improvements. A discussion of front tracking methods can be found in [47]. Three-dimensional front tracking algorithms have been developed for FronTier for solving problems in compressible gas dynamics [22, 20, 19]. The Hamilton-Jacobi solver for etching and deposition that has been added to FronTier builds upon these algorithms [18]. A recent thrust in three-dimensional front tracking involves a modi ed handling of the interface. The interface is periodically reconstructed based on the micro-topology of the portion of the interface within a rectangular cell block. This method, called \grid-based" front tracking, allows handling of complex topological issues, such as bifurcation of the interface, which has been demonstrated for smooth surfaces [19]. Front tracking without this reconstruction of the interface is known as \grid-free" front tracking, referring to the independence of the interface from a xed rectangular grid. A hybrid version of front tracking is also in use, using grid-free algorithms as much as possible, calling interface reconstruction routines to process complex intersection resolutions. The etching and deposition simulations in this work were performed with grid-free front tracking | no cell-based reconstruction of the surfaces.
3.1 The Interface The distinctive element in front tracking is the representation of important lower-dimensional features with a mobile grid embedded in the ambient space. A 30
static, rectangular mesh is then used to discretize the solution elsewhere. The geometric structure used for the mobile grid is called an interface. By using the interface to track important features such as shock waves and material interfaces, relatively coarse meshes can be used for both the interface and the rectangular grid. In addition, front tracking avoids the numerical di usion at sharp features of solutions that many numerical di erence schemes introduce. The structure of the interface in front tracking has been described earlier [30], while a mathematical de nition for the interface was presented in [32]. These works are followed here in describing the interface and its three-dimensional form used in this study.
De nition 6. Given an n-dimensional ambient space, an interface is a simplicial complex with dimension (at most) n ? 1. The elements of the interface are simplices of dimension k; k = 0; : : : ; n ? 1. Let Mk be the set of all simplices of dimension k in the interface. Each set Mk will be the union of k-dimensional manifolds. Faces of simplices in Mk which lie on the topological boundary of a manifold in Mk are themselves simplices in Mj ; j < k.
In three dimensions, simplices of dimension 2 are triangles, encoded as TRIS. Simplices of dimension 1 are line segments, encoded as BONDS. Simplices of dimension 0 are points, encoded as NODES. The two-dimensional manifolds of M are triangulated, oriented SURFACES, comprised of a connected set of triangles. The topological boundary of surfaces are formed by the one-dimensional manifolds of M , called CURVES. Curves are comprised of a connected, ordered set of BONDS. 2
1
31
Each boundary point of a curve must be a NODE in M . Surfaces and curves are assumed to be reasonably smooth. The complement of the interface is known as the interior [9]. A component function is used to identify the distinct regions of the interior according to the material present in that region. The orientation of surfaces is based upon this component function. The solution of the interior will be solved using a standard nite di erence or nite element scheme for most front tracking algorithms. For the etching and deposition simulations, it is assumed that all dynamics occur at the material surface; thus, no interior solver is required. The xed computational grid is still in use for the component function. 0
De nition 7. An interface is said to be untangled if all intersections of k-dimensional elements occur only along their boundaries and
the component function is consistent. If either condition fails, the interface is said to be tangled.
Thus, surfaces must intersect each other only along curves, and curves must intersect only at nodes. The self-intersection of an interface represents a bifurcation in the topology of the material interface. The failure of the consistency condition indicates an unphysical region, as the component information identi es di ering materials in the same location. When intersections or inconsistent components are detected, the interface must be returned to an untangled state, with adjustments 32
to the interface re ecting the topological changes. Currently, only limited types of tangles can be resolved in grid-free three-dimensional front tracking. Dynamic bifurcations in the topology are not yet supported for three dimensions.
3.1.1 Storage of topological information The interface structure and other structures for three-dimensional FronTier have been put forth in detail elsewhere (in particular, see [23, 22]). A summary of the approach is given here. Triangles in a surface, as well as bonds in a curve, are stored in doubly linked lists. Each triangle also has pointers to the neighboring elements connected to it, whether bonds or other triangles. Each surface has pointers to the lower-dimensional elements (curves and nodes) which form its boundary. Similarly, a curve stores pointers to its boundary nodes. A curve also has pointers to the surfaces of which it is a boundary element, and a node has pointers to all the curves containing it in the boundary. The vertices of each k-dimensional simplex are stored as pointers to points. Thus, pointer comparisons can be used rather than oating point comparisons to determine shared vertices or sides. This has the advantage of being less ambiguous (avoiding oating point errors caused by roundo ), as well as less being less expensive to compute. Also, a point need only be operated on once, and all elements which hold that point as a vertex will access the updated information.
33
3.1.2 Interface retriangulation The propagation algorithms for interface points typically depend upon geometric data given by the neighboring triangles. The accuracy and stability of the solution thus depend upon the uniformity of the triangles. The eciency of the computation is also a ected by triangle size as it a ects the allowable time step (discussed in Section 3.2). Triangles and bonds are free to compress or expand as regions of the interface move closer or farther apart, so periodic retriangulation is required to maintain the quality of the interface [22, 19]. Such retriangulation does introduce di usion into the solution, so it is desirable to perform retriangulation as seldom as possible. Some problems have a greater need for uniform meshes, so the frequency of retriangulation is left as a user-determined option. During retriangulation, all triangles are tested against user-de ned limits to nd those that are too large or too small, or that have unwanted characteristics such as an angle too small or a side too large. All triangles found to be outside the desired ranges are entered into processing queues. The elements in each queue are sorted according to length and operated on one at a time. Large triangles are split along their longest side, while small triangles are collapsed and removed along with their neighbor across the shortest side. Triangles with too small an angle are matched with a neighbor to form a quadrilateral with the shared side forming one diagonal. The two triangles are then replaced by the triangles formed by dividing the quadrilateral across the opposite diagonal. Each triangle that is operated on, as well as any neighbor that may be modi ed in the process, is removed from the processing queue. 34
3.1.3 Untangling the interface After the interface has been propagated, FronTier must resolve any intersections that have formed. The untangling of the interface is modeled after the method that has been robustly implemented in two-dimensional FronTier [21, 35]. The implementation of untangling interfaces in three-dimensional FronTier was presented in [19]. The three stages in the untangling process are brie y described here. The rst step is the detection of any intersections. All triangles must be compared with each other to nd any intersections. To ease the computational strain that a complete testing of each triangle against all others would create, the detection algorithm makes use of the topological grid | an auxiliary rectangular grid used for such maintenance purposes. A hash table is created listing the triangles partially contained by each cell. Individual triangle comparisons need only be made between triangles located in the same cell. When an intersection is found between two triangles, a new bond is inserted along the line of intersection. The placement of such bonds creates polygons bounded by the original triangle and the inserted crossing bonds. The second step in intersection resolution is to retriangulate the interface along these crossing bonds. This retriangulation may include the formation of new surfaces and curves so that all interface elements only intersect along boundaries. The interface now satis es the de nition of an interface set forth in De nition 6. The third and nal step is to remove any unphysical elements that remain in the interface. Inconsistent component information will indicate an unphysical surface in the interface. These unphysical regions must be identi ed and removed 35
based on global topology and geometry of the interface and the computational grid components.
3.2 Propagation of the Interface The numerical solution to the Hamilton-Jacobi equation is determined by advancing the interface along a series of time intervals. For each time step, a new interface is given by propagating the current interface according to the material
ux. Surfaces are propagated to new surfaces, curves to new curves, and nodes to new nodes. The time step is restricted to prevent points in the surface from propagating across the boundary curves of the surface. The time step is set to the minimum distance between a curve bond and the point opposite the bond in all attached triangles and divided by the maximum speed of the front. If after propagating the interface, FronTier is unable to resolve any resulting tangling of the interface or other complication, the propagation step will be redone with a smaller time step. This robustness technique is done recursively to enable FronTier to handle dicult interface operations in smaller pieces.
3.2.1 Surface propagation The rst step in propagating the interface is to advance the smooth regions of the interface, which are represented by triangulated surfaces. The ux must be computed for each vertex X interior to a surface. For each triangle T , we de ne the
ux C (T ) to be the ux at the center of T . The di erent components of the ux 36
| the source ux Cs(T ), the ejected ux Ce(T ), and the redeposited ux Cr (T ) | are calculated given the location of the center of T and the normal of T . The ux at the vertex X is then found by averaging the uxes for each triangle containing X as a vertex, weighted by the areas of the triangles. The material ux C (X ) gives the normal velocity of the surface at X . Propagating according to the method of characteristics would introduce a tangential component to the velocity [48]. We have chosen to propagate in the normal direction only in order to avoid the directional bias that tangential velocity would introduce to the interface. X then propagates to the point X + C (X )N (X )t. The normal N at the vertex X is computed by an area-weighted average of the normals of the triangles which share X as a vertex.
3.2.2 Curve propagation The next step in updating the interface is to propagate the interior portions of curves. The discussion of curve propagation is restricted here to curves formed by the intersection of two surfaces . To propagate such a curve, we recall the local representation of a curve from Section 2.2. A point on a curve is modeled as two half-spaces intersecting in a line tangent to the curve at that point. This intersection de nes a one-dimensional Riemann problem | a scale-invariant Cauchy problem. Using normal propagation, the two half-spaces are propagating at a rate determined 1
A two-material problem will have curves with three surfaces intersecting: the two exterior surfaces plus the interior surface dividing the two materials. Propagating this three-surface curve will require solution of a moving boundary problem. 1
37
by their ux, in the direction of their respective normals. The propagation of the curve is determined by the motion of the half-spaces. Thus, by consistency between the two sides of the curve,
V (X ) N = C (X; N ); 0
0
(3.1)
V (X ) N = C (X; N ): 1
1
The ux C (X; Ni) is computed using the uxes for the triangles of surface i that intersect at X . The equations of (3.1) form the Rankine-Hugoniot jump conditions [27] for the Riemann problem at the curve. They provide a unique velocity V (X ) save for the direction tangential to the curve. The choice of normal propagation, however, restricts V (X ) to be perpendicular to the curve. Thus, the velocity V (X ) is uniquely determined, with X propagating to X + V (X )t. When a curve point is propagated, the curve is tested against the appropriate admissibility condition, i.e. equation (2.9) for a convex curve or (2.10) for a concave curve. These conditions are tested by creating the upper or lower convex hull of the Hamiltonian for convex or concave curves, respectively. An algorithm for determining the convex hull of the Hamiltonian is given in [38]. If a curve propagates to an inadmissible curve, the correct solution requires the curve be split into a series of admissible curves and smooth surfaces. These curves and surfaces are determined by the chords and arcs in the convex hull. The current capability to handle bifurcations is limited. If the admissibility condition indicates the surface contains a rarefaction wave entering the curve, then the choice of normals in (3.1) is selected to represent the shock as determined by the convex hull. The algorithm thus correctly propa38
gates a shock with a rarefaction wave on both sides. More complicated bifurcations, such as a curve splitting into multiple curves connected by smooth surfaces, cannot yet be handled dynamically.
3.2.3 Node propagation Finally, the nodes of the interface must be advanced, as well. Solutions for two-dimensional Riemann problems of Hamilton-Jacobi equations were presented in [27] and are used for node propagation in FronTier. The algorithm for propagation of a triple-point node is presented here. Let X be an admissible node formed by the intersection of three surfaces. We model the node as the intersection of three planar segments, with normals N ; N , and N . The velocity of the node V (X ) is given by consistency with the velocity of the planar segments. Thus, 0
V (X ) Ni = C (X; Ni ); i = 0; 1; 2:
1
2
(3.2)
Solving this system of equations determines V (X ) uniquely, with X propagating to X + V (X )t. The admissibility conditions (2.11) and (2.12) must be satis ed for convex and concave nodes, respectively. These conditions are tested by creating the upper (lower) convex hull for convex (concave) nodes. If the node propagates to an inadmissible node, the correct solution requires the node be split into a series of admissible nodes, admissible curves, and smooth surfaces as described in [27]. As for propagation of curves, the choice of normals in (3.2) is governed by the convex hull 39
of the Hamiltonian along the curve entering the node. This allows for a rarefaction wave entering the node. Dynamic bifurcation of an inadmissible node is not yet supported by FronTier for three-dimensional etching and deposition simulations.
3.2.4 Visibility The algorithm used to determine if a point Y is visible from point X employs the topological grid. For each cell in the grid, there is a list of all the triangles partially contained in that cell. The visibility algorithm selects all cells through which the segment XY passes. For each such cell, the line XY is compared to the triangles in the cell to identify any possible intersections. An intersection between XY and a triangle indicates that a portion of the surface lies between X and Y and visibility is blocked. In addition, if point X is a surface point with surface normal N (X ), the surface must be oriented towards Y , i.e., N (X ) (Y ? X ) 0. Similarly, N (Y ) (X ? Y ) 0 must hold if Y lies on a surface. Visibility is applied to determining whether a point on the surface is a ected by an outside source of material, as well as if a triangle can receive resputtered material from another triangle.
40
Chapter 4 Numerical Simulations Algorithms for simulating etching and deposition were developed for the front tracking package FronTier. This chapter presents a series of results obtained using the three-dimensional FronTier code. Representative examples are given for various types of surface development. All Hamiltonians and normal velocity functions in this chapter are presented in the form of the reduced-coordinate system (2.6) { (2.8), where the surface is locally a graph. For ease of use, the circum ex notation, as in H^ , will be assumed for coordinates and functions in this chapter.
4.1 Isotropic Deposition We rst consider surface growth by isotropic deposition, where the deposition rate is independent of surface orientation. Isotropic deposition produces a uniform coating of the substrate. The low-pressure chemical vapor deposition (LP-CVD) of 41
6
H
4
2
0 -5
0
5
P
Figure 4.1: The Hamiltonian H (r (X )) = with a deposition rate of 1.
p
1 + r (X ) for isotropic deposition
a SiO glass lm has been observed to be isotropic in nature [55]. Also, the line-ofsight based simulations of Cale and Raupp predict that LP-CVD with a very low sticking probability for material (S < 0:05) is essentially isotropic [8]. The rate of deposition is set to 1, giving a Hamiltonian of H = jr (X )j = , (see Figure 4.1). The initial condition is a planar surface with a rectangular hole with length:width:depth ratio of 2:1:1 (Figure 4.2(a)). Figure 4.2(b) shows the trench during deposition, shortly before the walls intersect, closing the trench. The splitting of triangles when they grow too large enables the rounding of the initially sharp upper edges. The cutaway image of the interface in Figure 4.2(c) shows the rounded surface formed by deposition. 2
1 cos( )
42
(a) Initial rectangular hole
(b) Rectangular hole after isotropic deposition
(c) Cutaway image of rectangular hole after isotropic deposition
Figure 4.2: Isotropic deposition into a trench. 43
1.2
H
1.0
0.8
0.6
-5
0
5
P
Figure 4.3: The Hamiltonian H (r ) for the low-temperature etching of silicon oxide by CHF3 ions, as given by (4.1). The nonconvex nature of the Hamiltonian can lead to faceting of the surface.
4.2 Ion Etching The process of ion etching is often dependent on the incident angle of the ion beam with the surface [85]. For many materials, the normal velocity is a nonconvex function of the incident angle [16], which results in the formation of edges and sloped regions in the surface as it is etched away, a trait known as \faceting". The location and propagation of edges and corners in the surface is determined by the application of the admissibility conditions to the Hamiltonian. The propagation of a discontinuity in the surface gradient is determined by the convex envelope of a portion of the Hamiltonian.
44
4.2.1 Ion etching of oxide The etching of low-temperature silicon oxide by CHF3 ions provides an example of etching with a nonconvex Hamiltonian. The ion beams are assumed to have a unidirectional distribution. By taking experimental measurements of the etching rate at equally spaced angles and tting the results by a Fourier series , Ross determined the Hamiltonian in terms of the angle incident to the normal: 1
C () = 1:454743 cos() ? 0:464719 cos(3 ) + 0:015573 cos(5 ) H () = cos( )
?0:005669 cos(7 ) ? :010000 cos(9 ) + 0:010552 cos(11 ) ?0:006204 cos(13 ) + 0:005725 cos(15 ) (4.1)
if ? < < ; and C () = 0 otherwise [72]. The Hamiltonian has been scaled so that the etching rate in the normal direction is ?1 (see Figure 4.3). The form of the Hamiltonian, with a local maximum at the normal to the surface between two local minimums, is typical for amorphous materials [73]. The initial condition used for this FronTier simulation is a rectangular trench in a planar surface. The upper edge of the trench is inadmissible under the imposed conditions and bifurcates into two separate edges (shocks) connected by a smooth surface (rarefaction wave). The initial interface is preconditioned to introduce this bifurcation as shown in Figure 4.4(a). The length:width:depth ratio of the initial trench is 6:3:5. The e ects of the ion etching on the material surface are shown 2
2
An important consideration for interpolations is to maintain the same quality of convexity/concavity as the original data. Additional in ection points will create unphysical faceting of the surface. 1
45
(a) Initial interface with bifurcation of the top edge.
(b) Final interface.
(c) Cross-sections of the interface over time.
Figure 4.4: Ion etching of a rectangular trench with length:width:depth ratio of 6:3:5 according to the nonconvex Hamiltonian in (4.1). The process simulated is lowtemperature etching of silicon oxide by CHF3 ions. 46
in Figure 4.4(b). A time evolution series of cross-sections of this interface as the material is etched away is presented in Figure 4.4(c). The trench walls begin to curve and progressively become narrower, a process known as tapering. A rarefaction wave develops also between the facet edges at the top of the trench. The results agree with the two-dimensional shock-tracking simulation described in [38].
4.2.2 Comparison to other triangulated-mesh schemes Another Hamilton-Jacobi solver was designed by Barth and Sethian [5] for use on triangulated interfaces. Several numerical di erence schemes were implemented and applied to a variety of applications, including ion etching. Faceting of the surface was demonstrated using the nonconvex function
C = 4 cos () ? 5 cos() 3
(4.2)
for the normal velocity. The Hamiltonian for this function is H () = C () cos? () and is also nonconvex (see Figure 4.5). Under these conditions, the upper edges of the tower are admissible and do not bifurcate during the etching process as in Section 4.2.1. The introduction of a nonconvex Hamiltonian led to signi cant oscillations in those schemes in [5] which did not have a discontinuity capturing term. One scheme, the explicit Petrov-Galerkin scheme with discontinuity capturing, was able to accurately render the solution. The FronTier algorithm for etching was applied to this same problem. The initial interface is the rectangular tower structure presented in Figure 4.6(a). The 1
47
-1.000000
H
-2.000000
-3.000000
-4.000000
-5
0
5
P
Figure 4.5: The Hamiltonian H = cos? ()(4 cos () ? 5 cos()) for the ion etching experiment discussed in Section 4.2.2. 1
3
outer dimensions of the tower are 1.4 by 1.2 by 0.6. A rectangular hole is located in the center of the top of the tower, with dimensions 0.6 by 0.4 by 0.4. The tower structure after etching is pictured in Figure 4.6(b). Figure 4.6(c) shows a cut-away perspective in three dimensions, while Figure 4.6(d) displays a time series of cross sections. The cross-sections show smooth development of the sloping surfaces with sharp transitions at the facet edges. Unlike the CHF ion etching in Section 4.2.1, this example has no tapering (narrowing) of the trench walls. 3
4.3 Physical Vapor Deposition Sputter deposition, or physical vapor deposition (PVD), is a technique used to apply thin metal lms to semiconductor wafers. First, a plasma is used to ionize an inert gas. A metal target is then bombarded by these ions, which sputter atoms from the surface to be deposited onto the wafer. The sputtered material tends to 48
(a) Initial tower interface
(b) Tower interface after etching
(c) Cutaway after etching
(d) Time series of cross-sections
Figure 4.6: Ion etching of a structure according to the normal velocity in (4.2). The initial interface is a tower structure with a trench in its top surface. As the structure is etched away, the walls develop smooth sloping surfaces. 49
have a broad angular distribution, which leads to poor coverage of deep features. Variations on the PVD process attempt to provide more uniform deposition. An overview of di erent PVD systems for semiconductor manufacturing is provided by Rossnagel [74].
4.3.1 Simulation of physical vapor deposition We model the sputtered material by assuming the cosine distribution (Section 2.4.2) for sputtered particles as they reach the surface [39]. Particles are assumed to stick where they rst land, which is characteristic of PVD [7]. The ux at a point X on the surface is obtained by integrating over the portion of the source target visible to X . Using the view factor operator (2.14), the normal velocity function for a PVD process is Z
C (X ) = K (X; Y ) dA(Y );
(4.3)
with K de ned by (2.15) and A being the area of the overhead source. To approximate the e ects of a nite-sized source, the range through which a surface point can receive material is limited to 52 from the vertical [39]. Figure 4.7 shows the simulation of physical vapor deposition into a rectangular trench. The lower regions of the trench have less visibility to an overhead source, and so receive less material. An overhanging bulge thus forms at the top of the trench, visible in the cutaway in Figure 4.7(a). The visibility on the walls decreases near the corners, as there are two directions in which the incoming particles are blocked. As a result, the upper corners in particular propagate more slowly than 50
(a) Cutaway view of trench after PVD
(b) Overhead view of trench after PVD
Figure 4.7: Physical vapor deposition into a trench according to (4.3): (a) the upper regions of the trench receive more material, creating overhangs, while (b) the corners lag behind the overhangs, which grow into each other to form creases at the corners.
51
(a) Aspect ratio 4:1
(b) Aspect ratio 2:1
(c) Aspect ratio 1:1
Figure 4.8: The e ect of aspect ratio on physical vapor deposition. Three trenches of di erent widths are propagated for the same amount of time. The higher the aspect ratio of the trench, the less coating the walls and oor will receive. their surroundings. The growing overhangs atop the walls propagate into each other, forming creases in the surface that extend to the corner, as seen in Figure 4.7(b). Such creases do occur during physical vapor deposition; McVittie, et. al., present images of Ti/Al PVD which demonstrate such features [3]. The three-dimensional simulation described in [3] also exhibits crease formation. Improved semiconductor production requires structures to be packed more tightly together. Features must have higher depth-to-width ratios in order to t more elements onto the wafer. The broad distribution of PVD is unsuited for such work, because the deeper regions of a structure do not receive enough material for e ective coatings. The visibility modeling in FronTier captures this e ect, as seen in Figure 4.8. Three trenches of the same depth and length, but di erent widths, 52
are lled for the same amount of time. Each image shows both the initial trench and the deposited coating. The aspect ratios for the trenches are (a) 4:1, (b) 2:1, and (c) 1:1. The growth at the top of all three trenches is the same. However, a thinner layer is deposited on the oors of the narrower trenches. The walls of the narrower trenches are particularly a ected, to the point that the narrowest trench receives only a negligible coating on its lower half.
4.3.2 Collimated physical vapor deposition To reduce the angular spread of the sputtered material, a set of tubes called a collimator can be placed above the substrate [75]. Sputtered material that strikes a collimator wall will be deposited inside the collimator rather than reach the surface. Only those particles with a trajectory within some small angle to the normal (determined by the collimator geometry) will reach the surface. While this has the e ect of reducing the angular distribution of incoming particles, it also reduces the deposition eciency of the system. The e ect of adding a collimator to a PVD system was studied by repeating the PVD simulations, but reducing the angular range through which material can be received. Figure 4.9 shows the cross-section of a trench with aspect ratio 4:1, as well as the cross-sections of collimated PVD simulations into the trench. The simulations were based on the same conditions as Section 4.3.1, but with collimation angles of 52; 14, and 1 . The deposition rates of the experiments were normalized so that deposition in the vertical direction would be equivalent, and then all three simulations were run for the same length of time. Normal PVD, assumed to have 53
Figure 4.9: Collimated physical vapor deposition. Cross-sections of PVD into a trench with collimation angles of 52; 14 , and 1 . The lower collimation angles produce more deposition on the oor and thinner overhangs. a collimation angle of 52, produces wide overhangs and very little growth on the
oor. Collimated PVD limits the growth of the overhangs, while lling the trench at the bottom. This greater accuracy comes at a steep price in eciency, however, as most of the material is deposited inside the collimator and not on the substrate.
4.3.3 Ionized physical vapor deposition Another method of increasing the directionality of a PVD system is to ionize the particles that are sputtered from the target [74]. A second plasma is placed between the target and the substrate. Many of the atoms emitted from the target 54
will be ionized by this plasma. They can then be accelerated to the substrate with a near unidirectional distribution. Some particles will remain neutral and drift to the surface to be deposited with a broad angular distribution. Ionized magnetron sputter deposition was studied by Hamaguchi and Rossnagel, comparing a two-dimensional front tracking simulation to physical experiment [39, 40]. The model included the two-phase deposition of neutral and ionized metals. The ratio of ion:neutral atoms is de ned as : . The ionized particles are assumed to have a unidirectional distribution along the z-axis. Source visibility for a point X on the surface is de ned as (X ) = 1 if the source (assumed to be an overhead plane) is visible to X along the direction of ionic ux and 0 otherwise. Flux due to deposition of ions is then (X )(N (X ) z) . At this point we assume the ions are deposited at the point of impact without dislodging material from the surface. (Resputtering will be treated in Section 4.4.) The neutral particles are modeled as the PVD described in Section 4.3.1. The total ux is then given by +
Z
C () = (X )(N (X ) z) + K (X; Y ) dA(Y ); +
(4.4)
where A(Y ) is the area of the source target. By controlling the direction of the ionized atoms, the net angular spread of the deposition ux is decreased. Increasing the percentage of ionized atoms further decreases the angular distribution. The e ects of higher ionization ratios were studied by adjusting the ion:neutral ratio in the simulation. The initial surface is a trench with 4:1 aspect ratio (the length-to-width is 6:1). Figure 4.10(a) shows the cross-section of the initial trench, along with the cross-sections of the trench 55
(a) I-PVD with varying ion:neutral
ux ratios
(b) Overhead view of trench after I-PVD
Figure 4.10: I-PVD with increasing ion:neutral ux ratios. (a) Cross-sections of an initial trench and the results of I-PVD with ion:neutral ratios of 1:2, 1:1, 2:1, 3:1, and 4:1; the higher ratios lead to smaller overhangs and greater lling of the trench. (b) Overhead view of the trench after I-PVD with a 4:1 ion:neutral ratio.
56
after I-PVD with ion:neutral ux ratios of 1:2, 1:1, 2:1, 3:1, and 4:1. A higher ionization of material leads to smaller overhangs at the top of the trench and greater deposition on the oor. Figure 4.10(b) gives an overhead view of the trench after I-PVD with a 4:1 ion:neutral ratio. The ionized ux has lessened the disparity of propagation speed that occurred for PVD near the corners. The creation of creases at the corners is less dramatic with a highly ionized ux and occurs at a slower rate, as the vertically deposited ions ll in the creases as they form.
4.4 Resputtering When high energy particles strike the substrate, atoms from the substrate may be dislodged. Thus, in ion deposition processes, a scattering of surface particles may occur. This behavior is known as resputtering, where the particles sputtered from the target themselves in turn sputter material from the substrate. Redeposition occurs when material ejected from the surface in this manner then lands elsewhere on the surface. The modeling of these global e ects was discussed in Section 2.4. This section will present simulations that include resputtering and redeposition.
4.4.1 Ionized physical vapor deposition with resputtering We return to the example of ionized physical vapor deposition, but now the ion streams are allowed to sputter material from the substrate. Let Y () be the sputtering yield due to the combined impact of metal and inert gas ions, where the etching rate due to the sputtering is Ce() = Y () cos(). Then the ux for material 57
emitted from the substrate due to the impact of ions is ? (X )Y (). Sputtered material is assumed to scatter with a cosine distribution [39, 74]. A scattered atom is assumed to travel in a straight line and is deposited again if it lands elsewhere on the surface. From (2.16), the contribution of redeposited atoms to the ux is V ( (X )Y ()), where V is the view factor operator of section 2.4.2. Introducing the redeposition ux to (4.4) gives a global normal velocity of Z
C (X ) = ((X )(N z) ? (X )Y () + V (X )Y ()) + K (X; Y ) dA(Y ): (4.5) +
To determine the neutral ux at a point, we again integrate over a maximum range of 52 from vertical to represent the nite overhead source. The values for Y () that are used in this section are for the sputtering of Al. Experimental values for the etching rate of Al by a 300 eV Ar beam with a current of 0.32 mA/cm were given in [53] and tted with a complete cubic spline to arrive at the function shown in Figure 4.11. We follow [39] in assuming that this sputtering yield is valid for other energies and ion beams. (The physical experiments in [39] were performed with a low-energy, i.e., 120 eV, Al and Ar ion beam.) The function has been normalized so that the etching rate in the normal direction is 1.0. For a rst demonstration of resputtering and redeposition, consider a deposition stream of ionized atoms only | no neutral ux. The source thus has a unidirectional distribution. The regions with a vertical line-of-sight to the source will grow due to the source ux. Redeposition of ejected material, however, introduces a scattering e ect that will cause the directly shadowed regions of the substrate to grow as well. The structure in Figure 4.12(a) is a trench with overhangs on either side. Ion 2
58
Sputtering Yield Y
1.5
1.0
0.5
0.0 0
20
40
60
80
Degrees
Figure 4.11: The etching rate for Al [53], used to determine the sputtering yield during ion deposition [39]. The function is normalized so that Y (0) = 1. deposition of Al is applied to this structure with a deposition rate of 1.0, but a sputtering yield of Y (0) = 0:8. Particles ejected are assumed to stick where they next land [39]. Figure 4.12(b) shows the surface after deposition. While in the vertical direction, more material is deposited than sputtered, at some angles the amount of material sputtered is actually greater than the amount deposited. The top edges are etched away as a result, even though the top surface is growing. The image in Figure 4.12(b) shows the deposition growing on the oor and on the hidden walls. The image in Figure 4.12(c), shows the side walls develop the tapering typical of an etching process, but magni ed in the center of the trench by reception of resputtered material. The cross-sections in Figure 4.12(d) show the deposition that results underneath the overhangs due to resputtering as well as the net etching of the top edges. Next consider ionized PVD with both neutral and ionized ux, at a ratio of 1:1. 59
(a) Cutaway of initial overhanging interface
(b) Cutaway after ion deposition with resputtering
(c) Cutaway along opposite axis
(d) Initial and nal crosssections
Figure 4.12: Ion deposition of Al with resputtering (Y (0) = 0:8) into a trench with hidden regions. Resputtering of material inside the structure causes redeposition underneath the overhangs. The process is a mixture of both etching and deposition. 60
We return to the rectangular trench with a 4:1 aspect ratio, and allow for resputtering of material. The sputtering yield is that of Al (see Figure 4.11), normalized to Y () = 0:8. A cutaway view of the trench after deposition is shown in Figure 4.13(a), with cross-sections of the pre- and post-deposition interfaces given in Figure 4.13(b). Resputtering, and the subsequent redeposition of material, leads to a much more complete coating of the inside of the trench. In contrast to ionized PVD without resputtering (see Figure 4.10), the vertical walls are noticeably covered, and the edges along the oor collect more material. The resulting convex curvature of the trench walls is observable in experimental deposition with high sputtering yields [39, 40]. While resputtering does erode away the tops of the forming overhangs, the combination of visibility to the neutral source ux as well as the reception of resputtered material leads to faster surface growth at the top.
4.4.2 Sticking probability Next, the e ect of a sticking probability S < 1 is considered. Particles which have been resputtered are allowed to stick to their next point of impact with a probability of S . Particles which do not stick are assumed to rebound again with a cosine distribution [39]. If particles are restricted to only one rebound and must stick at the second landing, then the normal velocity is Z
Z
K (X; Y ) dA(Y ) + (1 ? S )V K (X; Y ) dA(Y ): source source Allowing for arbitrarily many rebounds leads to the equation C (X ) = S
C (X ) = S
Z
source
K (X; Y ) dA(Y ) + (1 ? S )V C (X ); 61
(4.6) (4.7)
(a) Trench after ionized PVD with resputtering
(b) Cross-sections of initial and nal interface
Figure 4.13: Ionized PVD with resputtering: (a) a trench is subjected to I-PVD with conditions as in Figure 4.10, except that resputtering and redeposition are now considered. (b) The redeposition results in a more complete coating along the walls and lower edges of the trench.
62
Figure 4.14: Ionized PVD with resputtering and variable sticking probability. The sticking probability for the three runs is 1.0, 0.75, and 0.5. Lower sticking probability provides less deposition in the lower regions of the trench. which is solved iteratively as a truncated Neumann series. The multiple-rebound model was applied to the ionized PVD with resputtering problem of Section 4.4.1. For one simulation the conditions were held exactly the same (i.e., S = 1:0), while two simulations were added with a sticking probability of S = 0:75 and S = 0:5. Note that the sticking probability is applied only to resputtered particles, not to the rst impact of neutral or ionized particles. The results are shown in Figure 4.14. The multiple bounces that a resputtered particle can make leads to less deposition inside the trench, as particles have more opportunity to escape. The lower edges of the trench in particular receive less material.
63
Chapter 5 Parallelization of the Interface for Three-dimensional Front Tracking In the search for higher resolution computations, two limiting factors are the computational time and the memory usage required to perform a numerical experiment. To improve the resolution of a three-dimensional mesh by n increases the memory required by O(n ). Because the time scale must also be re ned, the length of computation can increase by O(n ). These exponential dependencies quickly restrict the level of resolution that can be attained on available computers. In order to extend current computational capabilities, the FronTier algorithm has been parallelized for multiple-instruction-multiple-data (MIMD) architectures, using MPI as the parallel communication protocol [29, 24, 28, 31]. Parallelized versions of FronTier algorithms have been written for hyperbolic solvers and elliptic solvers in both two and three dimensions [14, 6, 34]. The two3
4
64
dimensional routines are mature. The three-dimensional algorithms were initially applied to problems in which the interface had no singularities in the surface; i.e., no curves or nodes [22]. As part of this work, FronTier has been parallelized for three-dimensional hyperbolic problems with more complex interfaces.
5.1 Domain Decomposition FronTier has been equipped to run on parallel machines using a distributed memory model. Each processor uses private memory storage. Processors operate instructions independently, save for any necessary communication, which is done synchronously. The assignment for each processor is determined by a xed-grid domain decomposition scheme [6, 22]. The computational domain of the problem is divided into rectangular subdomains of equal size, with one assigned to each processor. These computational subdomains intersect each other only along their boundaries. Each processor is responsible for operations on elements lying within its computational subdomain, including the portion of the interface contained within the subdomain boundaries, as well as all grid blocks of the interior. In order to correctly propagate all information within the computational subdomain, the solution of hyperbolic problems requires additional data from the regions bordering the subdomain [17]. To facilitate this, the range of information stored by each processor is extended into the neighboring subdomains by a bu er zone. The union of the computational subdomain and the bu er zone is referred to as the 65
Global Domain
Computational Subdomain
U
Virtual Subdomain
Figure 5.1: A global domain is divided into nine computational subdomains. The center computational subdomain is shaded dark. The corresponding virtual subdomain, containing a bu er region extending into the adjacent subdomains, is shaded light. virtual subdomain. A diagram of such a decomposition is shown in Figure 5.1. The bu er zone must be wide enough to allow updating of all the information in the computational subdomain through a full time step. In practice, the width of the bu er zone is set at least four grid-blocks wide. This provides the data needed for a propagation algorithm such as an explicit nite di erence scheme operating on the interior of the solution.
66
5.2 Communication of the Interface After a time step, each processor updates its bu er zone: the current bu er (containing information that may have been propagated incorrectly) is discarded and replaced by new information from the neighboring subdomains. This procedure, referred to as scatter-gather, is often used in domain decomposition problems [46]. Scatter-gather in FronTier requires that particular care be given to the interface. The interface in each virtual subdomain must be clipped down (i.e., restricted) to the computational subdomain. The bu er zones are formed by further restrictions that select the appropriate subrectangles from the computational subdomain. These bu er zones are then passed to the appropriate processors to be attached to the neighboring subdomains (Figure 5.2(a)). When new bu er information is received, it must be appended to the interface (Figure 5.2(b)). These operations must result in a new untangled interface (see De nitions 6 and 7) spanning the virtual domain, with consistent geometric, component, and logical information.
5.2.1 Removal and creation of bu er zones Updating the virtual domain begins with preparing the bu er zones of each processor's interface. The old bu er zones in the virtual subdomain must be discarded, and new bu ers must be created to send to the neighboring processors. Both discarding the old and creating the new involve one algorithm: to reduce the interface to a speci ed block. To discard the old bu ers, the interface is restricted to its computational subdomain. The bu er zones are created by restricting the 67
(a) Computational subdomain and neighboring bu ers
(b) Virtual subdomain (bu ers now attached)
Figure 5.2: (a) The interface as restricted to a given computational subdomain, with the four neighboring bu ers to be attached to complete the virtual subdomain. (b) The local interface for the given processor after the bu ers are attached.
68
interface to its intersection with the virtual subdomain of its neighbor. The virtual subdomains are updated one direction at a time [22]. Each processor will create the bu er for its neighbor on the left; this bu er is then transmitted. Each processor then receives the incoming bu er and attaches it to the local interface. The process is then repeated for the neighbor on the right. The down-up directions and bottom-top directions follow. In this way, no communication is needed between processors that adjoin diagonally. In clipping the interface down to a block of speci ed coordinates, all elements (triangles, bonds, nodes) lying at least partially inside the computational subdomain are retained. All elements lying entirely outside the computational domain are removed. These conditions result in the overlap of the reduced interface and the new bu ers. This overlapping region is used to merge the bu er to the local interface. A second result is that the new interface, after all bu er zones are updated, will be large enough to span the virtual subdomain. This is signi cant to the intersection-detection routines, which expect interface elements to be present in cells with component discontinuities. Triangles are examined rst and are removed if all points lie outside the computational subdomain. The clipping of bonds is dependent on the clipping of triangles. A bond serves to mark a discontinuity between two or more triangles. Thus, a bond is clipped from the interface if all or all but one of its adjoining triangles have been removed. Similarly, nodes are removed if the adjoining curves have been removed. The clipping routine does check to retain isolated nodes, such as the vertex of a cone, if they lie within the computational boundary. 69
Figure 5.3: Removal of the sections of the curve outside the computational subdomain (dotted lines) will result in four separate, disconnected curves (solid lines). The removal of portions of the interface may introduce local (not global) topological changes. If a curve crosses back and forth across the computational boundary, the removal of the bu er zone may divide the curve into multiple, disconnected parts (see the schematic in Figure 5.3). To accomplish this, if a bond is removed such that the resulting curve is disconnected, the curve is split into two new curves. This is done recursively, so that multiple crossings of the boundary may be handled. Figure 5.4 shows an example of such a topological change. In Figure 5.4(a), the interface contains a closed curve which crosses the computational boundary. A closed curve is a single loop, with no boundary nodes. (The interface does include a single node at a curve point for reference purposes.) In Figure 5.4(b), the bu er has been stripped from the interface, and the curve is now open. The curve is given 70
(a) Interface on virtual domain with a closed curve
(b) Clipping the interface opens the curve
Figure 5.4: Removing the bu er zone from the interface can lead to topological changes: an interface contains a closed curve in (a) which after removal of bu er triangles and bonds becomes an open curve with boundary nodes (b). a node at both its starting point and ending point. Figure 5.5 demonstrates a recursive example of curve splitting. In Figure 5.5(a), the curve in the interface crosses the computational boundary at multiple locations. In Figure 5.5(b), the interface has been clipped, and several individual curves remain. Each of these curves is a separate object in the new interface.
5.2.2 Attaching the bu er to the interface Once a bu er is received from a neighboring interface, it must be appended to the local interface with consistent logical and geometric information. First, the 71
(a) Interface over virtual subdomain
(b) Interface after bu er removal
Figure 5.5: The interface in (a) contains a curve that crosses the subdomain boundary multiple times. The removal of bu er triangles and bonds splits the curve into disjoint sections. Each section becomes a separate curve with boundary nodes.
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contents of the bu er are copied into the interface. The algorithm then proceeds to attach the nodes, then the curves, then the surfaces. If any element of the bu er overlaps an element of the interface, the structure native to the local interface will be kept. Any additional information from the new element will be added to the interface before the element is discarded. The identi cation of elements within the interface that correspond to elements in the bu er is achieved through a oating point comparison of point positions. It is desirable to limit the oating point comparisons as much as possible for reasons of both speed and accuracy. Only points near the boundary being extended will overlap, so the triangles in the bu er near the boundary are compared with the triangles in the interface near the boundary. A hash table of matching triangles is maintained until the bu er attachment is complete. Matching curves are also added to the hash table. A curve copied from the received bu er is compared with each curve in the local interface to nd any overlapping sections. If such a match is found, the bu er curve is merged into the local curve it overlaps. The curve is extended by any new points, and the node position is updated. Each new or overlapped bond of the curve is linked to any adjoining surface triangles, and the links for any curve-node or curve-surface attachments are updated. Merging the curves may lead to a new topology in the local interface. Figure 5.6 shows the inverse of the operation in Figure 5.4. The bu er in Figure 5.6(a) contains a curve to be merged with the curve in the local interface. In Figure 5.6(b), the two curves have been merged together, resulting in a closed curve. Extraneous 73
(a) Interface and bu er with overlapping curves
(b) Merged interface contains new closed curve
Figure 5.6: The inverse of the operation in Figure 5.4: two overlapping curves are merged together to form one closed curve. nodes are discarded. Figure 5.7 shows the inverse of the operation in Figure 5.5 . The bu er in Figure 5.7(a) contains multiple separate curves that will be merged with the curves in the local interface. In Figure 5.7(b), all the curves have been merged together into a single curve. A surface from the bu er must be merged with a surface in the local interface if the surfaces overlap at any triangle. To aid the identi cation of surfaces to be connected, each surface in the global interface is assigned an index number at initialization. When the interface is divided among the processors, each surface in the local interface retains the index number. In this way, searching for surfaces to connect is reduced to integer comparisons, instead of sorting through large numbers 74
(a) Interface and bu er with several overlapping curves
(b) Merged interface contains one single curve
Figure 5.7: The inverse of the operation in Figure 5.5: several overlapping curves are merged together to form one single curve.
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of oating point triangle comparisons to identify surfaces that overlap. When merging a bu er surface into the local interface, the triangle list for the bu er surface is attached to the end of the triangle list for the local surface. Triangles in the bu er that cross the computational boundary will duplicate a triangle in the local interface. These overlapping triangles are used to link the two surfaces together. The bu er version of a triangle will contain links to its neighboring triangles and bonds in the bu er. This information is copied into the local matching triangle. The bu er triangle is then discarded. If a bu er surface overlaps two or more local surfaces, they must all be merged into one surface. In Figure 5.8(a), the upper plateau of the bu er interface is a single connected surface that overlaps two surfaces in the local interface. The attachment of the bu er produces one single surface in the new interface (Figure 5.8(b)).
5.3 State Information at the Interface The interior of a three-dimensional FronTier grid is discretized into a regular grid of rectangular cells. The data stored in these cells is known as state information. A point of an interface surface will have state information for both sides of the surface: a left state and a right state. A point of a curve denoting an intersection of surfaces will have one state pair for each adjoining surface. The state information for a point along a curve is stored in the BOND TRI object, which represents a link between a curve bond and an adjoining surface triangle. As a result of this work, state information along curves is included in the scatter-gather update of interfaces. 76
(a) Bu er (left) and local interface (right)
(b) Interface after bu er has been attached
Figure 5.8: The bu er information in (a) is to be attached to the local interface. The two surfaces forming the upper plateaus in the local interface will both be attached to the corresponding surface in the bu er. The three surfaces are merged together to form one surface in the new interface (b).
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Extending parallel communication of state information to curves opens up new experiments for study using FronTier. To this point, three-dimensional parallel studies in FronTier focused on speci c problems with interfaces that did not require curves or nodes to represent surface intersections. An initial parallel application with curves in the interface is presented here. We take a problem from the realm of gas dynamics. A high-density gas jet is propelled into an ambient gas. As the jet plume evolves, it becomes turbulent, leading to bubbles breaking o from the surface. To capture this phenomenon numerically, a very ne mesh will be required | thus, the need for parallel implementation. The application of front tracking to gas dynamics is described in [9]. The equations of motion for an inviscid, compressible gas are given by the Euler equations:
t + r (v) = 0; (v)t + r (v v) + rp = 0; (E )t + r (( 21 jvj + H )v) = 0; 2
where is the mass density, v is the velocity of the uid particle, p is the pressure, E = jvj + e is the total energy, e is the speci c internal energy, and H = e + p is the speci c enthalpy. Assuming a calorically perfect ideal gas provides the polytropic equation of state which closes the system: 1 2
2
p = ( ? 1)[E ? jv2j ]; 2
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for some constant > 1. In the example shown in Figure 5.9, the same gas is used for both the jet and the ambient gas, with = 1:4. The ambient gas is initially motionless, with density = 1 and pressure p = 1. The jet has a density of = 10, pressure of p = 1, and an initial velocity v = (0; 0; 1). The initial perturbation for the jet (Figure 5.9(a)) is a semi-ellipsoid with height 2 and radius 2. The interface is composed of 3 surfaces: the planar surface forming the boundary of the ambient gas, the semi-ellipsoid at the jet-ambient contact, and the surface providing the boundary conditions for the jet. The three surfaces intersect in a closed, circular curve. Parallelizing this run requires the ability to handle the splitting and merging of this curve during communication. Figure 5.9(b) shows the results of a numerical experiment run with 4 processors. The ability to run such experiments in parallel will provide new avenues of study in gas dynamics and other elds.
79
(a) Initial interface for the highdensity gas jet
(b) Jet at time t = 14
Figure 5.9: A high-density polytropic gas jet injected into an ambient environment: the parallelization of this experiment requires curve splitting and merging.
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