X3D Moving Grid Methods for Semiconductor Applications*downloads.hindawi.com/journals/vlsi/1998/047283.pdfAs shown in the three examples presented, X3D grid optimization techniques

VLSI DESIGN1998, Vol. 8, Nos. (1-4), pp. 117-121Reprints available directly from the publisherPhotocopying permitted by license only

(C) 1998 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach SciencePublishers imprint.

Printed in India.

X3D Moving Grid Methods for SemiconductorApplications*

ANDREW KUPRAT *, DAVID CARTWRIGHT, J. TINKA GAMMEL,DENISE GEORGE, BRIAN KENDRICK, DAVID KILCREASE,

HAROLD TREASE and ROBERT WALKER

Los Alamos National Laboratory

The Los Alamos 3D grid toolbox handles grid maintenance chores and provides accessto a sophisticated set of optimization algorithms for unstructured grids. The applicationof these tools to semiconductor problems is illustrated in three examples: grain growth,topographic deposition and electrostatics. These examples demonstrate adaptivesmoothing, front tracking, and automatic, adaptive refinement/derefinement.

Keywords." Adaptive mesh smoothing, multimaterial grids, moving grids, moving surfaces,unstructured grids, moving finite elements

INTRODUCTION

3D grain growth modeling and 3D topographicsimulation have in common the requirement ofaccurately representing time dependent surfacemotion in a 3D volume. Problems involving fixedsurfaces such as parasitic parameter extraction forinterconnect modeling can benefit from adaptivegrid methods because they substantially reducesolution error from iteration to iteration. The LosAlamos X3D grid toolbox provides a set ofcapabilities including initial grid generation ofcomplex multimaterial geometries and grid opti-mization that preserves material interfaces. X3D

data structures and toolbox methods are designedas objects in order to be user accessible andextensible. X3D commands can be issued fromwithin an application driver program, and theexample applications use this feature to perform,as needed, grid reconnection, node merging andsmoothing. Additionally, all X3D data structuresare available to the application driver via calls toutility routines. Thus when an application detectsa non-routine event such as a topological changein a material region under deformation, a specialpurpose user routine can easily be incorporatedinto the system. These design features promote theseparation of the physical based simulation from

* Work supported by the U.S. Department of Energy.tCorresponding author.

117

118 A. KUPRAT et al.

the grid maintenance chores, but allow full accessto the grid data structures when required.

Gradient Weighted Moving Finite ElementsApplied to Metallic Grain Growth

In our first application involving the X3D toolbox[1], we use Gradient Weighted Moving FiniteElements [2] (GWMFE) to move a multiply-connected network of triangles to model theannealing of 3-D metallic grains. We assumeevolution of grain interfaces obeys the simpleequation

Vn #K,

where F is the normal velocity of the interface, # isthe mobility, and K is the local mean curvature [3].Gradient Weighted Moving Finite Elements mini-mizes

(Vn #K)2 dS

over all possible velocities of the interface vertices.(The integral is over the surface area of theinterfaces.) This leads to system of 3N ODE’s:

C (y) g(y),

where y is the 3N-vector containing the x, y, and z

coordinates of all N interface vertices, C(y) is thematrix of inner products of finite element basisfunctions, and g(y) is the right-hand side of innerproducts involving surface curvature. The ODE’sare integrated using an implicit variable time stepintegrator.As an example, we evolve a 5 grain micro-

structure in the confined geometry of an aluminuminterconnect on a semiconductor chip, and inFigure we show the smooth surfaces of the grainsat an intermediate time in their evolution undermean curvature. (The initial state for the timeevolution was obtained using Monte Carlo anneal-ing of a discrete effective model on a fixed lattice[4] and is not shown.) Visible in this exploded view

FIGURE 5 grain microstructure evolved under meancurvature motion using GWMFE.

FIGURE 2 Evolved 5 grain microstructure showing surfacegrid. Evolution has caused imbalance in grid density.

are triple points on the surface of the interconnect,as well as a triple line and tetrahedral point in theinterior. The corresponding surface grid is shownin Figure 2. It is clear that for this simulation to

continue, some of triangles must be annihilated toprepare for topological events such as the "pinch-ing off" of a grain. That is, a front trackingsimulation must involve a nontrivial amount ofgrid manipulation in order to successfully run tocompletion. In Figure 3, we show the effect ofmassage which is a grid manipulation command inthe X3D toolbox. As seen in the figure, themassage command has derefined the unstructuredmesh without significantly damaging the shapes ofthe grains. Indeed, the command takes as userinput a "damage" tolerance which gives themaximum acceptable amount of grain shapedeformation allowable in the derefinement pro-

X3D MOVING GRID METHODS 119

I,’.......>> M " ? ::’:-:"" ""?’’ ’/:";/)’’>’’,/’-.../".............7 ....,’ -.<<;............:: "’ /’: " ";;... ’.......

i//FIGURE 3 5 grain evolved grid a.erX3D massage com-mand. The grid has been derefined and refined according to .......:<..... I --/. .........:: , .. ........ w

and external boundaries.

cess. Also called is the recon command whichallows changing of connectivity of the associatedtetrahedral mesh in the interior of the volume. Stilla work in progress, we anticipate that with theavailability of these X3D grid manipulation toolswe will be able to run the annealing process tosteady-state. These microstructures will then pro-vide high quality three-dimensional models forelectromigration reliability simulation.

Finite Volume Electrostatic Calculationson a Solution Adapted Mesh

Problems involving fixed surfaces can also benefitfrom adaptive grids. To illustrate this, we show theeffects of solution adaptive grid generation forcalculation of electric fields in nontrivial geome-try- as occurs when attempting to extract para-sitic parameters for interconnect modeling. Figure4 shows the interior of a box with a conicalintrusion. We solve Laplace’s equation on thisthree-dimensional domain using a finite volumesolver. (Only the surface triangles are shown in thefigure; the calculation is done on unshown tetra-hedra that conform to the surfaces and fill thevolume.) As is well known, the electric fieldbecomes arbitrarily large near the tip of the cone.Displayed at the tip of the cone is an isosurface forthe component of electric field aligned with the

FIGURE 4 Surface triangles from an unadapted grid used tosolve Laplace’s equation on a unit cube with the bottom surfacepierced by a sharp cone. Displayed at the tip of the cone is anisosurface of the component of the electric field aligned with tileaxis of the cone.

FIGURE 5 Surface triangles after error-dependent gridadaption. Note improvement in electric field isosurface.Adaption was turned off on the surface of the cone to preventtoo much grid from disappearing into the tip of the cone wherethe electric field is infinite.

axis of the cone. As can be seen, this isosurface isnonsmooth due to lack of resolution in this area ofhighest solution error. In Figure 5 we show thesame view and electric field component isosurfacefor the solution on a grid that has been adapted toan a posteriori error estimate. The isosurface issmoother, indicating that adaptive error-based

120 A. KUPRAT et al.

smoothing has successfully attracted the solutionelements to the critical area near the tip of thecone. (However, to prevent the entire mesh fromdisappearing into the singularity at the tip of thecone, node movement on the surface of the conewas turned off.)The solution adaption was accomplished using

the X3D smooth command which adapts a mesh tominimize the/22 norm of the gradient of solutionerror, as calculated using an a posteriori errorestimate based on estimated second derivatives ofthe solution [5]. In order for the finite volumemethod to work, a Delaunay grid is required.Thus, after each adaptive smoothing iteration, theX3D recon command is called to adjust the meshconnectivity to restore the Delaunay condition onthe tetrahedral mesh. We note that the maximumelectric field on the adapted mesh of Figure 5 isapproximately ten times higher than that on theunadapted mesh of Figure 4. Thus, withoutincreasing the number of nodes used, the solutionadaption using X3D toolbox commands wasautomatically able to increase the quality of thesolution in a critical area.

area-weighted vector average of the contributionsfrom each of the triangles that share the node.

In low pressure simulations, the transport ofmaterial between the boundary plane and thesurface of a wafer is represented by straight-linetrajectories of particles. The flux of materialsarriving at a triangle on the surface interfacedepends on the source emanation rate, the distancebetween the source and the surface element, therelative orientation of the source and the surfaceelement, and the visibility of the surface as viewedfrom the source. In the simplest model, all thematerial arriving at the wafer surface stays there,implying that the only source elements are thosethat lie on the boundary plane. However, if somevapor phase materials arriving at the surface donot react there, they (or other species) will be re-

emitted, and in such cases, all interface trianglesserve as potential source elements. By specifyingthe surface chemistry (using the ChemKin andSurface ChemKin reaction software libraries),TopoSim3D incorporates the calculation of "stick-ing coefficients" in a natural way. If the rate of asurface reaction is slower on a surface element

3D Topographic Simulation

In simulating topographic etch and deposition,TopoSim3D separates the chemical and physicalprocesses from grid generation and maintenanceoperations. The flux calculation, source character-ization and chemical reaction mechanisms haveaccess to the needed X3D data structures and to theX3D geometry services. X3D is used to build theinitial 3D tetrahedral mesh that represents the waferand to accurately track the evolution ofthe materialinterfaces as they change with time. As material isdeposited or etched away from interfaces, thosetetrahedral faces (interface triangles) that lie on theboundary between two materials move in time.From a computation of the flux of materialsarriving at the surface of the interface triangles,we move triangle nodes after apportioning the areaof each triangle to each of the nodes it shares. Thevelocity of each node is then computed from an

(a)

FIGURE 6 Topographic deposition onto two materials withdifferent ’sticking coefficients’. View a shows the solid model ofthe wafer. View b gives the initial surface grid of the hole withthe front surface of the hole removed. View c shows anintermediate time step in the deposition. View d shows the finaltime step in the deposition. Note the differences in upper andlower material ’sticking coefficients’.

X3D MOVING GRID METHODS 121

FIGURE 6 (Continued).

than the arrival rate of reactants, mass balancerequires the excess material to be re-emitted fromthat interface triangle, which in turn becomes aneffective source element for other surface elements.As in grain growth modeling, the X3D grid

maintenance calls are used to refine, reconnect andsmooth the interface surface during the timeevolution. TopoSim3D uses the X3D data struc-

tures to help avoid folding problems and detecttopological events such as a pinch-off. Figure 6shows deposition on an overhang structure withmaterial-dependent sticking coefficients.

CONCLUSION

As shown in the three examples presented, X3Dgrid optimization techniques are well suited tosolving time-dependent and geometrically challen-ging problems occurring in semiconductor appli-cations.

References

[1] George, D. C. (1995). X3D User’s Manual, Los AlamosNational Lab Report, LA-UR-95-3608.

[2] Miller, K. (1997). A Geometrical-Mechanical Interpretationof Gradient-Weighted Moving Finite Elements, SIAM J.Num. Anal., 34, 67-90.

[3] Porter, D. A. and Easterling, K. E. (1988). Phase Trans-Jbrmations in Metals and Alloys, Great Britain: VanNostrand Reinhold, pp. 130-136.

[4] Kuprat, A. and Gammel, J. T. Modeling Metallic Micro-structure Using Moving Finite Elements, preprint, availableat http://xxx.lanl.gov/abs/physics/9705041.

[5] Bank, R. E. and Smith, R. K. (1997). Mesh smoothing usinga posteriori error estimates, SIAM J. Num. Anal., 34,979-997.

Author Biography

Andrew P. Kuprat received his Ph.D. in Mathe-matics from the University of California atBerkeley and is currently a staff member at LosAlamos National Laboratory. He is interested inadaptive mesh algorithms applied to problems insemiconductor process and device modeling, aswell as to other convection/diffusion problems inphysics. He is also interested in mesh generationand computational geometry.

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X3D Moving Grid Methods for Semiconductor Applications*downloads.hindawi.com/journals/vlsi/1998/047283.pdfAs shown in the three examples presented, X3D grid optimization techniques