On modelling thermal oxidation of Silicon II: numerical aspects

*Correspondence to: Thomas J. R. Hughes, Department of Mechanical Engineering, Stanford University, DurandBuilding, Stanford, CA 94305-4040, U.S.A. E-mail: [email protected] address: Schlumberger Cambridge Research Limited, Cambridge, U.K.tProfessorAResearch Associate

Contract/grant sponsor: Defence Advanced Research Projects Agency; contract/grant number: DABT63-93-C-0053

CCC 0029-5981/2000/020359}21$17.50 Received 1 April 1999Copyright ( 2000 John Wiley & Sons, Ltd.

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng. 47, 359}377 (2000)

On modelling thermal oxidation of SiliconII: numerical aspects

Vinay S. Raos, Thomas J. R. Hughes* and Krishna Garikipati A

Division of Mechanics and Computations and Center for Integrated Systems, Stanford ;niversity, Stanford,CA 94305, ;.S.A.

SUMMARY

The implementation of a new model to simulate the oxidation of Silicon (see Part I) is presented in thispaper. The implementation is done within a "nite element framework. The work involves representation ofthe Silicon}Silicon dioxide interface in a mesh-independent manner. The interface description and evolutionis accomplished using the level-set method. Formulation of a single "nite element containing multiplematerials is described. A discontinuously varying oxidant density "eld is modelled along with its reactionwith Silicon. Large expansion of Silicon dioxide is incorporated into the constitutive equations. A staggeredscheme to obtain the solution to the coupled set of equations is given. Copyright( 2000 John Wiley & Sons, Ltd.

KEY WORDS: thermal oxidation; continuum mechanics; "nite elements; discontinuous interpolations; levelsetmethods

1. INTRODUCTION

Very large-scale integrated circuits consist of millions of basic devices like transistors andcapacitors built on a single wafer of Silicon. These devices are electrically isolated from oneanother by means of &isolation structures' of Silicon dioxide. Silicon dioxide is also used asdielectric in capacitors and as gate oxide in metal oxide semiconductor devices. Construction ofSilicon dioxide structures is thus an important part of semiconductor manufacturing. Thermaloxidation of Silicon is one of the processes for creation of oxide structures. Thermal oxidation isthe process in which an oxidant (oxygen or steam) reacts with Silicon to form Silicon dioxide attemperatures ranging between 700 and 12003C. In this paper a numerical framework is proposedto solve the equations that model thermal oxidation. The displacements and stress "elds are thenused to characterize the behaviour of the device.

In the work of Ra!erty [1] and Peng et al. [2] the approach to oxidation simulation was asfollows. Two sets of equations were proposed. These were, the di!usion equation based on themodel of Deal and Grove [3] and the quasi-static momentum balance equation. The Deal}Grove

Figure 1. Interface: (a) along element edges; and (b) arbitrarily placed

model used Fickian di!usion and the reaction was modelled as a Neumann boundary conditionon the oxide domain. The momentum equation incorporated the expansion of Silicon dioxidethrough a one-dimensional model to arrive at a velocity which was applied as a Dirichletboundary condition. The formulation had velocity as the primary variable and the displacementincrements were computed by multiplying the velocity at the previous time step with the time-steplength. These equations were then solved as a set of coupled equations. The coupled solutionstrategy eliminated the need to iterate until a self-consistent solution was obtained.

These approaches have fundamental problems at the modelling level and at the numerical level.At the modelling level the mass balance of the oxidant is not considered. The mass balance law alsoprovides the Si}SiO

2interface motion speed. Expansion of any materials is a three-dimensional

phenomenon and must be modelled as such. Also, the expansion must be incorporated as a materialproperty in the constitutive equations to give the correct limiting conditions such as the freeexpansion and the fully constrained cases. The reaction of Silicon and Oxygen results in theformation of Silicon dioxide and is accompanied by an expansion of 120 per cent. This causes largedeformations which must be modelled with "nite deformation kinematics. The wafer-bendingexperiment of Yu et al. [4] highlights the shortcomings of the one-dimensional expansion modeland the small deformation theory. The bending cannot be predicted by the one-dimensionalmodel and deformation is large enough to warrant "nite deformation kinematics.

The mathematical modelling is considered in detail in Part I of the paper. A detailed derivationof the di!usion equation from the mass balance law is presented. The interface velocity derivationis described in a manner that conserves mass of reacting species. An expansion model is presentedwithin the "nite deformation theory. This model incorporates "nite expansions in a mannerconsistent with the kinematics of the motion. Finally, a thermodynamically consistent viscoelasticmodel based on experimental studies is given.

At the numerical modelling level, consider the description of two materials in terms ofa computational mesh. Figure 1 shows two ways in which a material interface can be handled.Case (i) assumes the interface to coincide with element edges. This makes the element formulationsimple. One idea would be to tag the elements of the two regions di!erently and have the "niteelement solver compute the sti!ness matrices at the element level for the two regions separately.This makes for a very di$cult formulation for the moving interface case. When the interfacemoves, the elements get distorted, although the domain (considering the displacement frozen)does not change. This would also require mesh checking algorithms to prevent serious degrada-tion of the mesh quality. From this stage it will be di$cult to recover the solution even with meshadaptation or remeshing. An added disadvantage is the inability to appropriately enforce theinterface conditions. This description is used in the computer program SUPREM [5], Law [6]

360 V. S. RAO, T. J. R. HUGHES AND K. GARIKIPATI

Copyright ( 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 47, 359}377 (2000)

and Rueda et al. [7]. Case (ii) presents an attractive alternative. The arbitrary placing of the elementboundary allows for a much simpler solution strategy. However, the element level formulation isdecidedly more complicated. The elements need to be enhanced with discontinuous shape functionsto handle the density pro"les. But, at the mesh level, no mesh validation algorithms are required.Remeshing may be performed solely for accuracy reasons and is not a stability requirement.

In this work the "nite element method is developed such that the Si}SiO2

interface can bedrawn arbitrarily across the mesh. Non-linear solution methods are used to solve the "nitedeformation kinematical models. This allows the use of a large expansion model. The timeintegration of the general viscoelastic model is done with an e$cient two-point recurrenceformula that is second-order accurate in time and unconditionally stable. The interface motion isdescribed by the level-set method that involves the solution of a non-linear advection equation.This is accomplished by using a stabilized Galerkin formulation.

In Section 2 a model to describe the di!usion}reaction process is presented brie#y and the "niteelement formulation to solve it is proposed. In Section 3 the mechanical problem is presentedalong with the "nite element implementation of the expansion model and constitutive equations.Section 4 brie#y outlines the staggered algorithm to solve the set of equations.

2. DIFFUSION}REACTION PROCESS

In this section the di!usion}reaction equations will be presented in the weak form, which is thesetting for the "nite element implementation of the equations. Interface description and evolutionusing the level-set method is described in Section 2.1. In Section 2.2 the enhanced gradientmethod to interpolate discontinuous variation of the density and the #ux is described.

Consider a body made of two materials occupying the region )3R3. Material 1 occupies thedomain )~ and Material 2 occupies the domain )`, where )~ and )` are open subsets of ).The oxidant is di!used in the domain )~. At the interface !, it reacts with Material 2 to formMaterial 1 and the interface advances into Material 2. The interface, !, moves in the direction ofits normal n with speed<

n. The mass balance equation is given below in terms of the density o and

a di!usive #ux T.Consider the space of weighting functions, V, given by

V"Mw :w3H1 over ), w"0 on L)N (1)

and the space of trial functions, S, given by

S"Mo :o3H1 over )$, o"g on L)$N (2)

Note that, o is continuous over the domains )` and )~ (Figure 2). It can be discontinuous acrossthe interface !. The weighting function, w, is continuous across the domain. In the developmentthat follows, it is necessary that w is continuous.

The weak form of the mass balance equation for each region is given below

(w, o,t))~!(+w, ov)L)~#(w, ov ' n~)!!(w, o~v ' n~)!

#(w, o~<n)!#(+w, T))~!(w, T ' n)L)~#(w, T~ ' n~)!"0 (3)

(w, o,t))`!(+w, ov)L)`#(w, ov ' n`)!!(w, o~v ' n`)!

#(w, o`<n)!#(+w, T))`!(w, T ' n)L)`#(w, T` ' n`)!"0 (4)

ON MODELLING THERMAL OXIDATION OF SILICON II: NUMERICAL ASPECTS 361


Figure 2. Domain representing the body ) with a discontinuity

In equations (3) and (4) the symbol ( ' , ' ) represents the inner product over the domain givenby the subscript. The inner product is computed by scalar multiplication if the arguments arescalars and the Euclidean inner product for vectors and tensors. The region ) is such that

)1 ")~X)`X! and the boundaries of the individual regions are given by L)$X!.Adding the two equations, we get

(w, o,t))3 !($w, ov)L)3 #($w, T))3 !(w, T ' n)L)3

#(w, Iov ' nJ)!!(w, IoJ<n)!!(w, IT ' nJ)!"0 (5)

where n is the unit outward normal on ! with respect to )~. We also denote )3 ")~X)`. Thesymbol I ' J denotes the jump in the argument across the interface !.

The constitutive equation for the #ux, T, is assumed to be the same as the Fickian di!usionmodel used by Deal and Grove [3]

T(x)"!D(x)+o(x) x3)3 (6)

where the di!usivity is given by D. The gradient of o is evaluated away from !, the surface ofdiscontinuity. This is consistent with the function space S that o belongs to.

The weak form stated in equation (5) is the complete form of the mass balance. Di!usion ofoxidant in the domain ) can be simpli"ed by applying the following assumptions.

The rate of change of the oxidant density at a given point is considered to be small. This impliesLo/Lt"0. In addition assume the velocity of the current point x is small, implying v"0. Thesetwo assumptions give the steady-state equation of mass balance as follows:

($w, T))3 !(w, T ' n)L)3 !(w, IoJ<n)!!(w, IT ' nJ)!"0 (7)

Finite element computations entail the projection of the solution space onto a "nite-dimensionalsubspace. A "eld projected onto the "nite-dimensional subspace is denoted by the superscript h.The weak form in the "nite-dimensional space is given by

($wh, Th))3 !(wh, Th ' n)L)3 !(wh, IohJ<n)!!(wh, ITh ' nJ)!"0 (8)

In equation (8), the "rst term is the weak form of the Fickian di!usion operator, the second term isthe Neumann boundary condition term and is non-zero only in regions where non-zeroNeumann boundary conditions are speci"ed, the third and the fourth term are the jumpconditions across the surface of discontinuity !. By employing a construction given in Chadwick



[8] the interface velocity is given by

<n"!ITh ' nJ/IohJ (9)

The "nite element implementation of the di!usion equation in regions where the "eld variable issmooth is well documented and the techniques given in the books by Hughes [9] and Zienkiewiczand Taylor [10] are used.

The mathematical description of a moving surface of discontinuity is given in the followingsection.

2.1. Level-set method

The level-set method of Sethian [11] is used to describe the evolution of the Si}SiO2

interface.A brief description of the method is presented here. This is followed by the numerical implementa-tion of the method which includes the weak form of the level-set equation, spatial stabilizationrequired for the advective operator and the time-stepping scheme.

Consider ) with surface of discontinuity, !, which can be represented as a level set of a surfacet. Note here that ! is a surface, hence it can be parameterized, at least piecewise, in ann!1-dimensional space. Similarly t can be parameterized, locally, as an n!1 manifold,assuming adequate smoothness of the surfaces.

The problem of tracking an n!1-dimensional surface is thus transformed to one for theevolution of a scalar "eld, t.

A family of hypersurfaces tt, parameterized by t represent the evolution. Without loss of

generality, !tcan be the zero level set of t

t. Mathematically, it can be stated as

!t"Mx :t

t(x)"0, x3)N (10)

Evaluating the derivative of t with respect to t and employing the chain rule, we get

LtLt

#a '$t"0 (11)

where a is the advective velocity. If the surface is assumed to advect in a direction normal to itselfthe velocity term, a, can be written as <

nn. <

nis the speed of propagation of the front. The normal

n is de"ned as n"$t/E$tE. Substituting in equation (11), we get

Lt/Lt#<nE$tE"0 (12)

In two dimensions or higher, equation (12) is non-linear. In one dimension it is a wave equation ofthe "rst order. In higher dimensions it can also be shown to support wave like solutions.

Speci"cation of the initial conditions for t is described next. It must be remembered that t isa mathematical construction from which the interface is extracted. Once the scalar "eld is known,the interface is obtained as its zero level set. One scalar "eld allows the two-material problem tobe represented. This follows by de"ning one region by t'0, the other by t(0, and the interfaceby t"0. Being hyperbolic in nature, equation (12) can support discontinuous initial conditionsand propagate them along the characteristics. The following initialization procedure has beensuggested by Sethian [11] and is based on the distance function.

The minimum distance to the interface from any point, x is set as the initial condition t(x, 0).When discretized, the scalar "eld, t, is to be initialized at the mesh points (nodes) and



interpolated to any desired point. Imposing continuity of the interface will automatically enforcethe continuity of the initial condition.

In mathematical terms, the initial condition is de"ned as

t(x, 0)"$d(x) (13)

where d(x)"miny3!0(Ex!yE) and the sign is decided based on the region in which x lies. !

0is

the interface location at time t"0. If the front divides the domain into two regions, one may beassigned &#' and the other &!'.

Stability of solution requires the normal to the shock plane to be in the same direction as theshock propagation. For the case at hand, it can be enforced by assigning Silicon dioxide to havet'0 and Silicon, t(0, because the interface moves into Silicon.

In the following section the numerical implementation is presented with the statement of theweak form, the stabilization procedure for advection and the time-stepping method.

2.1.1. Weak form of the non-linear advection equation. De"ne the weighting function space asfollows:

V"Mw : w3H1N

In a hyperbolic problem it is su$cient to prescribe Neumann boundary conditions on the in#owboundaries. The trial solution space, S, is identical to V, as there are no Dirichlet boundaryconditions. Thus

S"V (14)

The weak form is given as follows:Given d, ∀w3V

(w, tt)#(w, a '$t)"0 (15)

t(x, 0)"$d (16)

where a"<n$t/E$tE , is the advection velocity. The dependence of a on $t makes the equation

non-linear. In the one-dimensional case, however, it reduces to a linear, "rst-order wave equation.The weak form used here is of the form stated in the works of Mallet and Hughes [12]. The

spatial term is not integrated by parts. Hughes et al. [13] have shown that conservation of #uxescan be trivially satis"ed if the spatial term is integrated by parts. In this case however, thenonlinearity lead to excessively complicated terms.

Purely advective systems are known to give rise to spatial instabilities. This has beendocumented in Brooks and Hughes [14]. Spatial stabilization as suggested by Hughes andcoworkers is presented in the following section.

2.1.2. Spatial stabilization. In the numerical solution of advection dominated #ows spatialinstabilities may arise. These instabilities are controlled by adding residual based di!usion termswhich vanish as the solution is better represented. Two of the commonly used methods are:Streamlined Upwind Petrov}Galerkin (SUPG) (see [14]) and Galerkin Least Squares (GLS) (see[12, 15, 16]).

A very brief overview of GLS is provided here. Mathematical details about its stabilityand design of parameters are described by Hughes et al. [15]. Given a di!erential equation in



Plate 1. Pressure contours in dyn/cm2 after one hour of oxidation

Copyright © 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng.47 (2000)

Plate 2. Contour plot of σ11 after 100 minutes of oxidation. The x1 direction is to the right along the length of the wafer

strong form:

/,t#L/"f in )

where L is the spatial di!erential operator, the weak form is given by

(w, /, t)#a (w, /)"(w, f )

The term, a ( ' , ' ) is the weak form of the spatial di!erential operator L. It is obtained byintegration by parts of the inner product term (w, L/). The problem is assumed to have onlystrongly enforced Dirichlet boundary conditions, hence the boundary terms do not appear in theweak form (following integration by parts).

The stabilization term suggested by the GLS scheme is a least-squares operator containing theresidual. As the solution tends to the exact solution, the residual goes to zero. The stabilized weakform is given by

(w, /, t)#a (w, /)#(Lw, qR(/))"(w, f )

where R"/, t#L/!f. The least-squares form of the new term motivates the name, GLS. For

the problem under considerationL"a '$. It is a "rst-order non-linear di!erential operator. Thedesign parameter q is to be evaluated (see [15]). The &doubly asymptotic' approximation isemployed.

In the semidiscrete formulation employed herein the shape functions are purely spatial. Thematrix contributions of the various terms are given as follows:

mij"P)N

iN

j

kij"P)N

iakN

j,k

mJij"P)1 akNi,k

qNj

kIij"P)1 akNi,k

qalN

j,l

egggfgggh

sum over k, l"1, . . . , 3 (17)

Assembling the mass terms mij

and mJij

and sti!ness terms kij

and kIij

into matrices M and K,respectively, the matrix equation is given by

Mw0 #Kw"0 (18)

Equation (18) is a system of simultaneous ordinary di!erential equations. The integration schemeto solve for w is described in the next section.

2.1.3. ¹ime stepping. To solve the system of equations, the explicit forward-Euler method willbe used. Explicit methods are computationally e$cient as they do not involve solutions ofalgebraic systems of equations.

Consider the sti!ness term, K, in equation (18). Rewriting it in operator form

Rs"(w, a '$t(x, t))#(a '$w, qa '$t(x, t))"0 (19)



Figure 3. Discontinuity in one-dimensional example

where Rsis the spatial part of the total residual operator. At time t

n, let t(x, t) be denoted t

n,

assumed known. The forward-Euler algorithm gives

Aw,tn`1

!tn

*t B#Rs(t

n)"0 (20)

where Rs(t

n) is a non-linear function of t

n. The algebraic system corresponding to (20) can be

written as

Mw0n`1

!Mw0n#*tR(t

n)"0 (21)

If M is lumped, the equations uncouple. Mass lumping can be performed in several ways [9].

2.2. Enhanced gradient method

The knowledge of the Si}SiO2

interface can be used to construct very accurate "nite elementimplementations were the exact description of the jump can be obtained. This involves enhance-ments to the piecewise linear "nite element shape functions. In this section the enhancement tothe shape function is motivated with a one-dimensional example and is described in detail for thetwo-dimensional case. The method can be extended to compute the shape functions for thethree-dimensional elements also.

2.2.1. One-dimensional case. In Figure 3, there is a discontinuity in element number 3. Thebroken line represents the nodally exact piecewise linear interpolate of the solution. The solid lineindicates the exact solution. The di!erence between the solid and broken lines is the error.The linear shape functions need to be enhanced to incorporate this di!erence.

In the strong form of the di!usion equation, two interface conditions are required for thesolution of a problem that has discontinuities in #ux and concentration at the interface. The "niteelement implementation of the enhanced gradient method also needs two conditions to fullydescribe the enhancements to the shape functions. In the one-dimensional case, the two un-knowns can be the height of the jump and the fraction of the jump height above zero. The twounknowns are computed from two conditions that arise out of the physics of the problem and arecalled the segregation conditions. These are, the #ux segregation, given by the #ux jump at theinterface, and, the concentration segregation, given as the ratio of the density of the oxidant aboutthe interface.



2.2.2. Two-dimensional case. In one dimension, the segregation conditions are exactly enfor-ced at !, which is a single point and is known. In higher dimensions the weak enforcement of thesegregation conditions requires the integration of the conditions along the surface of thediscontinuity. Two-dimensional elements have, in general, two adjacent elements through whichthe surface of discontinuity passes. Continuity of the jump heights couples the jump unknowns.This would require assembly of the jump unknowns and the need for global solution as opposedto explicit evaluations within each element. In the following construction of the shape function,the jump height is assumed to be discontinuous across adjacent elements. This uncouples thejump heights and the unknowns can be statically condensed at the element level. In fact, they canbe determined analytically as described below.

The enhanced formulation will be discussed here for linear triangles. Consider Figure 4. TheLagrangian shape functions of a linear triangle are N"[m, g, 1!m!g]T. Let (m

0, g

0) be the

mid-point of the segment of the surface of discontinuity within the element.Using the level-set description of Section 2.1 to de"ne regions by t'0 or t(0, the

enhancements to the shape functions for the triangle are given by

Case I:

NI1"

igjgk

Nb#N

c(N

b#N

c) D(m0,g0)

t'0

0 t(0

(22)

NI2"

igjgk

!

Na

Na(m

0, g

0)

t(0

0 t'0

(23)

Case II:

NI1"

igjgk

Na

Na(m

0, g

0)

t'0

0 t(0

(24)

NI2"

igjgk

!

Nb#N

c(N

b#N

c) D(m0,g0)

t(0

0 t'0

(25)

Enforcing the segregation conditions exactly at the mid-point (m0, g

0), using a as the jump in o and

c as the fraction of the jump in the domain where t'0

!ks(NTc#acNI

1)"[(D

1!D

2)$ (NTc)#acD

1$NI

1!a(1!c)D

2$NI

2] ' n (26)

NTc!a(1!c)NTc#ac

"m (27)



Figure 4. Cases where the isolated nodes is in material 1 or material 2

where c is the vector of nodal densities, and ks, m, D

1and D

2are the following material properties:

reaction rate coe$cient, density ratio, di!usivities in the regions t'0 and t(0, respectively.The normal to the interface is given by n.

Solving for ac and a (1!c) in terms of c, the following expressions are obtained:

ac"lTc (28)

a(1!c)"qTc (29)

The vectors l and q are given by

l"1

FI 1n#mFI 2

n!k

sNI

1

ks

N1

N2

N3

#(D1!D

2)

N1,x

N1,y

N2,x

N2,y

N3,x

N3,y

n

!(1!m)D2

N1

N2

N3

[NI2,x

NI2,y

] n (30)

q" (1!m)

N1

N2

N3

!ml (31)

where FI 1n"!D

1$NI

1 'n and FI 2

n"!D

n$NI

2 'n .

In summary, the enhancement is given by NI1, NI

2, where NI

1'0 and NI

2(0. This de"nition

now allows the following representations for an element through which the interface passes

o"(N#NI1l#NI

2q )T c (32)

T"!D$(N#NI1l#NI

2q )T c (33)

Clearly, the enhanced shape function NIi

is linearly independent of the Lagrangian shapefunctions. This is obvious given that linear combinations of continuous linear polynomialscannot represent a discontinuity. Similarly, the gradients of the enhancements are also indepen-dent of the gradients of the regular part. Also, the constant #ux condition property is trivially



inherited from the Galerkin part of the enhanced shape functions. The above two propertiesimply satisfaction of the patch test in the sense of Taylor et al. [17]. For a review of theseconditions refer to Simo and Rifai [18].

The development of enhanced discontinuous interpolations may be found in Armero andGarikipati [19] and Simo et al. [20].

3. STRESS EQUILIBRATION

The reaction of Silicon with oxygen to form Silicon dioxide is accompanied by 120 per centexpansion. When the expansion is constrained, this causes stresses in the bulk Silicon dioxide andin the Silicon substrate.

In this section the local momentum balance law will be recalled followed by the description ofthe constitutive model. The expansion model will be described brie#y followed by the numericalintegration of the constitutive model and "nally the "nite element implementation of the stressproblem.

3.1. Local momentum balance law

The Lagrangian form of the balance equations takes the form

u5 "V

o0V0 "DIV[P]#o

0BH on )][0, t] (34)

where o0

is the material density, P is the "rst Piola}Kirchho! or the nominal stress tensor, B isthe body force vector, superposed dots signify material time derivatives, and DIV[ ' ] denotes thedivergence operator in the reference con"guration.

The initial-boundary value problem is given by equation (34), accompanied by the followinginitial and boundary conditions:

u (X, 0)"u0

on )

V(X, 0)"V0

on )

u(X, t)"u6 (t) on L)u

PN"h (t) on L)h

3.2. Constitutive equations

Assume the Helmholtz free energy function, (, is given by

(~(E, Q)";~(J)#(1 ~(E1 )!Q :E1 #(I(Q) in )~ (35)

(`(E)";`(J)#(1 `(E1 ) in )` (36)

where J is the determinant of the deformation gradient, E is the Lagrangian strain tensor and Q isthe tensor of internal stress-like variables [21]. The terms ( '6 ) involve the isochoric part of the



deformation gradient that is uncoupled multiplicatively into dilatational and isochoric parts, viz.

F1 "J~1@3F (37)

The term (Iis de"ned at equilibrium by

(I"!c(1 ~#Q : E1 (38)

Following standard procedures of thermodynamics, the constitutive equation is given by

S$

"L($/LE (39)

where S is the second Piola}Kirchho! stress tensor and

!

L(~

LQ: Q0 *0 (40)

which follows from the Second Law of Thermodynamics. For the complete description of a rate-dependent constitutive law the evolution of the internal variable needs to be speci"ed. This isdescribed in the following section.

3.2.1. Evolution equation for the internal variables. Linear evolution equations are postulatedfor the internal variable Q. In this section, Q will be replaced by Q

isignifying the ith viscoelastic

element (see Part I).This also enforces &standard solid' type of viscoelastic behavior [21, 22]:

Q0i#

1

qi

Qi"

ci

qiGDEVC2

L(1 ~LC1 DH (41)

Q Dt/0

"0 (42)

where ci3[0, 1] is subject to the restriction that

N+i/1

ci"1!c

=with c

=3[0, 1) (43)

and qi'0 is the relaxation time.

3.2.2. Expansion formulation. Assume that the expansion occurs in two stages. In the "rst stagean in"nitesimally small local region expands freely and in the second stage it is deformed bythe action of the surrounding regions. Mathematically, this leads to a multiplicative split of thedeformation gradient given by

F"F1 Fh (44)

This setting was proposed for thermomechanical problems by Lu and Pister [23]. The multiplica-tive split of the deformation gradient and the concept of the stress free state was proposedoriginally (in the context of "nite plasticity) by Lee and Liu [24].

As an example, consider a neo-Hookean model with quadratic logarithmic volumetricresponse. This has been called the neo-Hookean model with an extension to the compressible



regime in Simo [25]. The stored energy function is written as

((b1 )"12(i log[JM ])2#1

2k[JM ~2@3b1 : 1!1] (45)

where

12(i log[JM ])2"1

2(i log[J]!log[a3])2 (46)

12k[JM ~2@3b1 : 1!1]"1

2kCA

J

a3B~2@3

a~2b : 1!1D"1

2k[J~2@3b : 1!1] (47)

and JM "J/a. The component a3 (0, R), of the Jacobian J represents an arbitrarily chosenportion of the volume expansion ratio. In the present context, a"2)2 shall be used. Thisformulation is conveniently extended to the viscoelastic model.

3.3. ¹ime integration of the constitutive law

Integration of equation (41) is fundamental to an e$cient numerical implementation of theviscoelastic model. Being a rate-dependent model, the constitutive equations require time integra-tion. To evaluate S, the internal variable Q needs to be evaluated

Equation (41) is linear and "rst order. The integration can be carried out by the method ofvariation of parameters used in the solution of ordinary di!erential equations [26]. Timeintegration yields the familiar convolution equation

Q(t)"cPt

0

expCs!t

q Dd

dsGDEV CL(1 ~LE1 DHds (48)

A direct evaluation of equation (48) involves knowledge of the history over [0, t) for each timeinstant t. The problem is obviated by the following procedure leading to a numerical integrationalgorithm. Consider the convolution integral:

h"Pt

0

expCs!t

q Dd

dsqds (49)

The equation can be written for time tn`1

, noting tn`1

!tn"*t

hn`1

"Ptn

0

expCs!t

n!*t

q Dd

dsq ds#P

tn`1

tn

expCs!t

n`1q D

d

dsqds (50)

Splitting the exponential term in the "rst integral on the right-hand side and de"ning

hn"P

tn

0

expCs!t

nq D

d

dsqds (51)

the following relation is obtained

hn`1

"hnexpC

!*t

q D#Ptn`1

tn

expCs!t

n`1q D

d

dsqds (52)



Approximating the second term using the mid-point rule results in

hn`1

"hnexpC

!*t

q D#expC!*t

2q D Mqn`1

!qnN (53)

This approximation is second-order accurate. In addition, the convolution has been reduced toa two-point recurrence formula. At any given time step, only h

nand q

nneed to be stored.

Withh :"Q (54)

q :"DEVCL(1 ~LE1 D (55)

and multiplying by c, the recurrence relation for equation (41) is obtained.

3.4. Finite element formulation

The "nite element formulation used for the solution of the momentum equation, in conjuctionwith the level-set method, is described in this section.

The elements that are situated entirely in the Si or SiO2

domain are &single material elements'.The weak form of the standard displacement formulation of the "nite deformation kinematicalproblem with the viscoelastic constitutive equation is described in standard texts [22]. Finitedeformation kinematics engender non-linearities. To obtain linear systems of equations, consis-tent linearization is performed for each iteration of the Newton}Raphson method. The linearizedoperator is integrated with full Gaussian quadrature. This straight-forward method is used tocompute the contributions of the single-material Si and SiO

2elements to the sti!ness matrix.

Next we consider the case where an element has the material interface passing through it. Thelevel-set variable can be used to compute the zero level-set which de"nes the interface. For allthe elements where the level-set function changes sign, the exact location of the interface can becomputed. This construction divides the triangle into a triangle and quadrilateral, in general. Thedegenerate cases can have the zero level-set coinciding with an edge or passing through a vertex ofthe triangle. In either of these cases, the level-set function does not change sign.

Once the element with the interface passing through it has been divided into two regions, theintegration of the consistently linearized weak form with the appropriate constitutive equation ineach region can be carried out. The contribution of each subelement is added to form the sti!nessmatrix of the element which is then assembled in the usual manner to form the global sti!ness matrix.

In Ra!erty [1] it is observed that SiO2behaves as an incompressible material. The phenomena of

locking and checkerboarding are common manifestations of discrete approximations of incompressi-bility. These can be overcome by means of mixed interpolations. The method used here wasproposed by Simo et al. [25]. It is regarded as the non-linear extension of the B-bar method of Hughes[13]. Other methods such as that of Nagtegaal et al. [27] can also be derived from this formulation.These are some of the techniques that can be employed to overcome incompressibility [3].

4. COUPLED PROBLEM

The solution of the coupled problem is performed in a staggered fashion. Following the de"nitionof the initial oxide and silicon regions, the momentum balance equation is solved to compute



the stresses. The stress "eld is used to compute the di!usivity and the reaction rate coe$cient.Di!usivity and the reaction rate coe$cient depend on the hydrostatic pressure and the stressnormal to !, respectively, through Arrhenius relationships. This is described in Ra!erty [1] and isused in SUPREM [5] as well as Peng et al. [2]. The stress dependence in the present non-linearcontext has been outlined in Part I of this work. The di!usion equation is solved to compute theconcentrations in the domain and the interface speed. At the next time-step the interface positionis updated and the entire procedure repeated.

Momentum balance does not require the oxidant density to solve for the stresses. However, thedi!usion equation is still coupled to the momentum equation through the stress-dependentdi!usivity.

This procedure does not need iteration. The advantage of the staggered scheme is that theglobal matrix solution does not require all the variables to be solved for at once.

5. NUMERICAL EXAMPLES

In this section the calibration of the viscoelastic model in its simplest setting will be described. Theviscoelastic model will be used in three problems that will be presented. The "rst problem is thecylinder expansion experiment, the second is a local oxidation structure simulation and the thirdis a wafer bending experiment simulation.

5.1. Calibration

The experiments of Yu et al. [4] involved isothermal annealing of a wafer of silicon dioxidein a newly created experimental set-up. The setup measures the change in wafer curvaturewith respect to time after it has been oxidized in a dry oxygen ambient at a given temperature.Details of the procedure adopted to extract material properties for the standard solid visco-elastic model are presented in Garikipati et al. [28]. Here, it su$ces to summarize theprocedure: The standard solid viscoelastic model is particularized to the case of in"nitesimalstrains, since the thermal-mismatch strains which drive the isothermal relaxation experimentsremain well within the regime of the in"nitesimal theory. For the prevailing biaxial state of stressin the SiO

2"lm, a simple formula emerges for time-dependent stress relaxation. The simplest

standard solid model was considered, i.e. one with a single Maxwell element (see Paper I, Section4). A least-squares "t is performed at each of several temperatures to obtain i, k, c

1and q

1. The

experiments thus also provide information on the temperature dependence of the mechanicalparameters.

5.2. Cylinder expansion

Oxidation of Si cylinders was "rst performed by Kao [29]. The problem can be modelled astwo dimensional by considering a slice of the cylinder. This is done by using the plane-strainhypothesis.

One quadrant of the circular cross-section was used and the expansion and di!usion wereassumed to the isotropic. This renders the problem axisymmetric, but the problem was solved astwo dimensional to show that the method can be used with an unstructured mesh and produce



Table I. Material properties

Property Value

iS*O2

12)158 GPakS*O2

40)921 GPac1S*O2

0)9697qS*O2

132)49 siS*

141)67 GPakS*

73)047 GPaD

S*O21)3786]10~6 cm2/s

ks

!7)8752]10~3 cm/sD

S*1)3786]10~8 cm2/s

Figure 5. Progression of cylinder oxidation showing evolution of the interface and expansion of newly formed SiO2(outer

portion of each "gure)

good quality results. Material properties corresponding to mechanics and di!usion}reactionparameters appear in Table I.

Figure 5 shows the progress of the oxidation process. The interface position is plotted and canbe clearly seen to be circular even after several hundred time steps. The expansion is alsohomogeneous and shows no directional preference.

The problem was solved in 300 time steps. The mesh had fairly good quality triangles and thesize of the triangles did not vary over the domain.

No remeshing was required. The expansion is seen in the form of the enlarged triangles awayfrom the centre of the cylinder. Since the Si}SiO

2interface is not restricted to lie along the element

edges, the mesh quality does not deteriorate with moving boundaries.



Figure 6. Evolution of the Si}SiO2

domain

Table II. Material properties of nitride mask

Property Value

iS*3N4

88)067 GPakS*3N4

61)415 GPacS*3N4

0)9697qS*3N4

2926)9 sD

S*3N41)0]10~14 cm2/s

5.3. ¸ocal oxidation structure

Local oxidation is referred to as LOCOS in the process modelling literature. LOCOS struc-tures are widely used. In this example the Silicon substrate is elastic and has dimensions of 1 kmby 2 km. The pad oxide is 50 As thick. The nitride mask is 0)2 km thick. The oxide and nitride areviscoelastic. The di!usion depends on the hydrostatic stress and the reaction rate is also stressdependent.

Material properties for Silicon and SiO2correspond to those in Table I. The properties used for

nitride (Si3N

4) are presented in Table II.

The mesh was constructed once and no remeshing was required. No time step change wasrequired throughout the computation. Figure 6 shows the evolution of the Silicon dioxidedomain with time. The progression of the expanded region is evident.

Plate 1 shows the pressure contours.

5.4. =afer bending

In this computation the bending of a 5 km]1 km silicon wafer is presented. These dimensionsare not realistic in that actual wafers have dimensions of the order of 150 mm]0)5 mm. However,the simulation permits demonstration of the di!erence between the traditional oxidation simula-tion codes and the current one.

The oxidation of the top surface results in the formation of oxide and is accompanied byexpansion (see Plate 2). Material properties correspond to those in Table I and II. In anunexpanded state the oxide is in compression and the silicon is in tension. To achieve forceequilibrium the entire wafer will expand lengthwise. To achieve moment equilibrium across thecross-section the wafer bends away from the oxide.

The one-dimensional expansion model used in the standard oxidation simulation codes causesthe oxide to expand in a direction normal to the interface. This cannot produce any motion in thelengthwise direction, and these models cannot predict any bending of the wafer. This experimentalso highlights the need for "nite deformation kinematics. The curvature is su$ciently large to



render small deformation theory inapplicable. The "nite deformation theory allows large rota-tions and thus reduces the stresses observed in the simulations compared to small deformationtheory. In addition, to simulate this two-dimensional example, the plane strain hypothesis hasbeen employed implying non-zero stresses in the direction normal to the plane of the analysis.This is physically more realistic than the plane stress approximation traditionally used.

6. CONCLUSIONS

In this work the computational modelling of the oxidation process in Silicon has been performed.The equations have been derived from the basic conservation laws of mechanics, and numericalmethods to solve the di!usion}reaction process and the mechanical stress equilibration processhave been developed to overcome the problems faced in earlier models and to include the latestexperimental results.

The Si}SiO2

interface is modelled using the level-set method. The evolution of the interface isgoverned by a non-linear advection equation. This allows the description of the interface ina mesh-independent manner. The di!usion equation is solved using an enhanced gradient methodwith "nite elements to incoporate jumps in densities and #uxes. This allows very accuraterepresentation of the interface conditions and the reaction mechanism. The use of "nite deforma-tion kinematics renders the computation applicable to the large deformation regime. Theexpansion is modelled by incorporating it in the constitutive equation through a multiplicativesplit of the deformation gradient. The standard solid viscoelastic model is used to describe thestress}strain behaviour of Silicon dioxide. The time integration of the constitutive equations isdone entirely in the material frame thus trivially satisfying objectivity with respect to rigid bodyrotations.

The work done so far considers the process in the isothermal regime. Semiconductor process-ing is inherently a thermomechanical process. Thus, future work needs to be carried out to extendthe models to include temperature variations.

ACKNOWLEDGEMENTS

The authors would like to thank Prof. R. W. Dutton, Prof. Edwin Kan and Dr. Peter Gri$n formany rewarding discussions on the material. Prof. R. L. Taylor provided the "nite element codeFEAP in which the models were implemented. His support is gratefully acknowledged. We wouldalso like to acknowledge the "nancial support for the research provided by the Defence AdvancedResearch Projects Agency under contract DABT63-93-C-0053 to Stanford University.

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On modelling thermal oxidation of Silicon II: numerical aspects

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