On a Finite Element Method for a CompressibleViscous Flow in a MOCVD ReactorJairo DuqueDepartamento de Matematicas, Universidad del Valle, Cali, Colombia

Received 16 April 2004; accepted 15 September 2004Published online 1 December 2004 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/num.20061

We propose a stable finite element method for approximating the flow of a chemically reacting gas mixturein an MOCVD (metal-organic chemical vapor deposition) reactor. The flow is governed by the fullcompressible Navier-Stokes equations extended by transport equations for the chemical species, the energyequations and the equation of state, together with boundary conditions providing information on the reactorgeometry and experimental conditions. The equations form a semilinear system with a constraint for whichthe corresponding pressure term is not the Lagrangian multiplier. An application of our method to a realworld model of growth of GaAs shows the consistency with experimental data. © 2004 Wiley Periodicals,Inc. Numer Methods Partial Differential Eq 21: 764–769, 2005

Keywords: finite elements; saddle point problems; LBB-condition; MOCVD reactors

I. INTRODUCTION

The objective of the metal-organic chemical vapor deposition (MOCVD) modeling is theflow prediction and mass transfer prediction. Flow in MOCVD reactors can be satisfactorilypredicted by taking into account the detailed boundary conditions including heat conductionin the reactor walls, external and internal radiation, natural or forced convective heattransfer to the ambient, and three-dimensional (3D) effects whenever the various partialprocesses contribute to the formation of deposited films [1, 2]. Most numerical studies ofMOCVD transport processes published so far use finite difference and finite volumemethods. Prediction of complex flows at high numerical accuracy requires the use of veryfine grids. However, standard finite difference and finite volume methods are known toconverge slowly as the grid is refined. In this article we present a 2D model for laminar flowin a horizontal tilted MOCVD reactor, implemented in a finite element procedure. Bysystematically refining the grids it is possible to evaluate the discretization errors and makesure that they are negligible small on the finest grid. To test the validity of the finite element

Correspondence to: Jairo Duque, Departamento de Matematicas, Universidad del Valle, Cali, Colombia (e-mail:jjduque@univalle.edu.co)Contract grant sponsor: Vicerrectoria de Investigaciones, Universidad del Valle; contract grant number: CI-27626

© 2004 Wiley Periodicals, Inc.

discretization, calculations are compared to results from MOCVD calculations in [3]. Durstet al. [3] considers the growth of GaAs layers from mixture of Trimethyl-gallium (TMGa),arsine (AsH3), and hydrogen and compares his results with experimental data in [4, 5].

II. THE MODEL EQUATIONS

For low flow rates encountered in chemical vapor deposition reactors, the applicable steady stateequations can be expressed as follows [6]:

�2 div��Du� �2

3��� div u� � div��uuT � � �p � �g�, (2.1)

�div���T � � div��uCpT � � 0, (2.2)

div��u� � 0,

P0M� � �RT, (2.3)

with velocity vector u, temperature T, pressure p, gravitational acceleration vector g�, specificheat at constant pressure Cp, constant operating pressure P0, and deformation rate tensor Du,where Du is defined as 1

2(�u � �uT). The dynamic viscosity �, thermal conductivity �, and

density � of the gas are functions of the temperature T:

� � ��T �, � � ��T �, � � ��T �.

The transport equations (2.1)–(2.2) are to be complemented by appropriate boundary con-ditions at the walls of the reactor, the surface of the susceptor, the inflow, and the outflow. Figure1 shows the geometry of the reactor together with the susceptor’s position. The boundaryconditions for the equations of momentum are given by a uniform velocity profile at the inlet,tangential and normal velocities zero at solid walls, and zero velocity gradients at the outlet. Forthe energy transport equation we suppose uniform inflow temperature T0, zero temperaturegradient at the outflow, and constant susceptor temperature Ts. All other walls of the reactor aresupposed to be either adiabatic or of constant temperature.

III. THE FINITE ELEMENT METHOD

To solve the system with the finite element method, we need a weak formulation of Equations(2.1) through (2.3). Let � be the region in �2 given by the geometry of the reactor. Guided by

FIG. 1. Geometry of the reactor MOCVD.

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the similarity of our model with the compressible equations of Navier-Stokes at small Machnumbers (Mach � 0.1) we set

a��u, T; �v, �� � b�p; �v, �� � ��g� , v�, @�v, � � H01���2 � H0

1���, (3.1)

c��u, T; q� � 0, @q � L2���, (3.2)

with nonlinear forms

a��u, T; �v, �� � 2��Du, �v� �2

3�� div�u�, div v�

� ��u � �u, v� � ��p, v� � ���T, �� � � ��uCpT, �� �,

b�p; �v, �� � �p, div v�,

c��u, T; q� � �div��u�, q�.

Here for matrix valued functions S and T we use the notation

�S, T � � �i,j1

2 ��

sijtijdx.

To discretizise the weak problem, let �h {�} be a triangulation of � such that on �h ��the standard regularity condition (see [7]) is maintained, where h is the discretization parameterof the triangulation. To define the test space, we start with the piecewise linear test functions

Wh � vh � C��� h��On each � � �h�.

To satisfy the boundary conditions we define Vh Wh � H01(�h) and use Vh

2 for velocity, Vh fortemperature, and Xh Wh for pressure. Now define the following method of finite elements:

Find [u, T] � Vh2 � Vh and p � Xh such that

a��uh, Th; �vh, �h� � b�ph; vh� � ��g� , vh�, @�vh, �h � Vh2 � Vh, (3.3)

c��uh, Th; qh� � h2��ph, �qh� � 0, @qh � Xh, (3.4)

where in the choice of the stabilizing term h2(�ph, �qh) we follow Brezzi-Pitkanta [8] andHughes-Franca [9], who used it to stabilize the Stokes equation.

In the linearization

a�u, v� � b�p, v� � � f, v�,

c�u, q� � 0 (3.5)

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of the (nonlinear) MOCVD equations, b and c do not satisfy the standard b(q, v) c(v, q)condition, so we cannot use the standard saddle point theory. Nevertheless b and c comply toa weaker coupling. We formulate the following weaker condition: There is a matrix valuedfunction M � C2(�� )3�3 with the following properties:

m1�v�1, 2 a�v, Mv� � c�v�22, @v � H0

1���3, m1 � 0,

�b�q, Mv� � c�v, q�� c�v�2�q�2, � �v, q� � H01���3 � L2���. (3.6)

Moreover, for the bilinear form b we assume a more general form of the well-known Lady-zhenskaja-condition:

m2�q�2 supv�V,v�0

b�q, v�

�v�1,2� c�q��1,2, @q � L2���, m2 � 0. (3.7)

Duque [10] shows the finite element method (3.3)–(3.4) is stable and (u1, u2, T ) convergeswith O(h2), while the pressure p converges with O(h).

Theorem 3.1. Assume that the system (2.1)–(2.3) possesses a solution (u, T, p) � C2(�� )2

� C2(�� ) � C1(�� ), such that the linearized operator (3.5) satisfies conditions (3.6) and (3.7).Then, there exists an h0 � 0 such that the discrete system (3.3)–(3.4) has a solution ([uh, Th],ph) for 0 � h h0 with

�u � uh�2;�h�� � �T � Th�2;�h�� � h� p � ph�2;�h�� ch2,

where the constant c depends on the solution (u, T, p).Proof. For details see [10] and [11]. y

IV. THE NUMERIC SOLUTION

In this section we apply the finite element method of section III to the MOCVD model for theexample of GaAs growth as described in Durst et al. [3, case 5] to calculate the gas flow in thereactor. For the grid used in the calculation (see Fig. 2). In this case, the inflow temperature is300K, the susceptor temperature is 1050K, and the operating pressure is 1 atm. The boundaryconditions used in the calculation are as described in section II, with a top wall temperature inthe susceptor region of 650K and the walls in the entrance region and behind the susceptor areassumed to be adiabatic.

FIG. 2. Grid to the finite element method.

FEM FOR A FLOW IN A MOCVD REACTOR 767

Durst et al. [3] contains the following assumptions: The specific heat, normally a weakfunction of the temperature, is assumed to be constant, as well as the Prandtl number, Pr 0.71.Viscosity and thermal conductivity as functions of the temperature are described by

��T � � �0� T

T0� 0.678

, ��T � � �Cp

Pr,

where T0 is the temperature at the inlet. Table I contains the main flow parameters.The solution of the resulting set of coupled Equations (3.3)–(3.4) is based on a globalized

quasi-Newton method with line search. The calculated example (Fig. 3) shows reverse flow atthe leading edge of the heated susceptor with typical sharp temperature gradients in front of thesusceptor (Fig. 4), conditions which are considered to lead to uniform growth rates at thesusceptor.

V. CONCLUSIONS

In this article we present a finite element method to solve a model for 2D laminar flows in anMOCVD reactor. The method is stable and velocities and temperature converge with O(h2),while the pressure converges with O(h). Given the high precision necessary for the underlyingtechnical process and the well-known slow convergence of Navier-Stokes type equations bylinear elements due to the presence of the pressure gradient, we feel a need to accelerate theconvergence. Since computing time increments with the square of the number of nodes, the useof multigrid methods comes to mind. The finite element method presented adapts easily to amultigrid implementation with an expected linear time dependence, the problem lies in thedesign of the smoothing component. Detailed investigation of the smoothing properties ofdiverse methods will be the topic of forthcoming studies.

The author thanks the Vicerrectoria de Investigaciones of the Universidad del Valle for thefinancial support (CI-27626) and the Departamento de Matemicas for the opportunity to developthe numerical method and the computer programs used in this article. The triangulation used wasgenerated with the excellent public domain program Triangle.

FIG. 3. Streamlines for the flow in the MOCVD reactor.

TABLE I. Gas dynamics parameter.

u0 (cm/s) P0 (atm) Re Ra (Raleigh number)

2.0 � 10�2 1.0 9.55 29.016 � 102

768 DUQUE

References

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FIG. 4. Isotherms for GaAs growth: Temperature values are (A) 1041K . . . (M) 357K in steps of 57.6K.

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