I. INTRODUCTION

In manufacturing of semiconductor, rapid thermal processing (RTP), as a mainstream technique, has large advantage over batch furnaces processing with its smaller thermal mass. RTP is also the only way to perform certain crucial steps in the processing of compound semiconductor devices such as high-efficiency solar cells and high-mobility transistors. However, some problems such as non-uniformity of the thermal distribution hinder the applications of the RTP widely in the industry as shown by ^(1,2). Some resources of the non-uniformity of the temperature include the pattern on the surface of the wafer, the non-uniform distribution of the heating lamps and the heat dissipation on the edge. There are many works accounting for this problem and various strategies are proposed by ^(3,4). However, as the wafer size becomes larger, models of the system are required and more advanced control strategies are needed.

For the design of the measurement sensor and controller, the model of the system plays an important role. Numerous works such as ^(5-9) provide models of RTP systems. Two-dimensional model was used to simulate rapid thermal processing system where the dominant factors governing heat transfer and fluid flow inside the reactor were identified by ⁽⁵⁾. Then a finite-difference method was used to get the numerical solution of the system. Lord developed a numerical model in ⁽⁶⁾ that incorporated radiative and convective heat transfer and then thermal stresses were analyzed. Nonlinear numerical model was proposed by ⁽⁸⁾ and control method was developed.

The control performance can be improved by using the model knowledge in the control algorithm. Control approaches based on the model knowledge of the system have been applied to the RTP system. Apte and Saraswat proposed multivariable control in ⁽¹⁰⁾ based on an approximated model of the system. In ⁽¹¹⁾, a balanced reduced model is utilized for the design of the proportional-integral (PI) controller. An approximated Wiener model of RTP is constructed in ⁽¹²⁾ and multivariable control is provided. However, there is always uncertainty of model for RTP system. Adaptive iterative learning control is a more robust method which could learn the system parameters when updating the control law. Thus, it is suitable for RTP system with model uncertainty and operating in a repetitive mode. It enhances the performance by learning the system uncertainty iteratively. It is the first time that adaptive iterative learning controller is designed based on the finite-element-method for the RTP system in our work.

In this paper, we utilize the finite-element-method to obtain a finite-dimensional model of the system. Then, advanced adaptive iterative learning controller based on the finite-dimensional model is applied to regulate the thermal distribution of the RTP system. Compared to the spectrum-method based model of ⁽¹³⁾, finite-element-method based model can be obtained more readily especially when the boundary conditions of the system are complex. Although the accuracy of the finite-element-method based model is relatively low, the errors can be compensated by the adaptive learning of the controller iteratively as proposed in this paper.

The paper is organized as follows. In section 2, a partial differential model formulation using physical principles is presented. The methodology of finite-element-method for a finite-dimensional model is proposed in section 3. Section 4 proposes the adaptive learning of the control method and section 5 gives the simulation results using the proposed control method. Section 6 concludes the paper.

II. MODEL FORMULATION

The system we deal with in this work is the same as the one provided by ⁽¹⁴⁾ which is named as Steag CVD system. Fig. 1 illustrates a schematic diagram of the RTP system.

The energy balance of the system can be described as:

with the boundary conditions:

and initial condition:

where $T=(\bar{T}-T_{e})$ is the dimensionless wafer temperature with $\bar{T}$ representing the real temperature on the wafer, $x$ is the radius coordinate, $t$ is the time, $T_{ini}$ represents the initial dimensionless temperature distribution, $V_{j}$ is the view factor in the RTP system, whose structures are the same as in ⁽¹⁴⁾ and $u(j,t)$ is the power of the lamps in different zones where $j$ is the index of zones. In the RTP considered, 5 zones are used. The meanings and values of the system parameters can be found in Table 1.

III. FINITE-DIMENSIONAL MODEL

By using the finite-element-method, we can choose the following element function $\phi_{k}$ :

where $x_{k} \in [0,1]$ is a control point and is the index for the number of the element functions. The element functions with a homogeneous partition of the domain $x_{k}-x_{k-1}=x_{k+1}-x_{k}=h$, which is used in this paper are plotted Fig. 2.

Based on these element functions, we can project the state variable onto a finite-dimensional subspace

where $T_{N}$ is the approximated temperature in the subspace, $P_{N}$ is the projecting operator, and $T_{k}$ is the approximated state in the $k$th dimension. By projecting the system (1)onto the finite dimension subspace, we have

Then the following approximated subsystem is obtained:

In order to get the best approximation, we define the residual as

By using the Galerkin's method, we set the test functions as the element functions and make $\left(r, \phi_{k}\right)=0$ at $k=l,L ,N$ where (.,.) represents the inner product. $\left(r, \phi_{k}\right)=0$ is a fundamental assumption of the Galerkin's method which means that the residual of the system dynamics in the space of the element functions is zero. With this assumption, approximated system dynamics can be obtained. It has been proved that the approximated error could be driven into a small boundary thus it has been accepted as an effective approximated criterion for these kinds of problems. In this assumption, the definition of the inner product is given as $(f, g)=\int f(x) g(x)$ where $f$ and $g$ are square integrable functions. The space of the square integrable functions equipped with this inner product is a Hilbert space in which the RTP system dynamics considered are described.

With this assumption, the following reduced system model is obtained:

for $k=1,...,N$. Because

for $k=1,...,N$, we have the following reduced system

for $k=1,...,N$.

IV. ADAPTIVE LEARNING CONTROL METHOD

We can rewrite the finite-dimensional model as follows:

where

$$T=\left[T_{1}, \quad T_{2}, \quad L \quad T_{N}\right]^{\prime}$$ $$A=\frac{k^{\prime}}{R^{2} \rho C}\left[\begin{array}{ccccc} \frac{1}{h} & -\frac{1}{h} & 0 & 0 & 0 \\ -\frac{1}{h} & \frac{2}{h} & -\frac{1}{h} & 0 & 0 \\ 0 & -\frac{1}{h} & \frac{2}{h} & -\frac{1}{h} & \mathrm{M} \\ 0 & 0 & -\frac{1}{h} & \frac{2}{h} & -\frac{1}{h} \\ 0 & 0 & 0 & -\frac{1}{h} & \frac{1}{h} \end{array}\right]$$ $$F(T)=-\frac{F \varepsilon \sigma}{Z \rho C T_{e}}\left[\begin{array}{c} \left(\left(T_{e} T_{N}+T_{e}\right)^{4}-\bar{T}_{a}^{4}, \phi_{1}\right) \\ \mathrm{M} \\ \left(\left(T_{e} T_{N}+T_{e}\right)^{4}-\bar{T}_{a}^{4}, \phi_{N}\right) \end{array}\right]$$ $$B=\frac{\varepsilon}{Z \rho C T_{e}}\left[\begin{array}{ccc} \left(P_{N} V_{1}, \phi_{1}\right) & \mathrm{L} & \left(P_{N} V_{5}, \phi_{1}\right) \\ & \mathrm{O} & \\ \left(P_{N} V_{1}, \phi_{N}\right) & \mathrm{L} & \left(P_{N} V_{5}, \phi_{N}\right) \end{array}\right]$$

and

$$\boldsymbol{u}(t)=\left[\begin{array}{lll} u(1, t), & u(2, t), & \mathrm{L} & u(5, t) \end{array}\right]'$$

The derivation of the matrix $A$ is given in the appendix.

Based on the finite-dimensional model of the RTP system, we propose an adaptive learning control algorithm in this section to control the temperature on the wafer to a desired one $T_{d}$. The error is defined as $E=\left[e_{1} \mathrm{L} e_{N}\right]^{T}=T-T_{d}$. The design procedure of the ILC law is as follows:

1) The error dynamics can be obtained from the reduced model (13)as:

In this RTP application, we assume that and $A T+F(T)=\psi A T$ where $\psi$ is some unknown system parameter for the controller. In order to keep $E \equiv 0$, we must have which gives that

2) To estimate the unknown system parameter $\psi$ , we use a traditional proportional-type iterative learning algorithm as follows:

where $\hat{\psi}_{I}$ is the estimation of $\psi$ and $q$ is the proportional learning parameter.

3) A proportional-type control law is added to the control algorithm to guarantee that the control algorithm is effective when the estimated parameter $\hat{\psi}_{I}$ is zero.

As the result, the control strategy is proposed as:

with the updating laws

where $I$ represents the iteration number of the control, $r$ and $q$ are some positive learning parameters.

In this control algorithm, $\psi$ represents some unknown system parameter which is estimated using the iterative learning scheme (18)and $\hat{\psi}_{I}$ is its estimation in the $I$th iteration. With this estimated parameter, the adaptive controller (17)is designed and the accuracy of the temperature tracking can be improved iteratively. This is the basic idea of the proposed adaptive iterative learning control method. The structure of the proposed controller for heating the wafer in the RTP system is illustrated in Fig. 3.

In this algorithm, we require that $B$ has a right inverse, that is $BB^{T}$ is not singular. Fortunately, this is satisfied in the model of RTP system we consider.

V. SIMULATION

In the simulation, the temperature on the wafer is required to rise from 700K to 800K in a few seconds. In this simulation, 5 element functions of the finite-element-method are used as shown in Fig. 2. More element functions will enhance the accuracy of the approximated model while increase the computation complexity of the control design. Thus, there is a balance between the accuracy and the computation complexity. In this simulation, the number of the element functions is chosen to be 5, which could provide less temperature error while the computation complexity of the controller is acceptable. The efficiency of the finite-element-method is lower than the spectrum method as more model knowledge is used in the spectrum method. However, the benefit of the finite-element-method is that the choice of its element functions is independent of the boundary conditions of the system, thus this method is robust to the disturbance on the boundary. When the boundary conditions of the system are complex, the design of the spectrum method is not available and the finite-element-method can provide a robust model for the control design. In this simulation, we add a Gaussian noise $N(0,0.01)$ on the boundary. In the simulation, we also added an uncertainty around $t=3$ to the system as follows:

which represents some un-modeled system dynamics.

We apply the adaptive iterative learning control algorithm (17)to the thermal system with the control parameters set as $r=3$ and $q=1.8$. Adjusting the controller parameters $r$ and $q$, we could obtain a desired rate of decrease of the error index. However, there is a balance between the convergent speed and the stability of the system. Faster rate of decrease would cause the in-stability of the system. Thus, the control parameters should be carefully selected. In this simulation, control parameters are selected with the error index converging in an acceptable iteration convergent rate in the RTP application.

The thermal distribution with the control in the first iteration is shown in Fig. 4 while the one in the 21st iteration is presented in Fig. 5. The simulation results in Fig. 4 and 5 show that the proposed controller has a certain degree of robustness to noise and uncertainty. We can find from the results that the control performances are improved as the iteration goes as shown in Fig. 6 where the error distributions of the first iteration and 21st iteration are compared.

We also compare the proposed method to the proportional-integral-derivative (PID) controller wildly used in practice with a form as follows.

where $p=-1$ is its control parameter. The error distributions of PID controller is also illustrated in Fig. 6 where we can conclude that the proposed adaptive iterative learning controller based on the model of the finite-element-method can obtain less temperature errors compared to the traditional PID controller.

We define an error index to evaluate the performance of the controller as follows:

which represents the average temperature error at the last moment ($t=15$) of every iteration. The error indexes of different iterations are plotted in Fig. 7, whose physical meaning is that the average temperature error with the proposed controller will decrease to less than 2K. It can be derived from the simulations that, through the adaptive learning of the controller, the heating performances of the system are improved.

VI. CONCLUSION

A finite-element-method based adaptive iterative learning control approach is proposed in this paper for the thermal regulation of a RTP system. Firstly, a model in the form of partial differential equation is constructed using the physical principle. Then, a finite-dimensional model is obtained by using the finite-element-method. At last, an adaptive iterative learning controller is developed. The simulation results show that the proposed adaptive iterative learning control method can heat the wafer to a desired temperature effectively with smaller error.

APPENDIX (DERIVATION OF MATRIX A)

Matrix $A$ represents the matrix form of the first term of the right-handed side in Eq. (12) whose derivation is as follows.

We rewrite the Eq. (12) in a matrix form which gives that

According to the definition of the inner product, we have

The derivative of the linear piecewise element function can be obtained as

We can find that if $|i-k|>1$, then

In the case of $|i-k| \leq 1$, the integration result for $i=K$ is given by

while the result for $i=k-1$ is given by

and the result for $i=k+1$ is given by

With a homogeneous partition of the domain of the element function $x_{i}-x{i-1}=x_{i+1}-x_{i}=h$, the matrix $A$ in this RTP application can be rewritten as

$$A=\frac{k^{\prime}}{R^{2} \rho C}\left[\begin{array}{ccccc} \frac{1}{h} & -\frac{1}{h} & 0 & 0 & 0 \\ -\frac{1}{h} & \frac{2}{h} & -\frac{1}{h} & 0 & 0 \\ 0 & -\frac{1}{h} & \frac{2}{h} & -\frac{1}{h} & 0 \\ 0 & 0 & -\frac{1}{h} & \frac{2}{h} & -\frac{1}{h} \\ 0 & 0 & 0 & -\frac{1}{h} & \frac{1}{h} \end{array}\right]$$