RTP 시스템내의 웨이퍼 온도 균일도를 위한 모델링 및 모사

박 형진, 이 정석, 박 선원

한국과학기술원 화학공학과

Modeling and Simulation for Wafer Temperature Uniformity in RTP System

Hyungjin Park, Jeongseok Lee, and Sunwon Park

Dept. of Chemical Engineering, KAIST

Introduction

Generally, RTP (Rapid Thermal Processing) means a single wafer processing technology with  temperature ramp rates in 50∼100℃/s. RTP emerged as an attractive technology to replace conventional batch processes in the microelectronics industry for thermal processing of devices with submicron dimensional constraints (Roozeboom). Figure 1 shows the thermal cycles of the RTP system. As the wafer size increases, the temperature nonuniformity across the wafer surface becomes serious.

Figure 1. The representative temperature

trajectory of RTP.

This study considers the problem of controlling a 12 inch RTP system with 5 independently-controllable lamp zones. We present a systematic approach to the modeling of the RTP system. In this study numerical techniques are developed for using a thermal model to find the heat transfer characteristics of the RTP system and optimize performance of temperature control configurations.

RTP Model and Assumptions

Figure 2 shows a schematic diagram of hypothetical RTP system used in this study. It is assumed here that the model has 5 lamp zones, all lamp zones are flat, lamp power is uniform within each lamp zone, wafer is axisymmetric, and the temperature through the thickness of wafer is uniform. The interiors of the chambers are perfectly cylindrical and the wafers in them are perfectly cylindrical too. The wafer radius is 150□ (12 inch).

Figure . A schematic diagram of hypothetical RTP system.

The rate of thermal energy implies three heat transfer mechanisms, such as radiation, conduction, and convection. From the law of conservation of energy, we can obtain the energy ballance equation such as (1).

[eqn: scale 90 rho C_p partialT over partialt &= ~ k 1 over r PARTIAL over {partial r} left ( r {partial T} over {partial r} right ) - {2h} over z (T-T_gas ) + lim from { DELTA r ->0 } {bold Q_rad (r, bold T ) } over {2pi r z DELTA r } ### & rm with~boundary~conditions ##& ~~~~ {partial T} over {partial r} ~=~0 ~~~~~~~~at~~~r~=~0 , ~~~ and ##& ~~~ k {partial T} over {partial r} ~=~ Q_edge ``~~~~at ~~~ r~=~R .]

(1)

[eqn: rm &where it bold {Q_rad} `(r, bold T ) ~rm is~ a ~v ector~that~has~element it~Q_i ^rad , ##&and~~Q_i ^rad ~= SUM from { {j }=1} to {nlamp} L_ji P_j ~ + SUM from { k =1} to {nwafer} W_ki A_k epsilon_k sigma T_k ^{4 } ##&~~~~~~~~~~~~~+~~ Q _i ^wall ~-~ 2 A_i epsilon_i sigma T _i ^4]

For steady state simulation, we used FDM with the centered difference formula. And for the transient state simulation an ordinary differential equation (ODE) approximation can be derived from (1) which is a partial differential equation (PDE). Transient temperature profiles can be obtained using an ODE solver; ode45 of MATLAB□ was used here.

Steady State Optimization

We have considered the problem of choosing lamp power setpoints that result in the steady state temperature at each point on the wafer being close to some specified temperature Tset. The ultimate goal of steady state is to find lamp settings that minimize the worst-case difference between wafer temperature and Tset, that is

[eqn: rm {mi n i mize} ~~max atop { scale 65 0 <= r <= R } it LEFT | T^ss left(r, bold{P} right) - T^set RIGHT | }]

(2)

where Tss(r, P) is the steady state temperature at radius r with the constant lamp power vector P, subject to the constraint that each entry Pj of P satisfies 0≤Pj≤Pj,max (Norman). Suppose an RTP system is well described by the model developed previously. With very small error we can approximate (2) as minimizing

[eqn: max LEFT | T_i ^ss `( bold P` ) - it T^set RIGHT | } ~= ~ ||` T_i ^ss `( bold P` ) - it {bold T}^set ~||_ INF]

(3)

where [eqn: T_i ^ss `( bold P`)] is the steady state temperature of element i with constant lamp power vector P and Tset is a vector with all entries equal to Tset. Because [eqn: T_i ^ss `(bold P`)] is a nonlinear function of P, the problem of minimizing (3) is not a minimax-error problem. However, the problem can be solved in two steps, each of which involves solving a minimax-error problem (Norman).

Figure 3 shows results from LP method. The plots show the lamp power settings that minimize the worst-case difference between wafer temperature and Tset between 600 and 1100℃. The lamp settings are shown as fractions of maximum power and plotted using a log scale so that at any x-axis the spacing between the curves indicates the ratios between lamp settings at that point.

If the RTP system uses uniform lamp setting, WNU (worst nonuniformity : max | Ti - Tj | ) is 33.32℃ at 1 atm, 24.24℃ at the vacuum state. But using optimal lamp setting, nonuniformity can be reduced to 1.31℃ at 1 atm, 1.30℃ at the vacuum. state.

Figure . Optimal steady state lamp settings for Tset between 600 and 1100℃.

Figure . Radial temperature profiles using uniform lamp setting and optimal lamp setting at 1 atm.

Transient State Optimization

For the optimization of transient state temperature uniformity, we have to consider time in the optimal criterion of the steady state. So, the ultimate goal is to minimize the largest temperature error from the specified trajectory Tref at any point on the wafer and at any time between an initial time t0 and a final time tf.

We can construct a minimax problem which finds the optimal power trajectory in regard to each processing time to make wafer surface maintains the desired reference temperature trajectory.

[eqn: rm Minimize it LEFT~&||``bold LP (t_n ) - {bold Q}^{n eed} (t_n ) RIGHT ~|| _INF ~rm ##& rm s ubject~ t o ~it 0 <= {bold P} <= {bold P}^max `]

(5)

[eqn: {bold Q}^{n eed} left( t_n right)~=~{bold Q}`_{s.s.}^{n eed} left( {bold T }^ref ( t_n ) right) + {bold Q}`^add (t_n )]

(6)

Figure . Optimal lamp setting trajectories for Tref.

Differently from the steady state, Qneed of the transient state varies with time for the wafer temperature to follow Tref(t). That is to say, wafer needs additional heat to the heat required to maintain each temperature point during ramp up step.

We find optimal power trajectory through LP method presented previously. Following Figure 5 shows a lamp setting trajectory corresponding to Tref(t) using this scheme. The sample rate is 10 □. Each lamp setting begins at setting value of 600℃ setpoint.

Temperature profiles by this heating scheme are illustrated in Figure 6. The temperature profiles follow Tref(t) relatively well. And Figure 7 shows Tcenter(tn) - Ti(tn), which is the temperature relative error, plotted 3 dimensionally as a function of time and radial position.

Figure . Wafer temperature trajectories by optimal lamp setting trajectory (1 atm).

Figure . 3D plot of temperature error by optimal lamp setting trajectory (1 atm).

Schaper could control the wafer temperature with 45℃/s ramp rate in a 4 inch RTP system with less than 15℃ nonuniformity during the ramp step experimentally (Schaper). Compared with Schaper's experiments, the simulation results in Table 5.1 show uniformity of temperature during the ramp step for the 12 inch wafer system. As the ramp rate increases, the nonuniformity also increases. Because there exists a trade-off between characteristics of process and permitted limit of nonuniformities, so an optimum operation should be determined.

Conclusions

In this study, the modeling and simulation of multi-lamp zone RTP system for 12 inch wafer were implemented. Compared with 8 inch wafer system, 12 inch wafer system shows big nonuniformity in the case of simple heating. The temperature profiles are sensitive to small changes in the lamp zone illumination patterns. The optimization technique for temperature uniformity using this model equation was developed. These optimization results show that 12 inch RTP system can be used to achieve very low temperature nonuniformities during both the steady state and the transient state.

Acknowledgment

This work was supported (in part) by the Korea Ministry of Science and Technology and by the Korean Science and Engineering Foundation (KOSEF) through the Automation Research Center at POSTECH.

References

1. Norman, S. A., "Wafer Temperature Control in Rapid Thermal Processing," Ph. D. Thesis, Stanford University, 1992

2. Roozeboom, F., "Manufacturing Equipment Issues in Rapid Thermal Processing," Rapid Thermal Processing, Science and Technology, Chap. 9, Academic Press., New York, 1992

3. Schaper, C. D.,  Moslehi, M. S. and Kailath, T., "Modeling, Identification, and Control of Rapid Thermal Processing Systems," J. Electrochem. Soc., Vol. 141, No. 11, 1994