Ind. Eng. Chem. Res. 2007, 46, 3623-3628
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Optimal Feed-Forward Control for Multizone Baking in Microlithography Weng Khuen Ho, Arthur Tay,* Ming Chen, and Choon Meng Kiew Center for Intelligent Control, National UniVersity of Singapore, Singapore 117576
An algorithm for feed-forward control to improve the performance of a multizone baking system used for lithography in semiconductor manufacturing is derived in this paper. It uses linear programming optimization of the heat transfer in a multizone bake plate to produce a predetermined heating sequence. The objective is to minimize the temperature disturbance induced by the placement of a wafer at ambient temperature on the hot multizone bake plate, and the improvement is verified experimentally. 1. Introduction As shown in Figure 1, the microlithography sequence includes numerous baking steps, such as soft bake, post-exposure bake, and post-develop bake.1 In some cases, additional bake steps are used. Each of these bake steps serves different roles in transferring the latent image into the substrate. Of these, the most important (or temperature sensitive) is the post-exposure bake step. The post-exposure bake step is critical to current deep ultraviolet lithography. It is used to promote chemical modifications of the exposed portions of the photoresists. For such chemically amplified photoresists, the temperature of the wafer during this thermal step must be controlled to a high degree of precision for critical dimension control. The requirements call for the temperature to be controlled to within ( 0.1 °C at temperatures between 70 °C and 150 °C.1 A single-zone feed-forward controller was proposed in refs 2 and 3 for a single-zone bake plate. The key idea was to accurately model the temperature drop of the bake plate that is caused by the placement of a wafer at room temperature on the hot bake plate and precisely calculate the extra power required to eliminate the temperature drop. The single-zone feed-forward controller eliminated the temperature disturbance caused by the placement of a cold wafer on a single-zone bake plate.2,3 The latest state-of-the-art bake plate uses multiple-zone heaters for various purposes, such as achieving resist thickness uniformity (using different soft-bake temperature for different zones4,5) and improving critical dimension uniformity (using different post-exposure bake temperature for different zones6,7). However, when we apply the single-zone feed-forward algorithm from refs 2 and 3 on the multizone bake plate, complete elimination of the temperature disturbance could not be achieved. The algorithm did not take into account heat transfer between the zones. Because of the fact that the algorithm was developed for a single-zone bake plate, heat was applied to all zones with the assumption of no neighboring zones. As a result, overheating occurred, because of heat that was transferred from neighboring zones. This accounts for the 0.5 °C overheating at t ) 50 s and t ) 300 s in Figure 2 (see temperature curve, dashed line). In this paper, an algorithm for the multizone bake plate is developed. The new algorithm essentially eliminates the temperature disturbance, as shown in Figure 2 (see temperature curve, solid line). There is usually an error budget associated with the processing of the wafer. Because the wafer goes through many processing steps, errors introduced in each step leads to errors in the final * To whom correspondence should be addressed. Tel.: +65-65166326. Fax: +65-6779-1103. E-mail address:
[email protected].
Figure 1. Lithography sequence.
critical dimension. For a specified error tolerance, large errors in other processing steps can be compensated by reducing the temperature errors introduced in the baking step. 2. Multizone Bake Plate Thermal Model In this section, a physical model will be derived for a multizone bake plate, based on heat-transfer laws. As shown in Figure 3, the bake plate is formed by multiring zones, denoted as zones 1, 2, ..., m, from the inner zone to the outer zone, respectively. The wafer is considered to be formed by m zones also, with the same boundary as the bake plate. Because of good heat conduction of the metal and silicon, the temperature within each zone of wafer or bake plate is assumed to be sufficiently uniform. Thus, a distributed lumped model can satisfactorily describe the plant characteristics. Given the energy balance and the heat-transfer law, the bake plate can be modeled as
CwiT˙ wi(t) )
Tpi(t) - Twi(t) Tw(i-1)(t) - Twi(t) Twi(t) + + rai rw(i-1)i rwi Tw(i+1)(t) - Twi(t) (1) rwi(i+1)
CpiT˙ pi(t) ) pi(t) +
Twi(t) - Tpi(t) Tp(i-1)(t) - Tpi(t) + rai rp(i-1)i Tpi(t) Tp(i+1)(t) - Tpi(t) + (2) rpi rpi(i+1)
10.1021/ie061011p CCC: $37.00 © 2007 American Chemical Society Published on Web 04/27/2007
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Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007
At steady state, T˙ wi (∞) ) T˙ pi (∞) ) 0 and eqs 1 and 2 become
()
( )
( )
1 1 1 T (∞) ) T (∞) T (∞) rai pi Rwi wi rw(i-1)i w(i-1)
( ) ( ) ( ) ( )
1 T (∞) (3) rwi(i+1) w(i+1)
pi(∞) ) -
()
1 1 1 T (∞) T (∞) + T (∞) rai wi rp(i-1)i p(i-1) Rpi pi 1 T (∞) (4) rpi(i+1) p(i+1)
where
1 1 1 1 1 ) + + + Rwi rai rwi rw(i-1)i rwi(i+1) 1 1 1 1 1 ) + + + Rpi rai rpi rp(i-1)i rpi(i+1) Defining new variables (θwi(t) ) Twi(t) - Twi(∞), θpi(t) ) Tpi(t) - Tpi(∞), and ui(t) ) pi(t) - pi(∞)) and substituting eqs 3 and 4 into eqs 1 and 2 gives
Cwiθ˙ wi(t) )
( )
( ) ( ) () () ( ) ( ) ( )
1 1 θ (t) θ (t) + rw(i-1)i w(i-1) Rwi wi
1 1 θ (t) + θ (t) (5) rwi(i+1) w(i+1) rai pi
Cpiθ˙ pi(t) ) ui(t) +
1 1 θ (t) + θ (t) rai wi rp(i-1)i p(i-1)
1 1 θ (t) + θ (t) (6) Rpi pi rpi(i+1) p(i+1)
Equations 5 and 6 can be written as a state-space model: Figure 2. Comparison of bake-plate temperature disturbance caused by the placement of a cold wafer on the multizone bake plate: (s) multizone feed-forward algorithm, (- - -) single-zone feed-forward algorithm, and (‚ ‚ ‚) proportional-integral feedback control only.
Table 1. Parameters Used in eqs 1 and 2 parameter
definition/comment
i Cw Cp Tw(t) Tp(t) ra
parameters for zone i (i ) 1, 2, ..., m) heat capacity of the wafer (J/K) heat capacity of the bake plate (J/K) wafer temperature above ambient (K) bake-plate temperature above ambient (K) thermal resistance between wafer and bake plate (K/W) thermal resistance between wafer and surrounding air (K/W) thermal resistance between bake plate and surrounding air (K/W) thermal resistance between wafer zone i - 1 and zone i; rw(i-1)i ) ∞ for i ) 1 (K/W) thermal resistance between wafer zone i and zone i + 1; rwi(i+1) ) ∞ for i ) m (K/W) thermal resistance between bake-plate zone i - 1 and zone i; rp(i-1)i ) ∞ for i ) 1 (K/W) thermal resistance between bake-plate zone i and zone i + 1; rpi(i+1) ) ∞ for i ) m (K/W) heater power (W)
rw rp rw(i-1)i rwi(i+1) rp(i-1)i rpi(i+1) p(t)
where the parameters used in the equations are defined in Table 1.
x˘ ) Fx + Gu
(7)
y ) Hx
(8)
where
x ) [θw1 θw2 ‚‚‚ θwm θp1 θp2 ‚‚‚ θpm ]T u ) [u1 u2 ‚‚‚ um ]T
[ ]
y ) [θp1 θp2 ‚‚‚ θpm ]T 0
‚‚
0 1 G) C p1
0
‚
0
‚‚
‚
1 Cpm
0
[
0
H)
0 1
‚‚
‚
0 F)
0
‚‚
‚
0 0
[ ] P Q R S
1
]
[
and P, Q, R, and S are m × m matrices, given by P) 1 1 Rw1Cw1 rw12Cw1 1 1 1 rw12Cw2 Rw2Cw2 rw23Cw2 1 ‚‚ ‚ rw23Cw3
0
‚‚
‚
‚‚
‚‚
‚
0
‚ 1
[ ] [ ] rw(m-1)mCwm
1 ra1Cw1
Q)
[
R)
1 rw(m-1)mCw(m-1) 1 RwmCwm
0
‚‚ ‚
0
1 ramCwm
1 ra1Cp1
0
‚‚
‚
‚‚ ‚ 1 rp(m-1)mCpm
3. Multizone Feed-Forward Control
x(1) ) Φx(0) + Γu(0) y(1) ) H(Φx(0) + Γu(0)) x(2) ) Φ2x(0) + ΦΓu(0) + Γu(1) y(2) ) H(Φ2x(0) + ΦΓu(0) + Γu(1)) l
Y ) D + ΨU
0
‚
Assuming that the initial temperature of the wafer and bake plate, x(0), and the control signals u(0), u(1), ..., u(N - 1) are given, it is possible to solve eqs 9 and 10 using simple iterations.
The solution consists of two parts:
S)
‚‚
y(k) ) [θp1(k) θp2(k) ‚‚‚ θpm(k) ]T
y(N) ) H(ΦNx(0) + ΦN-1Γu(0) + ΦN-2Γu(1) + ‚‚‚ + Γu(N - 1))
1 ramCpm
1 1 Rp1Cp1 rp12Cp1 1 1 1 rp12Cp2 Rp2Cp2 rp23Cp2 1 ‚‚ ‚ rp23Cp3
u(k) ) [u1(k) u2(k) ‚‚‚ um(k) ]T
x(N) ) ΦNx(0) + ΦN-1Γu(0) + ΦN-2Γu(1) + ‚‚‚ + Γu(N - 1)
‚‚ ‚
0
0
]
Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007 3625
1 rp(m-1)mCp(m-1) 1 RpmCpm
]
D is dependent on the initial wafer and the bake-plate temperatures, and ΨU is the control signal.
[]
[ ][
y(1) y(2) Y) l y(N) u(0) u(1) U) l u(N - 1)
[ ]
HΓ 0 ‚‚‚ ‚‚‚ 0 ψ(1) HΦΓ HΓ 0 ‚‚‚ 0 ψ(2) Ψ) ) l l l l N-1 N-2 ψ(N) HΦ Γ HΦ Γ ‚‚‚ ‚‚‚ HΓ
[]
]
The objective is to eliminate the temperature disturbance caused by the placement of a cold wafer on the multizone bake plate. The feed-forward control algorithm derived in this section takes heat transfer between neighboring zones into consideration. Discretizing with sampling interval h, the state-space model of the multizone bake plate in eqs 7 and 8 is given by
HΦ HΦ2 D) x(0) l HΦN
x(k + 1) ) Φx(k) + Γu(k)
(9)
We note that the effect of the disturbance D caused by the ambient wafer temperature in x(0) can be eliminated if
y(k) ) Hx(k)
(10)
Y ) D + ΨU ) 0
where
When there is no constraint on the control signal, the disturbance can be rejected totally if we compute the control signal as
Φ ) eFh Γ)
∫0h eFh dηG
x(k) ) [θw1(k) θw2(k) ‚‚‚ θwm(k) θp1(k) θp2(k) ‚‚‚ θpm(k) ]T
U ) -Ψ-1D However, in practice, the power is subjected to lower and upper limits, i.e., u ∈ [umin, umax] and we optimize the objective function,
min
u∈[umin,umax]
max|D + ΨU|
(11)
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Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007
Figure 3. Schematic diagram of the multizone bake plate.
Figure 4. Photograph of the multizone bake plate.
The optimization problem of eq 11 is equivalent to the following linear programming problem:
[] ][ ] [ ]
U Min[0 ‚‚‚ 0 1 ] (objective function) e Ψ -1I U -D e (dynamic model) s.t. D -Ψ -1I e
[
U e Umax (upper control signal limit)
4. Experimental Results In this section, the feed-forward control strategy is demonstrated on a two-zone (m ) 2) bake plate for a 200-mm wafer. A photograph of the bake plate is shown in Figure 4. Two resistance temperature devices were used to measure the temperature. Room temperature was 24.5 °C, and the experiments were conducted at a setpoint of 90 °C with a sampling interval of h ) 1 s. For a two-zone bake plate, the model is reduced to
U g Umin (lower control signal limit) ψ(k)U + D(k) ) 0 (k ∈ [nf, ..., N]; disturbance eliminated from nf onward)
Cw1T˙ w1(t) )
Tp1(t) - Tw1(t) Tw1(t) Tw2(t) - Tw1(t) + ra1 rw1 rw12 (12)
where 1I is a column vector with all entries equal to 1, and
Cw2Tw2(t) )
Tp2(t) - Tw2(t) Tw1(t) - Tw2(t) Tw2(t) + ra2 rw12 rw2 (13)
Umax ) [u1_max u2_max ‚‚‚ um_max u1_max u2_max ‚‚‚ um_max ‚‚‚ ]T Umin ) [u1_min u2_min ‚‚‚ um_min u1_min u2_min ‚‚‚ um_min ‚‚‚ ]T where Umax and Umin are m × N rows vectors, ui_max and ui_min respectively denote the upper and lower bounds of the i heater power. For vectors V and w, V e w means that every element of V is less than or equal to the corresponding element of w. The parameter nf is chosen such that nf is valid while nf-1 is not.
Cp1T˙ p1(t) ) p1(t)
Tw1(t) - Tp1(t) Tp1(t) Tp2(t) - Tp1(t) + ra1 rp1 rp12 (14)
Cp2Tp2(t) ) p2(t)
Tw2(t) - Tp2(t) Tp1(t) - Tp2(t) Tp2(t) + ra2 rp12 rp2 (15)
where subscript “1” denotes the inner zone and subscript “2” denotes the outer zone. The parameters of the bake plate and wafer are given below.
Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007 3627
rw12 ) 0.013 K/W
For the wafer:
Inner-zone radius:
r1 ) 60 mm
Outer-zone radius:
r2 ) 100 mm dw ) 750 µm
Wafer thickness:
Surface area:
Aw ) πr22 ) 0.0314 m2 Aw1 ) πr21 ) 0.0113 m2
Surface area of the inner zone:
Surface area of the outer zone: Aw2 ) π(r22 - r21) + 2πr2dw ) 0.0206 m2 Heat capacity of the inner zone:
Cw1 ) 16.92 J/K
Heat capacity of the outer zone:
Cw2 ) 30.08 J/K
1 ) 11.949 K/W hpAp1
rp2 )
1 ) 2.286 K/W hpAp2
rp12 ) 0.17 K/W
hw ) 10.0 W/(m2 K)
Natural convection coefficient:
rp1 )
The thermal resistances rw12 and rp12 between the inner zone and the outer zone was calculated using eqs 3 and 4, with Tp1(∞) ) 66.2 °C, Tw1(∞) ) 64.25 °C, Tp2(∞) ) 65.5 °C, Tw2(∞) ) 64.22 °C, p1(∞) ) 19.11 W, and p2(∞) ) 35.56 W. Based on the aforementioned parameters, the bake-plate model is given as
[
-4.8389 4.5458 0.2864 0 2.5570 -2.8503 0 0.2864 F) 0.01349 0 -0.03008 0.01636 0 0.01349 0.009205 -0.02338 0 0 0 0 G) 0.002782 0 0 0.001565 0 0 1 0 H) 0 0 0 1
For the bake plate:
Air gap thickness:
la ) 70 µm
Natural convection coefficient:
hp ) 7.4 W/(m2 K)
Surface area of the inner zone:
Ap1 ) 0.0113 m2
Surface area of the outer zone:
Ap2 ) 0.0591 m2
Heat capacity of the inner zone:
Cp1 ) 359.4 J/K
Heat capacity of the outer zone:
Cp2 ) 638.9 J/K
Thermal resistances are given as follows:
ra1 )
la ) 0.206 K/W kaAw1
ra2 )
la ) 0.116 K/W kaAw2
rw1 )
1 ) 8.842 K/W hwAw1
rw2 )
1 ) 4.860 K/W hwAw2
[
ka ) 0.03 W/(m K)
Thermal conductivity of air:
[
0.269523 0.268836 Φ) 0.005309 0.003546
[
]
]
0.477930 0.112746 0.133872 0.478601 0.075318 0.171281 0.006305 0.971344 0.016767 0.008065 0.009431 0.978239 0.000188 0.000097 0.000097 0.000148 Γ) 0.002742 0.000013 0.000013 0.001548 x(0) ) [-63.99 -63.98 0 0 ]T
[
]
]
]
The optimal feed-forward control signal, U, was computed for the objective function (eq 11), using linear programming and the constraints u1_min ) -12.8, u2_min ) -41.7, u1_max ) 240, and u2_max ) 420. Note that a time delay of 4 s was present; this corresponded to the time delay for the heater power to reach the surface of the plate. Therefore, the feed-forward control signal U was applied 4 s before the placement of the wafer. Here, the steady-state powers were p1(∞) ) 12.8 W and p2(∞) ) 41.7 W. Besides the feed-forward control signals, two proportional-integral feedback controllers of the form
( (
Gc1 ) Kp1 1 + Gc2 ) Kp2 1 +
) ( ) (
1 1 ) 70 1 + sTi1 50 s
) )
1 1 ) 126 1 + sTi2 50 s
Table 2. Comparison of the Settling Time, Temperature Deviation, and Integrated Square Error for Multizone and Single-Zone Feed-Forward Control Algorithm Multizone Algorithm parameter settling time (s) temperature deviation (°C) integrated square error
Single-Zone Algorithm
Proportional-Integral Control Only
inner zone
outer zone
inner zone
outer zone
inner zone
outer zone
0 +0.06 -0.08 0.2079
0 +0.06 -0.09 0.1747
200 +0.45 -0.19 8.1604
108 +0.29 -0.14 1.9690
167 +1.26 -2.07 71.9116
120 +1.15 -2.04 50.1348
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were used for the inner-zone heater and the outer-zone heater, respectively. In discrete form, they are given as
Gc1(q-1) ) Gc2(q-1) )
70 - 68.6q-1 1 - q-1
126 - 123.48q-1 1 - q-1
The new algorithm eliminated the temperature disturbances at t ) 50 s and t ) 300 s, as shown in Figure 2 (see temperature curve, solid line). Table 2 shows the comparison of the multizone feed-forward algorithm, single-zone feed-forward algorithm, and proportionalintegral feedback control without any feed forward. The readings in Table 2 are the average of the two disturbances in Figure 2. The multizone feed-forward algorithm has reduced the temperature disturbance of the single-zone algorithm from a peak-topeak value of 0.6 °C (-0.19 to 0.45) to a peak-to-peak value of 0.15 °C (-0.09 to 0.06). There is an order-of-magnitude improvement in the integrated square error, from 1.9 to 0.2. The settling time, which is defined as the time required for the temperature disturbance to settle to less than (0.1 °C is reduced to 0 s. 5. Conclusion Requirements for critical thermal processing steps in microlithography call for the temperature to be controlled to within (0.1 °C. An algorithm for the feed-forward control of the multizone bake plate is given in this paper. It takes into account
the heat transfer between neighboring zones and almost eliminates the temperature disturbance caused by the placement of a cold wafer on the bake plate. Literature Cited (1) Quirk, M.; Serda, J. Semconductor Manufacturing Technology; Prentice Hall: Englewood Cliffs, NJ, 2001. (2) Ho, W. K.; Tay, A.; Schaper, C. D. Optimal Predictive Control with Constraints for the Processing of Semiconductor Wafers on Bake Plates. IEEE Trans. Semicond. Manuf. 2000, 13, 88. (3) Tay, A.; Ho, W. K.; Schaper, C. D.; Lee, L. L. Constraint Feedforward Control for Thermal Processing of Quartz Photomasks in Microelectronics Manufacturing. J. Process Control 2004, 14, 31. (4) Lee, L. L.; Schaper, C. D.; Ho, W. K. Real-Time Predictive Control of Photoresist Film Thickness Uniformity. IEEE Trans. Semicond. Manuf. 2002, 15, 51. (5) Ho, W. K.; Lee, L. L.; Tay, A.; Schaper, C. D. Resist Film Uniformity in the Microlithography Process. IEEE Trans. Semicond. Manuf. 2002, 15, 323. (6) Berger, L.; Dress, P.; Gairing, T. Global Critical Dimension Uniformity Improvement for Mask Fabrication with Negative-Tone Chemically Amplified Resists by Zone-controlled Postexposure Bake. J. Microlithogr., Microfabr., Microsyst. 2004, 3, 203. (7) Lee, H.-C.; Chen, C.-J.; Hsieh, H.-C.; Berger, L.; Saule, W.; Dress, P.; Gairing, T. Global CD Uniformity Improvement for CAR Masks by Adaptive Post-exposure Bake with CD Measurement Feedback. In Semiconductor Manufacturing Technology Workshop Proceedings, 2004; IEEE: Piscataway, NJ, 2004; pp 99-102.
ReceiVed for reView August 2, 2006 ReVised manuscript receiVed March 19, 2007 Accepted March 21, 2007 IE061011P
