ORIGINAL ARTICLE
Two-dimensional pheromone propagation controller applied
to run-to-run control for semiconductor manufacturing
Der-Shui Lee&An-Chen Lee
Received: 11 June 2011 / Accepted: 4 July 2012 / Published online: 28 July 2012 # Springer-Verlag London Limited 2012
Abstract This paper presents a new perspective on process control, called the two-dimensional pheromone propagation controller (2D-PPC), which considers the spatial informa-tion about disturbances of the process within a wafer to generate new predicted intercepts of the models for the subsequent use in time–effect controller (the exponentially weighted moving average, EWMA, in this study). The 2D-PPC assumes that the disturbances have their own behavior and affect others nearby in a wafer at a run; thus, it involves the “space-effect” among disturbances of the process at measurement positions within a wafer. The framework of the space–time controller (STC), which interlaces the time– effect controller and the space–effect 2D-PPC is con-structed, and the stability analysis and intrinsic character-istics of the STC are discussed. Simulations are conducted using two-dimensional anthropogenic disturbances generat-ed from fabrication data. The results show that the STC has better performance as compared to the conventional time– effect controllers. From implementation view point, since STC does not change the original code of time–effect con-troller, it can be easily implemented in the current process control loop by only adding an additional space-effect controller.
Keywords Two-dimensional pheromone propagation controller . Process control . Space-effect controller . Two-dimensional digital pheromone infrastructure . EWMA
1 Introduction
In semiconductor manufacturing, run-to-run (RtR) control adjusts the recipe slightly based on in-line measurements to even out disturbances. Statistical process control (SPC) [1] and the exponentially weighted moving average (EWMA) controller [2–8] are widely used in semiconductor RtR control. Predictor corrector control (PCC) [9,10] and double EWMA control [11–14] also have been proposed to im-prove the performance of EWMA when dealing with drift-ing processes. Recently, the output disturbance observer structure which provides a unified framework for the EWMA, the double EWMA and PCC controller is pre-sented. [15]. In addition to EWMA-based solvers, artificial neural networks [16–18], recursive least square technique [19], time series analysis [20], extended Kalman filter [21], and pheromone propagation controller (PPC) [22] are ap-plied in the RtR control. Besides, multivariate SPC [23,24], multivariate EWMA controller [25–27], and multivariate double EWMA (dEWMA) controller [28] were also devel-oped. No matter what kind of methodology is used in the RtR control, some researchers condense measurements to output quality characteristics, such as uniformity of CVD process [1,2,16,23], removal rate of CMP process [3,11,
17, 18], uniformity of CMP process [11, 17, 25], critical dimension of photolithography process [8], etching rate of etch process [9,19], aluminum deposition rate of aluminum sputter deposition process [15,20–22], and deposition thick-ness of diffusion process [27], for the process control; others employ control loops for every measurements, such as over 9,000 control loops for Y-markshift of lithography overlay control [4]; and still the others just control“output quality characteristics” without clear definition [5–7,10,12–14,24,
26, 28]. Nowadays RtR controllers employ the “time–ef-fect” characteristics among observed process data to calcu-late the recipe for the next run, which means disturbance of D.-S. Lee
:
A.-C. Lee (*)Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta-Hsueh Road,
Hsinchu City, Taiwan, People’s Republic of China e-mail: [email protected]
a wafer affecting the output quality characteristics at later runs is assumed. In other words, traditional RtR control employs performance indices along“time” axis and losses the spatial information among different measurements with-in a wafer for each run.
In this paper, we propose an algorithm that includes the “space–effect” among disturbances of the process at measure-ment positions within a wafer at a run. The concept comes from the observation that a disturbance of a wafer at certain location will affect its neighborhood area which may contain several measurement positions at a run. The new algorithm uses swarm intelligence by assuming that disturbance of the process at measurement position has their own behavior and affects others nearby within a wafer at a run. Because meas-urements within a wafer are a two-dimensional layout, our novel algorithm is called the two-dimensional pheromone propagation controller (2D-PPC). This study modifies the propagation-out ratio of digital pheromone infrastructure [22, 29] to achieve 2D-PPC. Under the two-dimensional digital pheromone infrastructure, the disturbances at different measurements within a wafer are modeled as a social insect colony. The interaction among disturbances is modeled by a propagation mechanism, which means that a measurement affects others nearby. Using 2D-PPC, new predicted intercepts of the models are generated for the subsequent use in time– effect controller (EWMA in this study).
The framework of the space–time controller (STC) is then constructed, which interlaces the time–effect controller and the space–effect 2D-PPC. For maximum coverage with minimum measurements, this study assumes that the wafer has M2measurements in the triangular coordinate
(hexago-nal grids) layout whose examples are shown in Fig.1. The subscript 2 of M2means the “two-dimensional” layout. In
Fig. 1, the arrow means that measurements affect others nearby. For easy interpretation, the authors employ triangu-lar grids with M2012 in the rest of this study and the
notations are shown in Fig.2. In Fig.2, measurements are divided into two circles; measurements 1–6 are in the outer circle and the others (7–12) are in the inner circle. Figure3
shows the concept of time–effect and space–effect control-lers. Figure3 is the concepts of the STC, which combines the MIMO (inputs are N2recipes and outputs are M2
mea-surement data) time–effect controller with 2D-PPC. In
Fig. 3, after forecasting disturbances by the MIMO time–
effect controller, the two-dimensional disturbances predic-tor, which is the realization of space–effect controller, modi-fies the forecasting disturbances of the MIMO time–effect controller spatially by 2D disturbances predictor for the next run. In other words, the STC interlaces time–effect and space–effects at a run and then the process recipe for the next run can be obtained by the forecast disturbances.
The stability analysis of the STC shows that larger sta-bility region is obtained as compared to the MIMO time– effect controller. Also, simulations are conducted to com-pare the output performance of three MIMO time–effect controllers, EWMA, PCC, and dEWMA, with and without 2D-PPC. The two-dimensional anthropogenic disturbances are generated from fabrication data of CMP process and the candidate controllers are examined by sum square error (SSE) in the simulation study. The simulation results show that the STC improves the output performance as compared to the MIMO time–effect controllers.
The rest of this paper is organized as follows. Section2
conducts the two-dimensional digital pheromone infrastruc-ture. Section3illustrates the STC structure. Section4 ana-lyzes the STC stability region and compares the intrinsic properties of STC with those of the time–effect controller. Section 5 shows the simulation results for the proposed controller. The final section draws the conclusion.
2 The two-dimensional digital pheromone infrastructure The concept of the space–effect 2D-PPC comes from the appearance that a disturbance of the process at a measurement position in a wafer will be affected by its nearby disturbances at measurement positions within the same wafer at a run and the space–effect will maintain to affect the next wafer in semiconductor fabrication. Corresponding to the food trail pheromones or alarm pheromones in the nature, the 2D-PPC employs “distur-bance pheromones” within the same wafer at a run in the manufacturing process. The 2D-PPC is an extension of PPC [22] and the digital pheromone infrastructures of PPC and 2D-PPC only differ from the shape of phero-mone basket and the propagation-out ratio of transition
…
4
M2= M2=12 M2=22 M2=36 …
Fig. 1 Examples ofM2
measurements in triangular coordinate (hexagonal grids) layout
functions. This section will introduce the two-dimensional digital pheromone infrastructure which includes pheromone basket, pheromone states, transition parameters and transition functions.
2.1 Pheromone basket
The pheromone basket is the pheromone propagation envi-ronment. The“two-dimensional” PPC is named by the fact that the shape of pheromone basket is a two-dimensional plane. The environment of two-dimensional digital phero-mone infrastructure is a tuple <B, N>, where B is a finite set of positions bm22 B : m2¼ 1; 2; . . . ; M2 within the phero-mone basket with size M2. Specifically, bm2of 2D-PPC maps the measurement position in the wafer. Then, N bmð 2Þ B is a finite set of neighbors of bm2 and N bmj ð 2Þj is the size of N bmð 2Þ. Suppose that (ρ1,ρ2,ρ3) in Fig. 2is the triangular coordinate (hexagonal grids) of position bm2 in the two-dimensional digital space [30], the coordinate of N bmð 2Þ is
ρ0 1; ρ 0 2; ρ 0 3 , where ρ0 1 ρ1 1 ρ0 2 ρ2 1 ρ0 3 ρ3 1 ρ0 1 ρ1 þ ρ0 2 ρ2 þ ρ0 3 ρ3 ¼1 8 > > > < > > > : ð1Þ
In addition, the two-dimensional digital pheromone in-frastructure assumes that the propagation relationship be-tween N bmð 2Þ and bm2 is irreflective, which means that bm2 will accept propagation inputs from N bmð 2Þ without preconditions.
The pheromone basket is initially empty, and then filled with M2 disturbances within the same wafer at each time
stamp (or run). In other words, the disturbances within a wafer can be treated as the external impulse input to the dimensional pheromone basket. The external input of two-dimensional pheromone basket is a finite set R2ðk; iÞ ¼
r2ðk; bm2; iÞ 2 L; Lð Þ : m2¼ 1; 2; . . . ; M2
f g , where k01,
2, …, is the run number of the manufacturing process, i∊ ℵ is the number of iterations (or propagations) in the tran-sition functions, and L∊ℜ is the global limit of the external inputs in the environment <B, N>. We use the notation ℵ for the set of natural numbers andℜ for the set of real numbers. Since the external input is the initial condition for launching transition functions of pheromone propagation, r2ðk; bm2; 0Þ maps to the disturbances at position bm2 at the run k and r2
k; bm2; i
ð Þ is 0 when i is larger than 0. 2.2 Pheromone states
Like [22, 29], the states in the two-dimensional phero-mone basket <B, N> are Q2(k, i) and S2(k, i), where Q2
k; i
ð Þ ¼ q2f ðk; bm2; iÞ 2 < : m2¼ 1; 2; . . . ; M2g is a finite set of the propagated inputs at run k and iteration i, and S2ðk; iÞ ¼ s2f ðk; bm2; iÞ 2 < : m2¼ 1; 2; . . . ; M2g i s a y1 y7 y8 y9 y10 y12 y11 y3 y4 y6 y5 x1 x2 x3 x4 x23x24 Wafer y2 y1 y7 y8 y9 y10 y12 y11 y3 y4 y6 y5 x1 x2 x3 x4 x23x24 Wafer y2 kthrun (k+1)thrun Forecast disturbance at a specific position within a wafer by a MIMO time-effect controller. Modify forecasting disturbance of the MIMO time-effect controller by space-effect controller.
Time-effect (process run)
Space-effect 1 ˆ ε 2 ˆ ε 4 ˆ ε 5 ˆ ε 7 ˆ ε 6 ˆ ε 8 ˆ ε 11 ˆ ε 9 ˆ ε 12 ˆ ε 10 ˆ ε 3 ˆ ε 1 ˆ ε 2 ˆ ε 4 ˆ ε 5 ˆ ε 7 ˆ ε 6 ˆ ε 8 ˆ ε 11 ˆ ε 9 ˆ ε 12 ˆ ε 10 ˆ ε 3 ˆ ε Measurement positions Fig. 3 Concept of STC, which
combines MIMO (M2012 and
N2024 in this example) time–
effect controller and space– effect (2D-PPC) controller, wherex1; x2; xN2are recipes, y1; y2; yM2are measurement outputs andb"1; b"2; b"M2are the forecasting disturbances
8 9 10 1 7 12 11 2 3 6 5 4 (1, -1, 0) (-1, 1, 0) (2, 0, -1) (0, 2, -1) (0, 0, 1) (0, 0, 0) (1, 1, -2) (1, 1, -1) (1, 0, 0) (0, 1, 0) (1, 0, -1) (0, 1, -1) 1
ρ
2ρ
3ρ
8 9 10 1 7 12 11 2 3 6 5 4 8 9 10 1 7 12 11 2 3 6 5 4 8 9 10 1 7 12 11 2 3 6 5 4 1ρ
2ρ
3ρ
finite set of the aggregated pheromones at run k and iteration i. Therefore, q2ðk; bm2; iÞ is regarded as the propagated input from N bmð 2Þ to bm2 at iteration i and run k. Similarly, s2ðk; bm2; iÞ is regarded as the aggregated pheromone of bm2 at iteration i and run k.
In addition, q2ðk; bm2; 0Þ ¼ 0 and s2ðk; bm2; 0Þ ¼ 0 are assumed to be the default initial conditions. While launching transition functions as shown in the following sections, Q2
k; i
ð Þ disseminates pheromones and S2(k, i) aggregates
pher-omones simultaneously.
2.3 Pheromone transition parameters
By [22, 29], two transition parameters of the two-dimensional digital pheromone infrastructure are the evap-oration parameter E2∈(0,1] and the propagation parameter
F2∈[0,1). In 2D-PPC, the propagation parameter F2
describes the effect of disturbance of the process on other nearby disturbance within the same wafer, and the
evaporation parameter E2 indicates that the importance of
the measurement data will“evaporate” with time. Because measurements within a chamber at a time stamp can be treated as a nondissipative system, the modified pheromone infrastructure uses E201 [22].
2.4 Transition functions
With the introduction of the two-dimensional pheromone basket <B, N>, the parameters E2and F2, the external input
R2(k, i), states Q2(k, i) and S2(k, i), transition functions of
2D-PPC can be developed. This section will modify the propagation-out ratio of transition functions in digital phero-mone infrastructure [22] to erase the boundary effect of a plane pheromone basket.
The propagation-out ratio at the frontier points of Fig.2
needs to be modified from F2=ð2 F2Þ [22] to F2=ð3 2F2Þ (proof is given in Appendix A). Then, the transition functions become
q2ðk; bm2; i þ 1Þ ¼ F2 3 ðr2ðk; bm2þ6; iÞ þ q2ðk; bm2þ6; iÞÞ; if m2¼ 1; 2; . . . ; 6 F2 32F2ðr2ðk; bm26; iÞ þ q2ðk; bm26; iÞÞ þ F2 3 ðr2ðk; bm2þ1; iÞ þ q2ðk; bm2þ1; iÞÞ þF2 3 ðr2ðk; bm21; iÞ þ q2ðk; bm21; iÞÞ; if m2¼ 8; 9; . . . ; 11 F2 32F2ðr2ðk; bm26; iÞ þ q2ðk; bm26; iÞÞ þ F2 3 ðr2ðk; bm2þ1; iÞ þ q2ðk; bm2þ1; iÞÞ þF2 3 ðr2ðk; b12; iÞ þ q2ðk; b12; iÞÞ; if m2¼ 7 F2 32F2ðr2ðk; bm26; iÞ þ q2ðk; bm26; iÞÞ þ F2 3 ðr2ðk; b7; iÞ þ q2ðk; b7; iÞÞ þF2 3 ðr2ðk; bm21; iÞ þ q2ðk; bm21; iÞÞ; if m2¼ 12 8 > > > > > > > > > > < > > > > > > > > > > : ð2Þ s2ðk; bm2; i þ 1Þ ¼ s2ðk; bm2; iÞ þ 1 F2 32F2 r2ðk; bm2; iÞ þ q2ðk; bm2; iÞ ð Þ; if m2¼ 1; 2;:::; 6: s2ðk; bm2; iÞ þ 1 F2ð Þ r2ð ðk; bm2; iÞ þ q2ðk; bm2; iÞÞ; if m2¼ 7; 8; . . . ; 12: ( ð3Þ
Figure4 illustrates the concept of Eqs.2 and3. In fact, one can apply transition functions in Cartesian coordinate system by modifying the propagation-out ratio from F2=ð3 2F2Þ to F2=ð4 3F2Þ in Eqs. 2 and 3 by same procedure in Appendix A. Then, the final propagation results of Eqs.2and3can be obtained by Eq. (A.8) in the AppendixB. For example, if M2is 12, the final propagation
results at bm2is s2ðk; bm2; 1Þ ¼ P2;1;m2 P2;2;m2 .. . P2;12;m2 2 6 6 6 4 3 7 7 7 5 T r2ðk; b1; 0Þ r2ðk; b2; 0Þ .. . r2ðk; b12; 0Þ 2 6 6 6 4 3 7 7 7 5; m2¼ 1; 2; . . . ; 12: ð4Þ where P2;1;m2; P2;2;m2; . . . ; and P2;12;m2 are the quantity of
positions 1–12 affecting position bm2 within the two-dimensional pheromone basket and the values of P2;1;m2, P2;2;m2; . . . ; and P2;12;m2 are listed at no. 1–12 of Table1.
3 Space–time controller
The STC interlaces the time–effect controller, such as EWMA or PCC and so on, with the space–effect controller, 2D-PPC, in pairs. Figure5is the scalar-form block diagram of EWMA with 2D-PPC when the layout of measurements is like Fig. 2. In Fig.5, the control loop of STC (EWMA with 2D-PPC) can be separated into four modules: MIMO plant, the time–effect controller, the space–effect controller and MIMO recipe generator. Sections 3.1 to 3.4 describe each of these modules.
3.1 MIMO plant
The linear regression model is assumed for the process model. Based on twelve measurements as in Fig. 2, this study assumes the process has 12 outputs and each output has its own disturbance, process gain and two inputs. The demonstrated MIMO plant is
Ykþ1¼ a þ b Xkþ "kþ1 k ¼ 0; 1; 2; . . . ; ð5Þ where Ykþ1¼ Y1½ ;kþ1 Y2;kþ1 Ym2;kþ1 YM2;kþ1 T; Xk¼ X1½ ;k X2;k Xn2;k XN2;k T; a ¼ a1½ a2 am2 aM2 T; b ¼ b1;1 b1;2 b1;n2 b1;N2 b2;1 b2;2 b2;n2 b2;N2 .. . .. . . . . .. . .. . .. . bm2;1 bm2;2 bm2;n2 bm2;N2 .. . .. . ... .. . ... bM2;1 bM2;2 bM2;n2 bM2;N2 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ; and "kþ1¼½"1;kþ1 "2;kþ1 "m2;kþ1 "M2;kþ1 T: In addition,
Y1;k;Y2;k; ;YM2;k 2 < M2measurements at the end of run k
X1;k;X2;k; ;XN2;k 2 < N2recipes (inputs) of a wafer at run k
a1;a2; ;aM2 2 < intercepts of the process b1;1;b1;2; ;bM2;N2 2 < system gains
"1;k; "2;k; ; "M2;k 2 < disturbances, which include noise and uncontrolled terms of run k.
3.2 The time–effect controller
The MIMO time–effect controller is a traditional con-troller, such as MIMO EWMA, MIMO dEWMA, MIMO PCC, and so on, applied to all measurement positions within a wafer. Since MIMO EWMA is now commonly used in semiconductor fabrication, this study employs it as a demonstrated MIMO time–effect con-troller and the forecast disturbance of the MIMO EWMA controller at run k+1, e"kþ1, becomes
e"kþ1¼½e"1;kþ1 e"2;kþ1 e"m2;kþ1 e"M2;kþ1 T ¼ Λ Yk a bbXkþ I Λð Þe"k
ð6Þ where Λ ¼ lI is the discount factor of MIMO EWMA. The Eq. (6) can be applied both in M2≥N2 and M2≤N2.
3.3 The space–effect controller
The infrastructure of the space–effect controller has de-scribed in Section 2. Then, the input of 2D pheromone basket is the forecasting disturbances of the MIMO time– effect controller. R2ðk; 0Þ ¼ r2ðk; b1; 0Þ r2ðk; b2; 0Þ .. . r2ðk; bm2; 0Þ .. . r2ðk; bM2; 0Þ 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ¼ e"1;kþ1 e"2;kþ1 .. . e"m2;kþ1 .. . e"M2;kþ1 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ð7Þ
And the forecast disturbance of 2D-PPC at run k+1, ^
"kþ1, can be obtained by substituting Eq. (7) into Eq. (4).
S1 S7 S8 S9 S10 S12 S11 S2 S 3 S4 S6 S5 Q1 Q5 Q6 Q2 Q9 Q11 Q12 Q7 Q8 Q3 Q4 Q10
: propagating out at the boundary points. : propagating out at the inner points.
: aggregating strength at the boundary points. : aggregating strength at the inner points.
(F2 3)Qbm2 (F2 3−2F2)Qbm2
(1−F2)Qbm2
( )
(1− F23−2F2)Qbm2
Fig. 4 Transitions with propagation parameterF2and size M2012 in a
b"kþ1 ¼ b"1;kþ1 b"2;kþ1 .. . b"m2;kþ1 .. . b"M2;kþ1 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ¼ P2;1;1 P2;2;1 P2;m2;1 P2;M2;1 P2;1;2 P2;2;2 P2;m2;2 P2;M2;2 .. . .. . . . . .. . P2;1;m2 P2;2;m2 P2;m2;m2 P2;M2;m2 .. . .. . . . . .. . P2;1;M2 P2;2;M2 P2;m2;M2 P2;M2;M2 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 e"1;kþ1 e"2;kþ1 .. . e"m2;kþ1 .. . e"M2;kþ1 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ð8Þ
Table 1 Notations and values of Eqs. (4), (8), and (13)
No. Notation Value
1 P2,1,1, P2,2,2, P2,3,3, P2,4,4, P2,5,5, P2,6,6 28bF7 2;kþ1219bF2;kþ16 þ162bF52;kþ1þ1566bF2;kþ14 2754bF32;kþ1972bF2;kþ12 þ4374bF2;kþ12187 3 5bF7 2;kþ155bF2;kþ16 þ96bF52;kþ1þ324bF2;kþ14 864bF32;kþ1þ1215bF2;kþ1729 2 P2,2,1, P2,6,1, P2,1,2, P2,3,2, P2,2,3, P2,4,3, P2,3,4, P2,5,4, P2,4,5, P2,6,5, P2,1,6, P2,5,6 7bF 7 2;kþ136bF62;kþ1þ108bF2;kþ14 81bF2;kþ13 3 5bF7 2;kþ155bF62;kþ1þ96bF52;kþ1þ324Fb2;kþ14 864bF2;kþ13 þ1215bF2;kþ1729 3 P2,3,1, P2,5,1, P2,4,2, P2,6,2, P2,1,3, P2,5,3, P2,2,4, P2,6,4, P2,1,5, P2,3,5, P2,2,6, P2,4,6, 2bF 7 2;kþ19bF2;kþ16 þ36bF2;kþ15 27bF2;kþ14 3 5bF7 2;kþ155bF62;kþ1þ96bF52;kþ1þ324Fb2;kþ14 864bF2;kþ13 þ1215bF2;kþ1729 4 P2,4,1, P2,5,2, P2,6,3, P2,1,4, P2,2,5, P2,3,6, 8bF7 2;kþ1þ24bF2;kþ16 18bF2;kþ15 3 5bF7 2;kþ155bF62;kþ1þ96bF52;kþ1þ324Fb2;kþ14 864bF2;kþ13 þ1215bF2;kþ1729 5 P2,7,1, P2,8,2, P2,9,3, P2,10,4, P2,11,5, P2,12,6, P2,1,7, P2,2,8, P2,3,9, P2,4,10, P2,5,11, P2,6,12, 11bF 7 2;kþ118bF2;kþ16 þ378bF2;kþ15 594bF2;kþ14 486bF2;kþ13 þ1458bF2;kþ12 729bF2;kþ1 3ð5bF7 2;kþ155bF2;kþ16 þ96bF2;kþ15 þ324bF2;kþ14 864bF32;kþ1þ1215bF2;kþ1729Þ 6 P2,8,1, P2,12,1, P2,7,2, P2,9,2, P2,8,3, P2,10,3, P2,9,4, P2,11,4, P2,10,5, P2,12,5, P2,7,6, P2,11,6, P2,2,7, P2,6,7, P2,1,8, P2,3,8, P2,2,9, P2,4,9, P2,3,10, P2,5,10, P2,4,11, P2,6,11, P2,1,12, P2,5,12, 14bF7 2;kþ1þ93bF62;kþ1108bF2;kþ15 216Fb42;kþ1þ486bF2;kþ13 243bF2;kþ12 3 5bF7 2;kþ155bF62;kþ1þ96bF52;kþ1þ324Fb2;kþ14 864bF2;kþ13 þ1215bF2;kþ1729 7 P2,9,1, P2,11,1, P2,10,2, P2,12,2, P2,7,3, P2,11,3, P2,8,4, P2,12,4, P2,7,5, P2,9,5, P2,8,6, P2,10,6, P2,3,7, P2,5,7, P2,4,8, P2,6,8, P2,1,9, P2,5,9, P2,2,10, P2,6,10, P2,1,11, P2,3,11, P2,2,12, P2,4,12, 4bF7 2;kþ1þ12bF2;kþ16 99bF2;kþ15 þ162bF42;kþ181bF2;kþ13 3 5bF7 2;kþ155bF62;kþ1þ96bF52;kþ1þ324Fb2;kþ14 864bF2;kþ13 þ1215bF2;kþ1729 8 P2,10,1, P2,11,2, P2,12,3, P2,7,4, P2,8,5, P2,9,6, P2,4,7, P2,5,8, P2,6,9, P2,1,10, P2,2,11, P2,3,12, 16bF 7 2;kþ172bF2;kþ16 þ108bF2;kþ15 54bF2;kþ14 3 5bF7 2;kþ155bF62;kþ1þ96bF52;kþ1þ324Fb2;kþ14 864bF2;kþ13 þ1215bF2;kþ1729 9 P2,7,7, P2,8,8, P2,9,9, P2,10,10, P2,11,11, P2,12,12, 22bF7 2;kþ1þ3bF2;kþ16 810bF52;kþ1þ2322bF2;kþ14 810bF2;kþ13 4374bF2;kþ12 þ5832bF2;kþ12187 3 5bF7 2;kþ155bF2;kþ16 þ96bF2;kþ15 þ324Fb42;kþ1864bF2;kþ13 þ1215bF2;kþ1729 10 P2,8,7, P2,12,7, P2,7,8, P2,9,8, P2,8,9, P2,10,9, P2,9,10, P2,11,10, P2,10,11, P2,12,11, P2,7,12, P2,11,12 28bF 7 2;kþ1228bF2;kþ16 þ495bF52;kþ1þ108bF2;kþ14 1620bF2;kþ13 þ1944bF2;kþ12 729bF2;kþ1 3 5bF7 2;kþ155bF2;kþ16 þ96bF2;kþ15 þ324bF2;kþ14 864bF32;kþ1þ1215bF2;kþ1729 11 P2,9,7, P2,11,7, P2,10,8, P2,12,8, P2,7,9, P2,11,9, P2,8,10, P2,12,10, P2,7,11, P2,9,11, P2,8,12, P2,10,12 8bF 7 2;kþ112bF62;kþ1þ234bF2;kþ15 621bF2;kþ14 þ648bF32;kþ1243bF2;kþ12 3 5bF7 2;kþ155bF62;kþ1þ96bF52;kþ1þ324Fb2;kþ14 864bF2;kþ13 þ1215bF2;kþ1729 12 P2,10,7, P2,11,8, P2,12,9, P2,7,10, P2,8,11, P2,9,12 32bF7 2;kþ1þ192bF62;kþ1432bF2;kþ15 þ432bF2;kþ14 162bF32;kþ1 3ð5bF7 2;kþ155bF2;kþ16 þ96bF2;kþ15 þ324Fb42;kþ1864bF2;kþ13 þ1215bF2;kþ1729Þ 13 G1, G2,…, G12 1 14 G25,13, G26,14,…, G36,24 1 15 G25,1, G26,2,…, G36,12 −1 16 W1, W2,…, W12 l z þ l 1=ð Þ
-+ -+ -+ -+ -+ + + + + + + + + + + + + + + + + + -+ -+ -+ + + + + + + + + + + + + + + + + + + + -+ + + + + + P2,2,1 P2,2,2 P2,2,3 P2,2,4 k , 2 , 1 ˆ ε P2,2,5 P2,2,6 P2,2,7 P2,2,9 P2,2,8 P2,2,10 P2,2,11 P2,2,12 P2,3,1 P2,3,2 P2,3,3 P2,3,4 k , 3 , 1 ˆ ε P2,3,5 P2,3,6 P2,3,7 P2,3,9 P2,3,8 P2,3,10 P2,3,11 P2,3,12 P2,4,1 P2,4,2 P2,4,3 P2,4,4 k , 4 , 1 ˆ ε P2,4,5 P2,4,6 P2,4,7 P2,4,9 P2,4,8 P2,4,10 P2,4,11 P2,4,12 P2,1,1 P2,1,2 P2,1,3 P2,1,4 k , 1 , 1 ˆ ε P2,1,5 P2,1,6 P2,1,7 P2,1,9 P2,1,8 P2,1,10 P2,1,11 P2,1,12 P2,5,1 P2,5,2 P2,5,3 P2,5,4 k , 5 , 1 ˆ ε P2,5,5 P2,5,6 P2,5,7 P2,5,9 P2,5,8 P2,5,10 P2,5,11 P2,5,12 P2,6,1 P2,6,2 P2,6,3 P2,6,4 k , 6 , 1 ˆ ε P2,6,5 P2,6,6 P2,6,7 P2,6,9 P2,6,8 P2,6,10 P2,6,11 P2,6,12 P2,7,1 P2,7,2 P2,7,3 P2,7,4 k , 7 , 1 ˆ ε P2,7,5 P2,7,6 P2,7,7 P2,7,9 P2,7,8 P2,7,10 P2,7,11 P2,7,12 P2,12,1 P2,12,2 P2,12,3 P2,12,4 k , 12 , 1 ˆ ε P2,12,5 P2,12,6 P2,12,7 P2,12,9 P2,12,8 P2,12,10 P2,12,11 P2,12,12 P2,8,1 P2,8,2 P2,8,3 P2,8,4 k , 8 , 1 ˆ ε P2,8,5 P2,8,6 P2,8,7 P2,8,9 P2,8,8 P2,8,10 P2,8,11 P2,8,12 P2,9,1 P2,9,2 P2,9,3 P2,9,4 k , 9 , 1 ˆ ε P2,9,5 P2,9,6 P2,9,7 P2,9,9 P2,9,8 P2,9,10 P2,9,11 P2,9,12 P2,10,1 P2,10,2 P2,10,3 P2,10,4 k , 10 , 1 ˆ ε P2,10,5 P2,10,6 P2,10,7 P2,10,9 P2,10,8 P2,10,10 P2,10,11 P2,10,12 P2,11,1 P2,11,2 P2,11,3 P2,11,4 k , 11 , 1 ˆ ε P2,11,5 P2,11,6 P2,11,7 P2,11,9 P2,11,8 P2,11,10 P2,11,11 P2,11,12 k
T
k , 2 υ k , 3 υ k , 4 υ k , 5 υ k , 6 υ k , 1 , 2 ˆ ε k , 2 , 2 ˆ ε k , 3 , 2 ˆ ε k , 4 , 2 ˆ ε k , 6 , 2 ˆ ε k , 8 , 2 ˆ ε k , 10 , 2 ˆ ε k , 11 , 2 ˆ ε k , 12 , 2 ˆ ε k , 5 , 2 ˆ ε k , 7 , 2 ˆ ε k , 9 , 2 ˆ ε + k , 1 υ1,k υY
1,k -+ + + -+ + + -+ k X -+ -+ kY
2, kY
3, kY
4, kY
6, kY
5, -+ + + -+ ˆ -+ + + -+ -+ + + -+ -+ + + -+ -+ + + -+ -+ + + -+ k , 7 υ k , 8 υ k , 9 υ k , 10 υ k , 11 υ k , 12 υ kY
7, kY
8, kY
9, kY
10, kY
11, kY
12, W1 W2 W3 W4 W5 W6 W7 W9 W8 W10 W11 W12 Time-effectcontroller Space-effect controller MIMO recipe generator MIMO plant 1 T T(ˆˆ ) ˆ ββ − β T 1 T ) ˆ ˆ (ββ−β or β
1
13
25
38
39
40
41
42
43
44
45
46
47
48
37
26
27
28
29
30
31
32
33
34
35
36
14
15
16
17
18
19
20
21
22
23
24
2
3
4
5
6
7
8
9
10
11
12
Outer circle Inner circlea
Fig. 5 Scalar form block diagram of a STC using EWMA as a time-effect controller with measurements layout in Fig.1and plant in Section3.1. a Block diagram. b Details of nodes 37-48 in Fig.5a
where the values of P2,1,1, P2,2,1,…, and P2;M2;M2are listed at no. 1–12 of Table1.
Figure 6a–f show an example of propagation results with different 2D propagation parameters F2 for a
two-dimensional pheromone basket in Fig. 2. Figure 6a is
the external inputs initially, and Fig. 6b–f illustrate the final propagation results with different F2. If F2 is 0,
the space–effect controller does not modify the forecast-ing disturbances of the MIMO time–effect controller, i.e., the 2D-PPC is disable. In addition, the propagation Fig. 5 (continued) 0 20 40 0 20 40 0 2 4 6 Positon (Y-axis) Positon (X-axis) E x te rn al inpu ts 0 20 40 0 20 40 0 2 4 6 Positon (Y-axis) Positon (X-axis) P ropa g ation r es u lt s 0 20 40 0 20 40 0 2 4 6 Positon (Y-axis) Positon (X-axis) P ropa g ation r es u lt s 0 20 40 0 20 40 0 2 4 6 Positon (Y-axis) Positon (X-axis) Pr o p ag at io n r es u lts 0 20 40 0 20 40 0 2 4 6 Positon (Y-axis) Positon (X-axis) P ropa ga ti on r es u lt s 0 20 40 0 20 40 0 2 4 6 Pr o p ag at io n r es u lts Positon (Y-axis) Positon (X-axis)
(a) External inputs (b) F2 = 0 (c) F2 = 0.25
(d) F2= 0.5 (e) F2= 0.75 (f) F2 =0.999
Fig. 6 An example of different propagation results with different 2D propagation parametersF2for a two-dimensional pheromone basket shown in
result approaches the mean of the external inputs as F2
approaches unity.
3.4 MIMO recipe generator
The recipe generator generates the recipe of the process for the next run. In Fig.5, we use the linear regression model to produce
T¼ba þ bb Xkþ1þb"kþ1; ð9Þ
where Xk + 1 is the recipe (input) of run k+1, ba is the
estimator of the interceptα, bb is the estimator of the system β and T ¼ T T T½ T
1M2is a given target values of measurements within a wafer. The recipe for run k+1, Xk+1,
is given based on different conditions: 1. If M2≥N2, Xkþ1¼ bbTbb 1 bbTðTba b"kþ1Þ ð10Þ (2) If M2≤N2, Xkþ1¼ bbT bbbbT 1 Tba b"kþ1 ð Þ ð11Þ (3) If M20N2,
One selects Eq. (10) to minimize the sum of the manip-ulated variables squared or Eq. (11) to minimize the sum of squares deviation from target [13].
In addition, ba and bb in Eqs. (10) and (11) are obtained from the linear regression model of the offline data and the parameterb"kþ1 comes from Eq. (8).
3.5 The controller parameter tuners
The controller parameters of STC are the EWMA discount factors,λ, in Eq. (6) and the 2D propagation parameter, F2,
in Eq. (8). The best controller parameters can be obtained by using historical (or training) data and examining all possible values of controller parameters. In this study, we utilized the ordinary tuning method of the traditional time–effect con-troller to obtain a set of the best propagation parameter F2, and discount factors,lF
2, of the time–effect controller for F2 by minimum sum square error, or
F2; lF2 ffi min F2;lF2 X m2 X k0 e2m2;k0 ! ð12Þ where e2
m2;k0is error at position m2and the k′th training data.
3.6 Control procedure
The STC interlaces the space–effect 2D-PPC and the time–effect controller in pairs. In this section, the con-trol procedure of STC is presented in the following sequence:
Step 1 Run time–effect controller, such as MIMO EWMA in Eq. (6) with the discount factor, lF
2, obtained from Eq. (12) at first two runs and obtain process error ek, where k01 and 2. Then do follow-up steps
from k is 3.
Step 2 Get forecasting disturbances for M2measurement
points, e"kþ1, by process error ek and the MIMO
time–effect controller.
Step 3 Set bF2;kþ1 to bF2 from Eq. (12).
Step 4 Modify the MIMO time–effect forecasting pro-cess disturbances at run k + 1 using Eq. (8), where e"kþ1 is obtained by step 2 and bF2;kþ1 is obtained by step 3.
Step 5: Determine the recipe Xk+1using Eqs. (10) or (11).
Then, obtain process error ek+1at run k+1.
Step 6: Let k0k+1 and repeat steps 2–5 to produce the follow-up recipes.
4 Stability analysis
EWMA is chosen as time–effect controller to analyze sta-bility region of STC. For the other time–effect controllers, same procedures can be applied. We first derive transfer function from the block diagram for MIMO plants and then discuss the stability region under different model mismatch, discount factor, and two-dimensional propagation parame-ter. Finally, intrinsic properties of EWMA and STC are compared.
4.1 Transfer function
The closed-loop transfer function of STC, recipe genera-tor, and plant can be derived from Fig. 5 directly by Matlab solution [31] with 36 nodes assigned (number 1–
36 in Fig. 5). However, while implementing [31] into Fig. 5, Matlab shows “maximum variable size allowed by the program is exceeded”. So, this study redraws a MIMO signal flow graph by reducing 36 nodes to 3 nodes. In Fig. 7, node A represents nodes 1–12 in
Fig. 5, node B represents nodes 13–24 in Fig. 5, and node C represents nodes 25–36 in Fig. 5. Furthermore, H1 is the path gain from T to node 1–12 in Fig. 5. H2
involves MIMO plant and controller. H3and H4are unity
process error. H5 is controller of STC. Then, the transfer
function from disturbance to output, Gdy, can be
obtained: Gdy ¼ I H2=ðI H5H4 H5H3H2ÞH5H3 ð13Þ where H1¼ G1 0 0 G2 0 0 .. . . . . .. . .. . . . . .. . G12 0 0 2 6 6 6 6 6 4 3 7 7 7 7 7 5( H2¼ b bbTbb 1 bbT when M2≥N2 and H2¼ bbbT bbbb T 1 when M2 N2, H3¼ G25;13 0 0 0 G26;14 0 .. . . . . .. . .. . . . . .. . 0 0 G36;24 2 6 6 6 6 6 4 3 7 7 7 7 7 5( H4¼ G25;1 0 0 0 G26;2 0 .. . . . . .. . .. . . . . .. . 0 0 G36;12 2 6 6 6 6 6 4 3 7 7 7 7 7 5 and H5¼ G1;25 G1;26 G1;36 G2;25 G2;26 G2;36 .. . . . . .. . .. . . . . .. . G12;25 G12;26 G12;36 2 6 6 6 6 6 4 3 7 7 7 7 7 5:
In Eq. (13), Giis the path gain of the ith forward path; Gi, jis the path gain from node j to node i in Fig.5;β and bb are
Propagation parameter ( )F2 Propagation parameter ( )F2 Discount factor ( )λ
Maximum
acceptable model mis
match . 0 0.25 0.50 0.75 1.00 0 0.25 0.50 0.75 0.99 0 10 20 30 40 50
(a)
M2>N2 Propagation parameter ( )F2 Propagation parameter ( )F2 Discount factor ( )λMaximum acceptable model mis
m atch . 0 0.25 0.50 0.75 1.00 0 0.25 0.50 0.75 0.99 0 10 20 30 40 50
(b)
M2<N2 Propagation parameter ( )F2 Propagation parameter ( )F2 Discount factor ( )λ Maximumacceptable model mis
match . 0 0.25 0.50 0.75 1.00 0 0.25 0.50 0.75 0.99 0 10 20 30 40 50
(a)
M2>N2 Propagation parameter ( )F2 Propagation parameter ( )F2 Discount factor ( )λMaximum acceptable model mis
m atch . 0 0.25 0.50 0.75 1.00 0 0.25 0.50 0.75 0.99 0 10 20 30 40 50
(b)
M2<N2Fig. 8 3D plot of stability region of STC in theouter circle, when
xouter¼ 1:5 xinner. a Stable region of STC in the outer circle, when
xouter¼ 1:5 xinner and M2>N2. b Stable region of STC in the outer
circle, whenxouter¼ 1:5 xinnerand M2<N2. c Stable region of STC in
the outer circle, whenλ00.3 and M2>N2. d Stable region of STC in the
outer circle, whenλ00.3 and M2<N2
T
Node A
Node B
Node C
H
2H
1H
5H
4H
3 ku
kY
T
Node A
Node B
Node C
H
2H
1H
5H
4H
3 ku
kY
0 10 20 30 40 50 0.0 0.2 0.4 0.6 0.8 1.0 Discount factor ( ) 0 . 1 2≅ F λ M axim u m accep tab le m o d el m is m atch . 2 . 0 2= F 4 . 0 2= F F2=0.6 8 . 0 2= F 0 . 0 2= F (EWMA)
(a)
Stable region of STC in the outer circle, when ξouter= 51.×ξinner and M2>N2.0 10 20 30 40 50 0.0 0.2 0.4 0.6 0.8 1.0 Discount factor ( )λ Maximum a cce p table mode l mis match . 2 . 0 2= F 4 . 0 2= F F2=0.6 8 . 0 2= F 0 . 0 2= F (EWMA) 0 . 1 2≅ F
(b)
Stable region of STC in the outer circle, when ξouter= 51. ×ξinner and M2<N2.0 2 4 6 8 10 12 14 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Model mismatch ( ) 0 . 1 2≅ F inner ξ Maxim um accep table m o del m is m atch 2 . 0 2= F 4 . 0 2= F F2=0.6 8 . 0 2= F 0 . 0 2= F (EWMA)
(c)
Stable region of STC in the outer circle, when λ=0.3 and M2>N2.0 10 20 30 40 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Model mismatch ( ) 95 . 0 2= F inner ξ M aximum accep table mo d el mi sma tc h 2 . 0 2= F 4 . 0 2= F F2=0.6 8 . 0 2= F 0 . 0 2= F (EWMA)
(d)
Stable region of STC in the outer circle, when λ=0.3 and M2<N2.0 10 20 30 40 50 0.0 0.2 0.4 0.6 0.8 1.0 Discount factor ( ) 0 . 1 2≅ F λ M axim u m accep tab le m o d el m is m atch . 2 . 0 2= F 4 . 0 2= F F2=0.6 8 . 0 2= F 0 . 0 2= F (EWMA)
(a)
Stable region of STC in the outer circle, when ξouter= 51.×ξinner and M2>N2.0 10 20 30 40 50 0.0 0.2 0.4 0.6 0.8 1.0 Discount factor ( )λ Maximum a cce p table mode l mis match . 2 . 0 2= F 4 . 0 2= F F2=0.6 8 . 0 2= F 0 . 0 2= F (EWMA) 0 . 1 2≅ F
(b)
Stable region of STC in the outer circle, when ξouter= 51. ×ξinner and M2<N2.0 2 4 6 8 10 12 14 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Model mismatch ( ) 0 . 1 2≅ F inner ξ Maxim um accep table m o del m is m atch 2 . 0 2= F 4 . 0 2= F F2=0.6 8 . 0 2= F 0 . 0 2= F (EWMA)
(c)
Stable region of STC in the outer circle, when λ=0.3 and M2>N2.0 10 20 30 40 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Model mismatch ( ) 95 . 0 2= F inner ξ M aximum accep table mo d el mi sma tc h 2 . 0 2= F 4 . 0 2= F F2=0.6 8 . 0 2= F 0 . 0 2= F (EWMA)
(d)
Stable region of STC in the outer circle, when λ=0.3 and M2<N2.Fig. 9 Stability region of STC (EWMA with 2D-PPC) for the given conditions in Section4.2. a One set of anthropogenic disturbance. b Anthropogenic disturbances of 27 successive runs, where 12 measurement positions are conducted for each wafer
Table 2 Controller structure comparison of time–effect controller (EWMA) and STC (EWMA with 2D-PCC) Concept Controllers in this study Controller parameters Pole number Zero number
Controller type Allowable stable region for all controller parameters
λ F2
Time–effect controller
MIMO EWMA IfΛ0λI, 0≦λ≦1
NA min (M2, N2) min (M2, N2) MIMO EWMA is a MIMO
integral controller. IfΛ0λI, ξλ≤2.b ¼ xbband
STC MIMO EWMA
with 2D-PPC
IfΛ0λI, 0≦λ≦1
F200 The space-effect controller is disable and the control loop is equal to the time-effect controller IfΛ0λI,
0≦λ≦1
0<F2<1 min (M2, N2) min (M2, N2) STC is a MIMO integral controller and has P
min Mð 2;N2Þ 1
M1poles and
zeros (M1varies by different
controllers and is 1 for EWMA)
Stable region increases with the growth of F2
IfΛ0λI, 0≦λ≦1
F2→1 min (M2, N2) min (M2, N2) When F2approaches to 1, the 2D-PPC is equal to a M2moving average filter among M2disturbances of EWMA
defined in Sections3.1and3.4. The values of notations in Eq. (13) are listed at no. 13–19 of Table1.
4.2 Stability region
The stability region of Fig.5 can be obtained by checking the pole locations of the transfer function, Eq. (13). We examine the stability region of STC in terms of model mismatch (b ¼ xeb) with different values of λ and F2. If all
of the poles are within the unit circle, the point (λ, F2,ξ) is a
stable point.
To simplify analysis,βouterandβinnerdenote the system
gains of the measurement points at the outer circle (positions
1–6) and at the inner circle (positions 7–12) in Fig.2. Then, β becomes,
b ¼ bouter binner
: ð14Þ
In Eq. (14), the size ofβouterandβinnerare both M2ð =2Þ
N2. Next, this study assumes that model mismatch of the outer and inner circle,ξouterandξinner, are
bouter¼ xouterbbouter ð15Þ
binner¼ xinnerbbinner ð16Þ
(a)
One set of anthropogenic disturbance.-1500 -1000 -500 0 500 1 3 5 7 9 11 13 15 17 19 21 23 25 27 Run (No.) D is tur ba nc e ( n m )
Position 1 Position 2 Position 3 Position 4
Position 5 Position 6 Position 7 Position 8
Position 9 Position 10 Position 11 Position 12
(b)
Anthropogenic disturbances of 27 successive runs, where 12 measurement positions are conducted for each wafer. 0 10 20 30 0 10 20 30 -3000 -2000 -1000 0 1000Disturbance (nm)
Position (Y-axis)
Position (X-axis)
Fig. 10 Anthropogenic disturbance in the simulation
Table 3 The settings of the MIMO plant and model for the simulations
Settings
The first MIMO plant (M2>N2)
a ¼ a½ 1 a6 a7 a12 T ¼ 6700:26 6700:26 4304:63 4304:63½ T bm2;n2 n201 n202 bm2;n2 n201 n202 bm2;n2 n201 n202 m201 −1,356.6 1,182.2 m205 −1,356.6 1,182.2 m209 85.54 −1,458.8 m202 −1,356.6 1,182.2 m206 −1,356.6 1,182.2 m2010 85.54 −1,458.8 m203 −1,356.6 1,182.2 m207 85.54 −1,458.8 m2011 85.54 −1,458.8 m204 −1,356.6 1,182.2 m208 85.54 −1,458.8 m2012 85.54 −1,458.8
The second MIMO plan (M2<N2)
a ¼ a½ 1 a6 a7 a12 T ¼ 6; 700:26 6; 700:26 4; 304:63 4; 304:63½ T bm2;n2 n201 n202 n203 n204 n205 n206 n207 n208 m201 −111.55 106.64 −109.33 105.20 −110.98 102.12 −107.77 99.75 m202 −109.51 104.93 −111.77 103.38 −103.23 105.14 −104.72 103.59 m203 −108.29 102.98 −107.17 102.72 −108.16 98.90 −106.50 106.42 m204 −112.08 99.27 −110.12 107.35 −105.29 107.28 −107.90 107.89 m205 −103.48 99.30 −112.42 104.43 −110.55 106.06 −106.60 105.24 m206 −103.49 99.97 −111.15 106.55 −108.94 107.88 −109.81 100.23 m207 −109.17 106.72 −106.11 101.37 −107.61 105.47 −112.74 101.19 m208 −104.01 103.39 −110.39 107.76 −108.33 102.90 −105.12 101.90 m209 −106.53 102.53 −111.30 99.87 −111.33 100.97 −108.55 106.24 m2010 −105.45 107.66 −105.27 106.07 −105.21 100.04 −109.91 100.62 m2011 −104.18 107.12 −112.36 102.43 −107.11 106.80 −112.55 103.18 m2012 −106.70 103.79 −111.33 101.47 −106.27 100.66 −110.05 108.12 bm2;n2 n209 n2010 n2011 n2012 n2013 n2014 n2015 n2016 m201 −104.29 103.44 −108.49 104.38 16.31 −118.08 10.62 −111.87 m202 −111.41 104.92 −112.40 98.80 8.33 −118.34 10.84 −114.80 m203 −110.03 101.38 −110.93 100.77 10.39 −119.68 14.70 −119.29 m204 −110.87 100.49 −104.24 105.46 15.99 −116.28 7.85 −115.25 m205 −107.31 102.72 −107.95 100.69 15.96 −118.26 17.05 −115.25 m206 −103.51 102.19 −107.42 104.31 9.39 −113.18 8.35 −120.98 m207 −109.00 107.36 −110.03 104.42 7.16 −116.02 12.36 −120.16 m208 −108.29 99.09 −110.20 102.20 13.59 −121.12 12.54 −111.86 m209 −111.65 107.16 −104.63 105.13 12.90 −120.48 13.99 −117.09 m2010 −109.96 107.54 −105.10 101.29 7.83 −119.38 16.58 −120.76 m2011 −103.88 106.69 −111.60 101.51 9.19 −121.00 15.23 −118.80 m2012 −103.09 100.63 −104.63 103.13 15.47 −118.17 15.85 −119.31 bm2;n2 n2017 n2018 n2019 n2020 n2021 n2022 n2023 n2024 m201 9.85 −121.22 13.57 −121.38 12.58 −114.67 8.47 −115.36 m202 8.26 −117.69 9.88 −118.88 9.48 −119.41 8.18 −117.01 m203 12.73 −116.30 7.63 −114.26 8.92 −115.75 14.43 −121.52 m204 11.23 −119.47 16.83 −116.52 16.83 −120.77 7.32 −117.71 m205 10.59 −112.10 16.94 −118.52 13.02 −115.21 12.47 −111.92 m206 7.98 −120.33 14.19 −114.37 12.66 −119.06 10.67 −117.51 m207 8.89 −114.38 14.34 −119.14 8.79 −120.32 13.03 −111.57 m208 15.42 −116.01 11.00 −120.22 11.41 −121.17 8.33 −117.14 m209 7.99 −116.84 13.25 −116.71 16.22 −119.72 16.74 −116.60 m2010 13.94 −117.01 16.07 −121.39 7.97 −119.60 11.29 −111.97 m2011 11.49 −118.76 12.53 −116.31 12.58 −119.30 8.89 −119.20 m2012 16.44 −117.72 8.85 −117.21 10.22 −119.99 8.99 −112.18
Thus, there are four parameters,ξouter,ξinner,λ, and F2, in
Eq. (13); when two of them are given, stability region among the three other parameters can be presented in a 3-D plot. For example, whenξouter01.5×ξinnerandλ00.3, the
maximum acceptable model mismatches of STC in the outer, bx2;outer, shown in shaded curved surface, for different F2 and λ are presented in Fig. 8a–b, where the stability
region is under the shaded curved surface. The cross-sections perpendicular to the F2−λ plane and parallel to
the bx2;outer l plane in terms of various F2 are shown in
Fig.9a–b. In Fig.9a–b, bx2;outerdecreases with the growth of λ but increases with the growth of F2. In Fig.9c–d, bx2;outer
decreases with the growth of ξinner but still increases
with the growth of F2. In sum, it shows that STC
increases the stability region at the outer circle by the growth of F2 as compared to EWMA, which is the region
under the curve when F200. Note that the conclusion is also
applied to the stability regions at the inner circle by the same token.
The intrinsic properties of STC and EWMA are given in Table2 including the number of poles and zeros of STC examined from Eq. (13), EWMA and the controller type. The MIMO EWMA is a MIMO integral controller, while STC varies with the 2D propagation parameter F2. When F2
is zero, the space–effect controller is disable and the control loop is equal to MIMO EWMA. When F2is not zero, the
number of pole or zero of STC becomes P
min Mð 2;N2Þ
1 M1
and one of the zero positions is 1. Thus, STC preserves the property of EWMA in step disturbance rejection. As F2
of STC approaches to 1, the 2D-PPC is equal to a M2
moving average filter among M2 disturbances of
EWMA.
5 Simulation results
To do the simulation, this study makes 27 sets of an-thropogenic disturbance from fabrication data to examine output performance of the six candidate controllers. Figure 10a is a set of anthropogenic disturbance and Fig. 10b is the anthropogenic disturbances of 27 succes-sive runs, where 12 measurement positions are conducted for each wafer. This paper conducts two simulations for two types of MIMO plants (M2> N2 and M2< N2) to
compare output performance of the candidate controllers. In the first example, with different model mismatch at the measurement positions in the inner and outer circles, Table 4 Simulation results of the first MIMO plant (M2>N2) for Example 1
SSE (nm) of controlled outputs MIMO EWMA MIMO EWMA with 2D-PPC Improvement (%) MIMO PCC MIMO PCC with 2D-PCC Improvement (%) MIMO dEWMA MIMO dEWMA with 2D-PPC Improvement (%)
3.64E+07 3.59E+07 1.27 3.64E+07 3.31E+07 9.05 3.64E+07 3.59E+07 1.34
Controller parameters F2 NA 0.18 NA NA NA NA NA NA NA F2,s NA NA NA NA 0.99 NA NA 0.23 NA F2,d NA NA NA NA 0.45 NA NA 0.00 NA λ 0.48 0.51 NA NA NA NA NA NA NA λ1 NA NA NA 0.00 0.00 NA 0.48 0.5 NA λ2 NA NA NA 0.48 0.52 NA 0.00 0.01 NA
Table 5 Simulation results of the second MIMO plant (M2<N2) for Example 1
SSE (nm) of controlled outputs MIMO EWMA MIMO EWMA with 2D-PPC Improvement (%) MIMO PCC MIMO PCC with 2D-PCC Improvement (%) MIMO dEWMA MIMO dEWMA with 2D-PPC Improvement (%) 9.13E+07 6.75E+07 26.06 9.13E+07 6.16E+07 32.52 9.13E+07 6.49E+07 28.89 Controller parameters F2 NA 0.19 NA NA NA NA NA NA NA F2,s NA NA NA NA 0.76 NA NA 0.63 NA F2,d NA NA NA NA 0.65 NA NA 0 NA λ 0.45 0.41 NA NA NA NA NA NA NA λ1 NA NA NA 0.45 0 NA 0.45 0.37 NA λ2 NA NA NA 0 0.34 NA 0 0.01 NA
one investigates the effect of the model mismatch on the performance. In the second example, an additional im-pulse disturbance at run 10 of measurement position 6 is added to investigate the effect of disturbance on the performance. The process target T was 4,500 nm and the settings of two MIMO plants are listed at Table 3.
This section compares the performances of MIMO EWMA, MIMO dEWMA, and MIMO PCC with and with-out 2D-PPC for the two types of MIMO plants. In MIMO
dEWMA and MIMO PCC with 2D-PPC, Λ10λ1I and
Λ20λ2I are discount factors of the shift and drift terms,
respectively and F2,sand F2,d are the two 2D propagation
parameters of the shift terms and the drift terms of a wafer independently. Finally, this study employs SSE to evaluate the output performance.
5.1 Performance comparison with different model mismatches within a wafer
The first simulation compares output performance of twelve candidate controllers with ξinner01.2 and ξouter01.5. The
controller parameters can be obtained from anthropogenic disturbance by Eq. (12). Then, the optimal control parameters are applied to the same simulated data. Con-troller parameters with output performance for the two MIMO plants are listed in Tables 4 and 5 in which simulation results are divided into three groups: (1) EWMA and EWMA with 2D-PPC, (2) dEWMA and dEWMA with 2D-PPC, and (3) PCC and PCC with 2D-PPC. It is observed that the output performances of STC for three groups are all better than those of tradi-tional MIMO time–effect controllers.
5.2 Performance comparison with an additional impulse disturbance within a wafer
To show the advantage of space–effect controller, this example adds an additional impulse disturbance 1,000 at run 10 of measurement position 6. The simulation assumes that model mismatches of the all measurement positions are 1.2. The optimal controller parameters is also obtained from Eq. (12). Controller parameters with Table 6 Simulation results of the first MIMO plant (M2>N2) for example 2
SSE (nm) of controlled outputs MIMO EWMA MIMO EWMA with 2D-PPC Improvement (%) MIMO PCC MIMO PCC with 2D-PCC Improvement (%) MIMO dEWMA MIMO dEWMA with 2D-PPC Improvement (%)
3.05E+07 3.01E+07 1.53 3.05E+07 2.81E+07 7.81 3.05E+07 3.01E+07 1.53 Controller parameters F2 NA 0.34 NA NA NA NA NA NA NA F2,s NA NA NA NA 0.03 NA NA 0.34 NA F2,d NA NA NA NA 0.99 NA NA 0.00 NA λ 0.27 0.31 NA NA NA NA NA NA NA λ1 NA NA NA 0.00 0.00 NA 0.27 0.31 NA λ2 NA NA NA 0.27 0.35 NA 0.00 0.00 NA
Table 7 Simulation results of the second MIMO plant (M2<N2) for example 2
SSE (nm) of controlled out-puts MIMO EWMA MIMO EWMA with 2D-PPC Improvement (%) MIMO PCC MIMO PCC with 2D-PCC Improvement (%) MIMO dEWMA MIMO dEWMA with 2D-PPC Improvement (%) 4.53E+ 07 4.01E+07 11.46 4.53E +07 3.73E+07 17.50 4.53E+ 07 4.01E+07 11.46 Controller parame-ters F2 NA 0.66 NA NA NA NA NA NA NA F2, s NA NA NA NA 0.83 NA NA 0.66 NA F2, d NA NA NA NA 0.93 NA NA 0 NA λ 0.45 0.35 NA NA NA NA NA NA NA λ1 NA NA NA 0.45 0.00 NA 0.45 0.35 NA λ2 NA NA NA 0.0 0.39 NA 0.00 0.00 NA
output performance for the two MIMO plants are listed in Tables 6 and 7. One observes that, when the wafer has an additional impulse disturbance, the STC improves the output performance over the MIMO time–effect controller in all three groups for the given plants.
6 Conclusions
In this paper, the 2D-PPC is developed to realize space– effect controller, and then it is interlaced with the MIMO time–effect controller to establish the framework of the STC. From implementation view point, since STC does not change the original code of time–effect controller, it can be easily implemented in the current process control loop by only adding an additional
space–effect controller to obtain new modified intercept for the subsequent use in the MIMO time–effect con-troller. The advantages we gain are that STC not only preserves the property of the MIMO time–effect con-troller but also increases the stable region and performs better as compared to time–effect controller.
Appendix A
To overcome the end effect of the transition function [22], we modify the propagation-out ratio at the frontier points from F2=ð2 F2Þ [22] to Γ. Without loss of generality, this study takes the shape of the two-dimensional pheromone basket is shown as Fig. 2 and R2(k,0) is the 12×1 matrix of 1s as an example. The
transition functions are
q2ðk; bm2; i þ 1Þ ¼ F2 3 ðr2ðk; bm2þ6; iÞ þ q2ðk; bm2þ6; iÞÞ; if m2¼ 1; 2; . . . ; 6 Γ r2ð ðk; bm26; iÞ þ q2ðk; bm26; iÞÞ þ F2 3 ðr2ðk; bm2þ1; iÞ þ q2ðk; bm2þ1; iÞÞ þF2 3 ðr2ðk; bm21; iÞ þ q2ðk; bm21; iÞÞ; if m2¼ 8; 9; . . . ; 11 Γ r2ð ðk; bm26; iÞ þ q2ðk; bm26; iÞÞ þ F2 3 ðr2ðk; bm2þ1; iÞ þ q2ðk; bm2þ1; iÞÞ þF2 3 ðr2ðk; b12; iÞ þ q2ðk; b12; iÞÞ; if m2¼ 7 Γ r2ð ðk; bm26; iÞ þ q2ðk; bm26; iÞÞ þ F2 3 ðr2ðk; b7; iÞ þ q2ðk; b7; iÞÞ þF2 3 ðr2ðk; bm21; iÞ þ q2ðk; bm21; iÞÞ; if m2¼ 12 8 > > > > > > > > > < > > > > > > > > > : ðA:1Þ s2ðk; bm2; i þ 1Þ ¼ s 2ðk; bm2; iÞ þ 1 Γð Þ r2ð ðk; bm2; iÞ þ q2ðk; bm2; iÞÞ; ifm2¼ 1; 2;:::6 s2ðk; bm2; iÞ þ 1 F2ð Þ r2ð ðk; bm2; iÞ þ q2ðk; bm2; iÞÞ; ifm2¼ 7; 8; :::; 12 ðA:2Þ
Next, Eq. (A.2) can be rewrote as
Q2ðk; i þ 1Þ S2ðk; i þ 1Þ ¼ T11 T12 T21 T22 Q2ð Þk;i S2ð Þk;i þ U11 U12 U21 U22 R2ð Þk;i R2ð Þk;i ; ðA:3Þ where Q2ð Þ ¼k;i q2ðk; b1; iÞ q2ðk; b2; iÞ .. . q2ðk; b12; iÞ 2 6 6 6 4 3 7 7 7 5; S2ð Þ ¼k;i s2ðk; b1; iÞ s2ðk; b2; iÞ .. . s2ðk; b12; iÞ 2 6 6 6 4 3 7 7 7 5; R2ð Þ ¼k;i r2ðk; b1; iÞ r2ðk; b2; iÞ .. . r2ðk; b12; iÞ 2 6 6 6 4 3 7 7 7 5;
T11 ¼ 0 0 0 0 0 0 F2 3 0 0 0 0 0 0 0 0 0 0 0 0 F2 3 0 0 0 0 0 0 0 0 0 0 0 0 F2 3 0 0 0 0 0 0 0 0 0 0 0 0 F2 3 0 0 0 0 0 0 0 0 0 0 0 0 F2 3 0 0 0 0 0 0 0 0 0 0 0 0 F2 3 Γ 0 0 0 0 0 0 F2 3 0 0 0 F 2 3 0 Γ 0 0 0 0 F2 3 0 F2 3 0 0 0 0 0 Γ 0 0 0 0 F2 3 0 F2 3 0 0 0 0 0 Γ 0 0 0 0 F2 3 0 F32 0 0 0 0 0 Γ 0 0 0 0 F2 3 0 F 2 3 0 0 0 0 0 Γ F2 3 0 0 0 F 2 3 0 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ; T12¼ 01212; T21 ¼ 1 Γ 0 0 0 0 0 0 0 0 0 0 0 0 1 Γ 0 0 0 0 0 0 0 0 0 0 0 0 1 Γ 0 0 0 0 0 0 0 0 0 0 0 0 1 Γ 0 0 0 0 0 0 0 0 0 0 0 0 1 Γ 0 0 0 0 0 0 0 0 0 0 0 0 1 Γ 0 0 0 0 0 0 0 0 0 0 0 0 1 F2 0 0 0 0 0 0 0 0 0 0 0 0 1 F2 0 0 0 0 0 0 0 0 0 0 0 1 F2 0 0 0 0 0 0 0 0 0 0 0 0 1 F2 0 0 0 0 0 0 0 0 0 0 0 0 1 F2 0 0 0 0 0 0 0 0 0 0 0 0 1 F2 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ;
T22¼ I1212; U11¼ T11; U12¼ 01212; U21¼ 01212; and U22¼ T21:
The final propagation result of V12×1(k) obtained from Eq.
(A.8) in the AppendixBmust be the 12×1 matrix of ones.
V121ðkÞ ¼ v1ðkÞ v2ðkÞ v3ðkÞ v4ðkÞ v5ðkÞ v6ðkÞ v7ðkÞ v8ðkÞ v9ðkÞ v10ðkÞ v11ðkÞ v12ðkÞ 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ¼ 1 2F2þ ΓF2 3 1 Γ ð Þ F2ð 3Þ 1 Γ ð Þ F2ð 3Þ 1 Γ ð Þ F2ð 3Þ 1 Γ ð Þ F2ð 3Þ 1 Γ ð Þ F2ð 3Þ 1 Γ ð Þ F2ð 3Þ 3 1ð þ ΓÞ F2ð 1Þ 3 1ð þ ΓÞ F2ð 1Þ 3 1ð þ ΓÞ F2ð 1Þ 3 1ð þ ΓÞ F2ð 1Þ 3 1ð þ ΓÞ F2ð 1Þ 3 1ð þ ΓÞ F2ð 1Þ 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ¼ 1 1 1 1 1 1 1 1 1 1 1 1 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ðA:4Þ
Solving Eq. (A.4) yields,
Γ ¼ F2
3 2F2 ðA:5Þ
Relatively, when R2(k, 0) is the 12×1 matrix of ones, Eq.
(A.8) becomes lim z!1ðz 1ÞZ H2f ðk; iÞg ¼ lim z!1ðz 1Þ zI Tð Þ 1 UI241¼ 0121 1121 : ðA:6Þ
Equation (A.6) shows that the final propagation result S2(k, ∞) is also a 12×1 matrix of ones. Thus, the modified
transition functions not only obey the energy balance law but also avoid the end effect.
Appendix B
Because states Q2(k, i) and S2(k, i) are updated
simulta-neously, the transition functions can be rewritten in matrix form: Q2ðk; i þ 1Þ S2ðk; i þ 1Þ ¼ T11 T12 T21 T22 Q2ð Þk;i S2ð Þk;i þ U11 U12 U21 U22 R2ð Þk;i R2ð Þk;i ; ðA:7Þ where Q2ð Þ ¼k;i q2ðk; b1; iÞ q2ðk; b2; iÞ .. . q2ðk; b12; iÞ 2 6 6 6 4 3 7 7 7 5; S2ð Þ ¼k;i s2ðk; b1; iÞ s2ðk; b2; iÞ .. . s2ðk; b12; iÞ 2 6 6 6 4 3 7 7 7 5; R2ð Þ ¼k;i r2ðk; b1; iÞ r2ðk; b2; iÞ .. . r2ðk; b12; iÞ 2 6 6 6 4 3 7 7 7 5; T11 ¼ 0 0 0 0 0 0 F2 3 0 0 0 0 0 0 0 0 0 0 0 0 F2 3 0 0 0 0 0 0 0 0 0 0 0 0 F2 3 0 0 0 0 0 0 0 0 0 0 0 0 F2 3 0 0 0 0 0 0 0 0 0 0 0 0 F2 3 0 0 0 0 0 0 0 0 0 0 0 0 F2 3 F2 32F2 0 0 0 0 0 0 F2 3 0 0 0 F32 0 F2 32F2 0 0 0 0 F2 3 0 F32 0 0 0 0 0 F2 32F2 0 0 0 0 F2 3 0 F2 3 0 0 0 0 0 F2 32F2 0 0 0 0 F2 3 0 F 2 3 0 0 0 0 0 F2 32F2 0 0 0 0 F2 3 0 F 2 3 0 0 0 0 0 F2 32F2 F2 3 0 0 0 F32 0 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ; T12¼ 01212;
T21 ¼ 3 1Fð 2Þ 32F2 0 0 0 0 0 0 0 0 0 0 0 0 3 1ðF2Þ 32F2 0 0 0 0 0 0 0 0 0 0 0 0 3 1Fð 2Þ 32F2 0 0 0 0 0 0 0 0 0 0 0 0 3 1Fð 2Þ 32F2 0 0 0 0 0 0 0 0 0 0 0 0 3 1Fð 2Þ 32F2 0 0 0 0 0 0 0 0 0 0 0 0 3 1ð F2Þ 32F2 0 0 0 0 0 0 0 0 0 0 0 0 1 F2 0 0 0 0 0 0 0 0 0 0 0 0 1 F2 0 0 0 0 0 0 0 0 0 0 0 0 1 F2 0 0 0 0 0 0 0 0 0 0 0 0 1 F2 0 0 0 0 0 0 0 0 0 0 0 0 1 F2 0 0 0 0 0 0 0 0 0 0 0 0 1 F2 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ;
T22¼ I1212; U11 ¼ T11; U12¼ 01212; U21¼ 01212and U22 ¼ T21:
In Eq. (A.7), Q2(k, ∞) and S2(k, ∞) can be obtained with
the z-transform and the final value theorem H2ðk; 1Þ ¼ lim z!1ðz 1ÞZ Hf 2ðk; iÞg ¼ lim z!1ðz 1Þ zI Tð Þ 1U R2ðk; iÞ R2ðk; iÞ ¼ 0121 V121ðkÞ : ðA:8Þ where H2ðk; iÞ ¼ QS22ðk; iÞ k; i ð Þ , T¼ T11 T12 T21 T22 , U¼ U11 U12 U21 U22 and V121ðkÞ ¼ v1ðkÞ v12ðkÞ½ T . Note Q2(k, ∞) in Eq. (A.8) will converge to the matrix of 0 and
S2(k, ∞) will converge to V12 × 1(k). In addition, the
z-transform of the external input Z{R2(k, i)} is equal to
R2(k, 0) since R2(k, i) is an impulse at i00 by definition in
Section2.1. Thus, V12×1(k) is a function of F2and R2(k, 0)
for a specific M2. The final propagation results can be
obtained analytically using Eq. (A.8).
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