The theory of sliding-mode control (SMC) has been developed for a long time and has been known as an effective control method for many advantages. One of the most unique aspects of sliding mode is the discontinuous nature of the control action whose primary function of each of the feedback channels is to switch between two distinctively different system structures (or components) such that a new type of system motion, called sliding mode, exists in a manifold. This intriguing system characteristic is claimed to result in superb system performance which includes insensitivity to parameter variables, and complete rejection of disturbances [35].
However, a serious problem is also generated due to this characteristic, namely, chattering. Fortunately, many methods were proposed to reduce the chattering phenomenon. In this paper, the sliding-mode theory is employed to operate CMP process, i.e. to set the removal rate during a single wafer (within wafer) polishing. In this chapter, VSC and SMC will be introduced briefly and the presentation of controller design for the conceptual CMP model will be followed by some simulation results and discussions.
3.1 Variable Structure Control
Variable structure control (VSC) with sliding mode control (SMC) was first proposed and elaborated in the early 1950’s in the Soviet Union by Emelyanov and several co-researchers. This method did not catch researchers’ attention widely until the publication of the survey paper in 1977. Significant interest on VSC and SMC
has been generated in the control community worldwide [35]. To illustrate the advantage of VSC, a concise and acceptable example is generally used, that is, linear second order system [37]. Consider the second order system
u, exceeding zero (not across t-axis). Traditionally, the design is to choose a fixed value ‘a’ for acceptable response. According to VSC theory, the best performance will be derived easily by switching the system between two subspaces. Fig. 3.1.1 is all possible situations in traditional design and Fig. 3.1.2 shows the system response via VSC design. The system which is shown in Fig. 3.1.2 can also be called variable structure system (VSS). The VSS is defined as followed: one system which consists of more than two subsystems (a set of subsystems) together with suitable switching logic. In the example above, the value ‘a’ which is equal to zero or larger than two form two subspaces. Besides, the switching logic/condition is magnitude of ‘x’, ‘a’
is zero or two while ‘x’ is larger or smaller than ε, respectively.
To date, VSC has developed into a general design method being examined for a wide spectrum of system types including nonlinear system, multi-input/multi-output systems, and stochastic systems. In addition, the objectives of VSC have been greatly extended from stabilization to other control functions. The most distinguished feature of VSC is its ability to result in very robust control systems, in many cases invariant control systems result. Conceptually, the term “invariant”
means that the system is completely insensitive to parametric uncertainty and external disturbances [36]. Today, research and development continue to apply VSC to a wide variety of engineering systems.
Recently, the VSC and SMC are almost synonymous nouns, and the key difference between VSC and SMC is the existence of sliding-mode in the system.
From the viewpoint of mathematics, sliding-mode is a system behavior in
“hyperspace.” (The main feature is described in the beginning of this chapter.) It should be mentioned that the system performance will not be influenced by any matched disturbances5, and the control strategy provides an effective and robust means for certain classes of nonlinear systems subject to modeling uncertainties in the sliding mode. In next section, more detailed descriptions of SMC will be introduced.
3.2 Sliding Mode Control
SMC is a robust nonlinear control algorithm that employs discontinuous control to enforce a system state trajectory on some prescribed sliding surface. The SMC includes several different continuous functions that map plant state to a control surface, and the switching among different functions is determined by plant state which is represented by a switching function. In general, a variable structure system can be described by
( ) (
x u x)
f
x = , (3.2)
where x R∈ n is the state variables of the system, u
( )
x is the control input, and the timing of switching is governed by x (a specific form of combination of state variables, namely, switching function). In addition, the equilibrium point of the system is supposed to be x=0, which is the system design target (the control goal).For a single input system, the switching condition of control input is represented by
5 Matched disturbances mean that the disturbances which can be affected by system control input directly.
( ) ( ) ( )
s collocate the switching condition, and divides the system state space into three sub-spaces: s
( )
x >0, s( )
x =0, and s( )
x <0. It is important that the hyperspace,( )
x =0s , must be continuous and contain the equilibrium point x=0, i.e. system in this specified hyperspace must be stabilizable. The main objective of the system design is to generate a sliding mode in the hyperspace s
( )
x =0. Note that s( )
x is also called sliding function, and s( )
x =0 is the so-called sliding-surface6.The most important thing in SMC design is to generate (specify) a sliding-mode (sliding-function). There are two procedures in sliding-mode generation: (1) When system is outside of hyperspace s
( )
x =0, it must be ensured that the trajectory of system states would reach the hyperspace in a finite time th; (2) After the trajectory getting into the hyperspace, the trajectory will never depart from the hyperspace and move toward the equilibrium point x=0. The system behavior of first procedure is called approaching-mode, and the second one is named sliding-mode. Fig. 3.2 illustrates this generation of sliding-mode. In order to ensure the procedures mentioned above being workable, one condition must be satisfied, that iss σ s
s<− (3.4)
where s≠0 , and σ is a positive constant which can guarantee the system trajectories to hit the sliding surface in a finite time. Note that (3.4) is also called
“reaching/approaching and sliding condition.” Further looking into the behavior of system in the sliding mode, if the switching condition (3.3) can be realized perfectly,
6
i.e. the infinite frequency of switching exists, the ideal sliding mode exists, too.
Unfortunately, the system in real world can not provide such a high-frequency switching. To display the situation of system in the sliding mode schematically, a very small layer7 covering the sliding surface is often employed, as shown in Fig. 3.3.
To design a controller via SMC, the procedure can be concluded into two steps8, that is: (1) Specify a sliding function s
( )
x which will let the system to move toward control goal while the system is in the sliding mode; (2) Determine the control input u which will force the system to hit the sliding surface and generate the sliding mode in a finite time. For a simple single-input system in controllable canonical form( )
t, the disturbance of system. There are several ways to choose sliding function, for the simple case, the easiest way is setting sliding function as( )
x =cxs (3.6)
where c=
[
c1 c2 . . . cn]
. Through some calculation, the value of ‘c’ will be derived, and then the sliding function is obtained. At the second step, based on the important condition (3.4), the control input can be determined as
7 In SMC design, the setting of layer namely sliding-layer is an important and effective way to reduce so-called chattering phenomenon.
8 Note that the two steps are concluded based on the procedures of generating sliding-mode mentioned above.
(
c a)
x ...(
c a)
x(
δ( )
t, σ)
sign( )
s xa
u=− 1 1− 1+ 2 2− − n−1+ n n− x + (3.7) where δ
( )
x t, is the maximum value of disturbance d x( )
t, and σ is the value to be designed. Note that the value of δ is determined by estimating or deciding among the measured data. The control input (3.7) contains an ideal switching function sign( )
s , which can not be realized in real world. Besides, the ideal switching function sign( )
s is the main reason of existence of chattering phenomenon.The easiest and effective way to reduce chattering is to replace sign
( )
s by saturation functionwhere ε is the thickness of sliding-layer mentioned above, and the system trajectory will be limited in s <ε. It should be mentioned that employing of sliding-layer would reduce the chattering but it can induce the loss of precision of the controller.
This controller is actually a continuous approximation of the ideal relay control.
The consequence of this control scheme is that invariance property of SMC is lost.
The system robustness is a function of the width of the boundary/sliding layer.
Robustness is decreased as width is increased. Next, dynamic tuning of CMP process based on SMC is presented.
3.3 Operation Profile Setting
Before designing the setting of removal rate of CMP process, some preliminary research results and statement will be recounted first.
3.3.1 Problem Statement
In copper CMP, copper dishing and oxide erosion are critical problems and have to be solved urgently due to the serious influence on RC constant and surface uniformity, especially the copper dishing. (see section 2.2.2.)
From the research by K. Wijekoon et al. [23], the operation parameters (applied downward pressure, P, and relative speed between wafer carrier and platen table, V) have direct influence on copper dishing and oxide erosion, as shown in Fig. 3.4.
Furthermore, in the investigations by Y.-C. Kao et al. [38], a robust operation region is located on the base of copper removal rate curve. Note that the robust operation region means low-sensitivity to parameters change. In their works, some operation-relevant models and an overall strategy of polishing are established, and the strategy setting is subject to the copper dishing (Note that the strategy, soft landing, is related to the work by J.-B. Chiu et al. [24]. The conceptual description of the strategy is shown in Fig. 3.5). Based on these results, better operation profile of the process might be a candidate to improve CMP process, especially in reducing the copper dishing. In other words, the copper dishing might be suppressed by accurately tuning the process parameters during a single wafer polishing.
Furthermore, the works by V. Nguyen et al. [14] [32], G. Fu and A. Chandra [33] and Y. Lin [42] reveal that the copper dishing is a function of over-polish time and process parameters. Consequently, the feasibility of this idea is greatly strengthened. In detail, it is clear that the process parameters, like P and V, play important roles in the forming of dishing besides the feature size, polish time and properties of pad, in reference to equation (2.9). Equation (2.12) reveals that dishing will be also suppressed directly during overpolish stage if the blanket wafer removal rate drops down. From many investigations, the removal rate in CMP process can be related to
process parameters including the parameters which can be operated in process.
One of methods of reducing critical problems occurred in CMP process is to optimize the recipe of wafer processing through setting of these process parameters.
In this thesis, the other way is considered to improve the performance of CMP process, that is, dynamic tuning of the operation parameters during CMP process run in an appropriate form/scheme.
Generally speaking, to maintain high throughput high removal rate recipe has to be used, but that can cause more damages in most cases. On the other hand, the recipe of low removal rate often cause better surface profile but that will make low throughput, too. Obviously, to achieve balance between high throughput and good surface profile is an important topic. In the discussion in section 2.3, very few works have been done via the strategies of control theory except the work by J.-B. Chiu et al.
[24].
In the works of J.-B. Chiu et al. [24], the control law of time optimal control, is employed, and an analogy between the soft landing of a spacecraft and the CMP process is illustrated for CMP operation. The purposes of citing this analogy is for reducing the copper dishing and handling the uncertainties, the undetermined initial thickness of copper (measurement uncertainties), and the bang-bang control law is used to implement this idea. Fig. 3.6 shows their simulation results. Note that the timing of stopping the process is determined by the endpoint detection. From the first plot of Fig. 3.6, it is clear that the removal rate has to increase to 9000 A/min at once while the process is starting, and drop to 2000 A/min while the process is going for 40 seconds. There are two reasons of setting second step (based on the concept of soft landing), one is for better surface profile and the other is for handling uncertainties. Looking into the ability of uncertainties handling, using lower removal rate can be a “conservative” method. In general, the over-polishing is
necessary to remove the barrier layer (in copper CMP) and make sure that copper residue is cleared entirely. However, many damages may occur if the over-polishing is not controlled well. For this reason, to reduce the damages which might occur at the final stage of the process, lower removal rate recipe has to be employed.
Additionally, it is known that this strategy is employed in practical production, too.
Even though this method seems to be pretty good, two questions remains: (1) How small is the value of second step (after 40 seconds) will be enough? (2) Do the unstable/discontinuous situations, like the period of rising and dropping the removal rate immediately, will cause any undesired damages?(e.g. vibration of platen or turbulence of slurry) To avoid these two problems, a more continuous/smooth operation strategy will be established via SMC algorithm. Furthermore, the mechanism of operation profile will be proposed to not only solve the above two problems but derive better planarization (dishing and step-height reduction).
Note that it should be not sufficient to improve the performance by adjusting mechanical loads (P and V) only. From the investigations on CMP process mentioned in section 1.2, the chemical loads should be important factors, like PH value, etcher, inhibitor and oxidizer. In this thesis, only the oxidizer (H2O2) concentration will stand for chemical load, i.e. the oxidizer concentration is the only chemical factor which is discussed in this thesis. (see chapter 4)
3.3.2 Operation Profile Setting
There are two steps in SMC design: first, a stable sliding mode/surface/function has to be specified, and the system will move along sliding surface toward equilibrium point (control goal); next, a control input which will force the system to hit sliding surface in a finite time has to be derived. From Chen and Chang [37],
there are three methods which are generally used in designing sliding function, Transformation Matrix method, Eigenstructure Assignment method and Lyapunov-based method. Similarly, in establishing control law, there are Hierarchical Sliding-mode Control and Integral Sliding-mode Control and so on. In this work, a convenient and powerful method which is also proposed by Chen and Chang [40], namely, Virtual Eigenvalue method is used in controller design.
In this conceptual model of CMP shown in (2.15), just one sliding-mode eigenvalue and one virtual eigenvalue instead of two set of them have to be selected because of the two by two system matrix. After choosing eigenvalues first, according to linear control theory, a state feedback control input by pole-placement method can be obtained as
−kx
=
u (3.9)
where k is the value to be designed. Note that this method can assign the eigenvalues of A−bk to any desired value (Of course, it is always negative for stabilizing systems). Then, A−bk can be diagonized as
where J and Ω represents sliding-mode eigenvalue and virtual eigenvalue, respectively. Consequently, the left eigenvector corresponding to the virtual eigenvalue has to be determined
Ω, respectively. Up to present, the sliding function is derived
=cx
s (3.12)
where c is the left eigenvector corresponding to the virtual eigenvalue Ω in Eq. (3.14).
To establish the control input, let the control input be u
u= kx− + ′ (3.13)
where u’ is the term with switching function. Based on the concept of approaching-mode mentioned before, differentiate Eq. (3.12) and u’ will be derived, that is
(
δ( )
t, σ) ( )
sat( )
s,εu′=− c x + cb −1 (3.14)
where δ
( )
x t, represents the maximum value of disturbance d( )
x t, and sat( )
s,εis saturation function defined in Eq. (3.8).
Stability
Because the design of SMC is to force the system to hit the specified sliding-mode in a finite time and make it stay there forever, to analyze the stability of the system under SMC control, only the part of system which has entered the sliding mode needs to be considered. In other words, the stability of the system is equal to the stability of the system in specified sliding-mode.
To analyze the stability of the sliding-mode control, an important concept has to be employed, that is, equivalent-control [37]. It is known that the trajectory of the system is continuous even in the sliding-mode. However, the continuous trajectory in the sliding-mode control is made by a discontinous switching condition. The relationship between the continuous trajectory and the discontinuous switching condition is proposed by Filippov in 1988, that is
(
,u)
= f(
,u+)
+(
−)
f(
,u−)
f x eq µ x 1 µ x (3.15)
where the meaning of u+ and u- is the same to (3.3). Fig. 3.7 shows a brief but comprehensive explanation. Then, the stability of the sliding-mode can be proved through the concept of equivalent-control. The detailed proof is shown in appendix
Ⅰ.
3.3.3 Results
Some simulations are implemented by MATLAB software package. These simulation results are the preliminary demonstration of feasibility of the controller.
Analysis
In SMC design, the first thing which has to be taken care is the convergence to the sliding surface as shown in Fig. 3.8. In Fig. 3.8, it is clear that the system reach the sliding-surface (s=0) in a finite time, about 0.6 minutes. This result represents that the design is successful, i.e. the control law (3.14) works. Additionally, to visualize whole performance of the system, phase-plane analysis is a powerful tool.(Note that the method is restricted to second-order or first-order systems.) Fig.
3.9 shows the phase-plane plot of the system, and it reveals that the thickness (horizontal axis) decreases continuously and the variation of material removal rate (vertical axis) is smooth. Besides, Fig. 3.10 shows the plot of control input (3.13) and it also varies smoothly. The last one is Fig. 3.11, and it shows time history of removed copper thickness and (copper-) material removal rate during CMP process separately. In Fig. 3.11, the more detailed profile of these two important parameters are presented. The CMP process will finish in about 1.7 minutes (102 seconds), and
the removal rate will decrease to zero at about the same time.
Discussions
To summarize entire process of CMP under SMC control: Initially, the system starts from an idle situation, the platen speed and the carrier speed increase loading separately , the down pressure is increased gradually, the temperature and the oxidizer concentration of slurry is increased proportionably. After about 0.2 minutes (12 seconds), all parameters tend to decrease their value to lower the removal rate. Note that the rate of change is lower than the period of first stage (increasing removal-rate).
The process keeps on going until about 1.7 minutes, and then the platen and the carrier rotation speed go down to a lower value; so do the down pressure and the chemistry of slurry. Finally, all parameters decrease to very low values, and the CMP process is accomplished at the same time.
Certainly, the mechanism mentioned above is just a concept, but the description does contain some possible ways to achieve/realize the mechanism of “dynamic tuning” of operation, i.e. tuning platen and carrier rotation speed, varying down pressure, changing the temperature and oxidizer concentration of slurry and so on.
Besides, the distinguishing feature of operation by dynamic tuning comes out from the description, that is, the variations of parameters are all continuous and smooth. In the operation via SMC design, not only the violent switching is avoided but the lower loads of parameters are employed. It will make this method much more applicable
Besides, the distinguishing feature of operation by dynamic tuning comes out from the description, that is, the variations of parameters are all continuous and smooth. In the operation via SMC design, not only the violent switching is avoided but the lower loads of parameters are employed. It will make this method much more applicable