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On the interaction between measurement strategy and

control performance in semiconductor manufacturing

An-Jhih Su

a

, Cheng-Ching Yu

a,*

, Babatunde A. Ogunnaike

b

aDepartment of Chemical Engineering, National Taiwan University, Taipei 106-17, Taiwan bDepartment of Chemical Engineering, University of Delaware, Newark, DE 19716-3110, USA

Received 4 January 2007; accepted 11 July 2007

Abstract

Manufacturing in the high revenue semiconductor industry involves a highly capital intensive process consisting of more than 300 steps. To ensure stable process operation and ultimately meet the exacting requirements on final product quality, the typical advanced IC fabrication process requires many on-line sensors and off-line metrology tools for acquiring process and product information neces-sary for effective monitoring and control. However, the high cost associated with these measurement devices has made the economics of metrology a major factor in the industry’s quest for world-class manufacturing. In this paper, we first introduce various measurement data types and describe how they feature within an ideal fab-wide control architecture; subsequently, and from a process control point of view, we derive various run-to-run controllers, carry out stability analyses, and analyze control system performance. These results are then applied to the problem of rational metrology strategy selection where the effects of various metrology strategies on control system performance are systematically analyzed. In particular, if control performance takes priority over economics, we present results for deter-mining maximum tolerable sampling intervals, maximum tolerable delay, and measurement priority.

 2007 Elsevier Ltd. All rights reserved.

Keywords: Semiconductor manufacturing; Measurement strategy; Controller design; Run-to-run control

1. Introduction

Semiconductor manufacturing is a highly capital inten-sive venture with high revenue, where maintaining compet-itiveness demands high equipment efficiency, tight quality control, and a relentless and continuous reduction of mate-rial cost. A key factor in achieving these goals is the mea-surement system, the set of sensors and analyzers needed to acquire critical measurements and information about the process and product. Such information, if processed appropriately can be used to create knowledge, increase equipment efficiency and accelerate yield improvement, ultimately generating increased profit.

1.1. Measurements classification

To construct a device on a wafer, the process flow involves several cycles of lithography, etch, chemical vapor deposition (CVD), chemical mechanical polishing (CMP), etc. as shown schematically in Fig. 1 [19,17]; and many modern microelectronic fabrication plants (‘‘fabs’’ for short) are equipped with world-class information technol-ogy (IT) infrastructure for collecting and storing lots of pro-cess and product information. Such data is typically used for fault diagnosis and classification, or to provide informa-tion as feedback in run-to-run control to maximize opera-tion efficiency of the equipment. The measurement data collected in the typical fab may be classified as follows:

(1) Real-time equipment data: typically from an in situ sen-sor, and used as a feedback signal for on-line control (for example, on-line temperature measurements can

0959-1524/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2007.07.005

*

Corresponding author. Tel.: +886 2 3366 3037; fax: +886 2 2362 3040. E-mail address:ccyu@ntu.edu.tw(C.-C. Yu).

www.elsevier.com/locate/jprocont Journal of Process Control 18 (2008) 266–276

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be used for set point tracking for a given temperature program in a CVD tool). They also can be used to mon-itor the condition of the equipment, and to build a ‘‘health index’’ to guide decisions on the preventive maintenance. For instance, measurements of chamber

pressure can be monitored to check for the existence of a leak and thus determine when maintenance is needed. Typical sampling time ranges from seconds to millisec-onds[18].

(2) Geometric properties data: measured by metrology tools. A wide variety of semiconductor manufactur-ing equipment is used to fabricate wafers with desired geometric properties, such as layer thickness and trench structures. The metrology tools used to deter-mine such properties can be divided into two catego-ries: (i) Integrated metrology tools: these are combined with manufacturing equipment and can be configured to measure the quality of every one, or one out of several wafers, (e.g., one out of 4–6 wafers). Such quality information can be used for feedback control on a wafer-to-wafer basis. Typical sampling time for feedback control ranges from 5 to 10 min. (ii) Stand-alone metrology tools: these measure the quality of several wafers in a lot, batch-wise, and therefore can only be used for lot-to-lot feedback control. The lots of wafers are typically delivered to the cluster of stand-alone metrology tools to wait in a queue for measurement (Fig. 1). There will obviously be time delays associated with such measurements. Typical sampling time for lot-to-lot control ranges from hours to one day[4,6]. (3) Wafer acceptance tests (WAT) data: provide

infor-mation on many important electric properties that indicate whether or not the device will function cor-rectly after the completion of a metal layer or a struc-ture (Fig. 1). Because many steps are required to complete the construction of a structure, WAT data are available for feedback only after a long time delay. Typical sampling time ranges from days to a week[14,16,20].

Fig. 1. Multilayer configuration leading to repetitive characteristics in manufacturing. It uses the same module to execute the same function with the metrology tool(s) shared by same module; the WAT test is performed at the completion of each layer; yield is determined toward the end of the entire process.

Notations

CMP chemical mechanical polishing Deff effective delay

dt disturbance sequence in production domain dt disturbance sequence in sampled-run domain GC controller transfer function

GP process transfer function

j imaginary root

IMC internal model control

IMA integrated-moving-average time series

Nm measurement delay

Ns sampling interval

KP process gain

b

KP estimated process gain

KF forward loop gain

KF,opt optimal value of KF

k ramp slope

PI2 proportional-plus-double-integral controller q1 discrete domain (backward operator) S sensitivity function

s Laplace domain

yt quality sequence in production domain y

t quality sequence in sampled-run domain

at white noise sequence in production domain at white noise sequence in sampled-run domain sf time constant of IMC filter

h IMA coefficient

h* IMA coefficient in sampled domain with Ns

sampling interval r2 sequence variance

n process gain mismatch (n¼ KP= bKPÞ

x frequency

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(4) Yield data: measure the percentage of acceptable dies in a wafer, a quality measurement directly related to revenue. Note that a wafer contains multiple useable products, called dies, and the number of dies is deter-mined by the wafer diameter, e.g., 8 and 12 in. Yield data are usually not available until almost one month into production[20].

Fig. 2 shows a schematic depiction of the time-scale characteristics and measurement complexity of the various categories of measurement data. The observed yield depends on WAT results which in turn depend on many geometric properties; each geometric property is itself determined by the recipes used in each manufacturing equipment.Fig. 2also indicates the strong structural rela-tionship between these variables. Consequently, the desired WAT properties are used to design the target of metrology for equipment run-to-run (R2R) control, and this desired metrology target is in turn used to design tool recipes in what is known in the terminology of process control as a cascade control structure. The R2R control loop is designed to compensate for variability in the equipment while the WAT feedback loop sets the correct metrology target for the lower level controller. Both control loops suf-fer from the same problem: the measurements have signifi-cant time delays, causing signifisignifi-cant deterioration in the closed-loop stability characteristics and ultimately the con-troller performance. To address this problem, it is neces-sary to augment available, but delayed measurements with ‘‘virtual metrology’’ and ‘‘virtual WAT’’ [14], as shown inFig. 3. By virtual metrology, we mean the estima-tion of yet unavailable metrology data from available lower level data, such as temperature, pressure, batch time, etc. Thus the ‘‘virtual metrology’’ unit serves as a soft sensor, providing estimated measurements that can be used for inferential control [23]. Similarly, ‘‘virtual WAT’’ implies the estimation of electrical properties from available metrology data. Including the measurement types and con-trol mechanisms, such an ideal fab-wide concon-trol structure is

shown inFig. 4. A similar structure has also been proposed in [4], with the crucial difference that the structure in [4] contains no measurement delay compensation mechanisms comparable to the one we are proposing here.

1.2. Control

The R2R control strategy remains popular in industrial practice and has, for years, attracted a lot of research atten-tion. Del Castillo[2]summarizes results on the design and performance of the most popular R2R controllers: EWMA and double-EWMA. Chen et al.[15] discuss the effects of sequencing on incoming wafers for run-to-run control; and Qin and Good [8] analyze the stability of double-EMWA controllers in the presence of metrology delay, and extended the results to MIMO controllers [9]. Qin [26]also discusses the fab-wide control structure for electri-cal parameters.

Our current work is motivated by the following ques-tions: (i) what is the effective metrology delay in discrete systems; and (ii) how does the measurement strategy (e.g., sampling interval) influence stability and control per-formance? Some of these issues have been partially con-fronted in previous studies. For example, Tseng and Hsu [10]discuss the statistically appropriate number of samples Measurement Complexity Electric properity Yield Thickness Resistance CD Overlay Particles Metrology Temperature Pressure Flowrate Voltage Recipe WAT

10-3 sec sec min hr day week month

Fig. 2. Measurement complexity and sampling frequencies and data availability times at the recipe, metrology, and WAT levels.

10-3 sec sec min hr day week month

Virtual Metrology Set Measure Virtual WAT Yield WAT Recipe Metrology Set Measure Update Update

Fig. 3. Coordination between measurements at different time-scales.

Photo Etch CVD CMP

Control Control Control Control

VM M VM M VM M VM M WAT Virtual WAT FF FF FF FF FB FB FB FB MSet MSet MSet MSet

M:metrology, VM:virtual metrology,MSet:metrology setpoint

FB:feedback, FF:feedforward Keys:

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required to obtain an accurate model, and hence guarantee the stability of R2R control; Jula et al.[6]compare the eco-nomic impact of in situ, in-line, and off-line metrology sys-tems, while Nurani et al. [7] provide an integrated framework for designing an optimal defect sampling strat-egy for wafer inspection. Lensing [24] mentions dynamic sampling in metrology to measure just enough to charac-terize systematic sources of variance and correct them using wafer level control. Moyne [27] discusses financial return-on-investment (ROI) analysis of run-to-run control. However, none of these papers explicitly relate control per-formance issues to sampling strategies in a comprehensive manner.

1.3. Measurement strategies and controller performance In order to reduce cost, a common measurement option is to employ one stand-alone metrology tool that is shared by several pieces of equipment. However, this results in metrology delay for R2R control. Under these conditions, determining the maximum tolerable delay that will guaran-tee control stability and performance is clearly an impor-tant issue. Similarly, for equipment with an integrated metrology tool, the critical question is: how frequently should a wafer be sampled for feedback measurement in order to ensure that control performance and high throughput objectives are met?

In the fab, the measurement delay and sampling interval are typically not constant but change depending on many factors. Our goal in this paper is to identify the relationship between measurement delay or sampling interval and the control performance, and illustrate how these results can be used to determine appropriate measurement strategies.

In this study, we consider there are no product-relative terms, since high-mix product problem is very common nowadays [25]. Without these terms, it is more clearly to realize the influence of sampling. The rest of the paper is organized as follows: in Section2, we define the variables used in this paper, and derive an expression for the effective time delay in a R2R control system; we also discuss how to derive appropriate R2R controllers. In Section3, we ana-lyze control performance, explicitly dealing with the effects of sampling intervals and effective delay. The application of these results to the rational design of measurement strate-gies are presented in Section 4 where various concepts including maximum tolerable sampling intervals, tolerable delay, and measurement priority are introduced and dis-cussed. Conclusions follow in Section5.

2. Effective time delay in R2R control 2.1. Definition

As mentioned earlier, there are two types of basic strate-gies employed in equipment R2R control: lot-to-lot and wafer-to-wafer. The following analyses of stability and per-formance can be applied to either type. For simplicity, we

use the term ‘‘runs’’ to refer to wafers in the case of wafer-to-wafer control, and lots in the case of lot-to-lot control.

Sampling interval, Ns: the number of runs between sam-ples. For example, the specific value Nsmeans that after a measured run, the next run to be selected for measurement is the Nth

s run; alternatively, that one out of every Ns runs will be measured. One may increase the sampling interval (i.e., making fewer measurements per lot) to achieve higher throughput rate (no processing interruption from measure-ment delay); however, because the sampling interval has a significant effect on stability and control performance, it should be chosen judiciously. Therefore, it is important to determine the maximum tolerable sampling interval required to guarantee quality. Nsis an integer with a mini-mum value of 1.

Metrology delay, Nm: the number of runs between when the measurement is made and when the metrology data is available for feedback. The specific value Nm means that data from a measured run will be available at the next Nthm run. Because of this definition, the metrology delay should not only include the measurement time, but also the queue time and material transportation time. It should be noted that even if the measurement can be completed immediately, the result can only be applied to the next run for feedback control. Thus, Nm, also an integer, has a minimum value of 1. For some modules with several sequential processing steps (e.g., washing, baking, main process), it is possible to have several wafers held simulta-neously in the module; however, the manipulated variable (tool settings) can only be applied to the wafers at the entrance of the module (i.e., fresh wafers entering the mod-ule). In this case, the metrology delay is the difference in the number of runs between the measured wafer and the wafer at the entrance.

2.2. Determination of effective time delay

It is well-known that the presence and magnitude of a time delay are important determinants of the characteris-tics of a control loop. In general, if each run is sampled (so that Nm= 1), the delay in the control loop is equivalent to the metrology delay. However, when the sampling inter-val is greater than 1, the effective time delay in the control loop, Deff, is a function of sampling interval and metrology delay as follows Deff ¼ roundup Nm Ns   ; ð1Þ

where because this number must be an integer, the indi-cated function, roundup(.), means that the computed num-ber must be rounded up to the next higher integer. From the perspective of the sampled-run domain, Deff may be interpreted as meaning that the data available for use at the current sampling instant is from the previous Dth eff ‘‘measured-run’’.Fig. 5shows how the system representa-tion in the producrepresenta-tion domain is converted into the sam-pled-run domain for analysis; here, GC is the controller,

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GP the process, d the disturbance, y the metrology or quality data, and q is the standard discrete system back-shift operator. The ‘‘star’’ variables d* and y* refer to the sampled-run domain versions of the original variables d and y in the product domain (See Section 3 for more detail.).

It is more convenient to carry out closed-loop analyses in the sampled-run domain rather than in the original dis-crete-time domain; specifically, closed-loop stability and controller performance analyses are more easily carried out in the sampled-run domain. We use following two examples to illustrate the meaning of effective delay with constant or variable Nmand Ns.

Example 2.1. Consider the wafer-to-wafer control config-uration shown in Fig. 6a, with Ns= 3 (i.e., we sample every three wafers), and with a metrology delay that varies from 1 to 3 (1 6 Nm63). We obtain from Eq. (1) that Deff= 1 for all three values of Nm. In this case there is only a unit delay in the sampled-run domain control system.

Example 2.2. Consider another wafer-to-wafer control strategy, this time in a CMP equipment with an integrated metrology tool that has a delay of three wafers. To main-tain high throughput, a decision has been made that only six wafers in a lot of 25 wafers will be measured. A process engineer, believing that the process is less stable early in the lot, decides to take more samples during this period; specif-ically, the 2nd , 3rd , 5th , 10th , 20th and 25th wafers are chosen to be sampled. It is easier to analyze the effective delay via Fig. 6b, which shows clearly that in the sam-pled-run domain, the delay is 1 or 2. Because the delay is variable, one must tune the controller conservatively on the basis of the worst case. However, if we rearrange the sampling to be uniform, so that Ns= 4, by applying Eq. (1) the result will be an effective delay of only 1. From a GP GC yt +_ Ns m N qdt + + GP GC yt +_ eff D

q

dt* + + *

Fig. 5. The control system in: (a) product domain and (b) sampled-run domain. To sampled-run domain Nm=1 Nm=2 Nm=3 Ns=3

System delay :

q

-1

D

eff

=1

Wafers flow y* t y*t-1 y*t-2 y*t-3 Nm=1 Nm=2 Nm=3 Ns=3

:

y* t y*t-1 y*t-2 y*t-3 1 2 3 4 5 6 7 8 9 10

q

-1

q

-2

q

-2

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process control point of view, this latter system that has only a unit delay will have better control performance and enjoy a wider range of stable controller parameters than the previous control system.

2.3. Controller derivation and stability

In R2R control of semiconductor manufacturing, the most popular strategies are the EWMA controller, a pure integral controller[12] designed to deal with shift or step-like disturbances, and the double EWMA controller, a pro-portional-double-integral controller [11] designed to deal with drift or ramp disturbances. These control strategies have been discussed extensively elsewhere as off-the-shelf controllers available for use in semiconductor manufactur-ing; here we will discuss instead how to derive appropriate controllers with no preconceptions, and subsequently how to ensure robust stability.

On the premise that rational control system design should be based on sufficient knowledge about the process and the input signal types (setpoints to track and/or distur-bances to reject), one can use such information along with the internal model control (IMC) principle[13](or equiva-lently direct synthesis concepts[21]) to derive appropriate controllers required to achieve specified performance objectives. In particular, it is customary in semiconductor manufacturing to model the batch-to-batch operation as a pure gain process, i.e., GP= KP, with bGP¼ bKPas the best estimate of the true process gain. It is easy to show that according to the IMC principle (or via direct synthesis), the standard feedback controller for a pure gain process subject to shift disturbances (type-1), is

GCðsÞ ¼ 1 b KP

 1

sfs; ð2Þ

where sfis a tuning parameter that determines the speed of the desired closed-loop response. If sfis small, the closed-loop system is aggressive but less robust; conversely, the closed-loop system is more robust but also more sluggish for larger values of sf. In discrete form, Eq.(2) is

GCðq1Þ ¼ 1 b KP

 1

sfð1  q1Þ: ð3Þ

This is a pure integral controller, identical in form to the EWMA controller, confirming why EWMA controllers deal effectively with step-like disturbances.

Many pieces of process equipment in semiconductor manufacturing are subject to drift disturbances, such as the decay of polish pad efficiency in a CMP tool, or undesired deposition on side walls in a CVD tool. For sim-plicity, it is customary to idealize this degradation of effi-ciency as a ramp disturbance to a time-invariant process model instead of using a more complicated time-varying process model. The IMC design for dealing with such a ramp (type-2) disturbance results in a feedback controller of the form GCðsÞ ¼ 1 b KP 2sfþ 1 s2 fs2 ð4Þ or in discrete form GCðq1Þ ¼ 1 b KP 2 sf 2 sf  1 s2 f   q1 1 2q1þ q2 : ð5Þ

This is a proportional-double-integral controller (PI2) with the same form as a double EWMA controller but with only one tuning parameter, sf. For more compli-cated process models and disturbances, this same proce-dure can be used to derive R2R controllers appropriate to the known information, no matter how complicated. Note how the nature of the process model and the disturbance structure determine the appropriate control-ler, emphasizing the importance of understanding the process and disturbance structure before implementing feedback controllers. One should not just choose an existing controller arbitrarily without incorporating such knowledge.

We now consider the stability characteristics of Eqs.(3) and (5). FromFig. 5a, the closed-loop characteristic equa-tion is

1þ GCðq1Þ  GPðq1Þ  qDeff ¼ 0; ð6Þ

whose roots must all be located inside the unit circle in the complex plane for stability. In the following derivation, plant/model mismatch is allowed (because of the simplicity of the pure gain model) with KPas the true but unknown process gain, and bKP as the model estimate. In combina-tion with sf, the controller tuning constant, the static part of the forward transfer function is defined as

KF ¼ KP b KPsf

The characteristic equation can be solved analytically for the ultimate gain value required for the process to be on the verge of instability, using the frequency approach. Let q = ejx; the ultimate gain (KF,u) and ultimate frequency (xu) of the forward transfer function (GCGPqDeffÞ can be found by solving the following two equations when GCis chosen to be an EWMA controller

argðGCGPqDeffÞx¼xu¼  tan1

sin xu 1 cos xu    Deffxu¼ p; ð7Þ GCGPqDeff    x¼xu¼ KF;u 2 sinðxu=2Þ ¼ 1; ð8Þ

where x is frequency. After some algebraic manipulation, the ultimate frequency is obtained as: xu= p/(2Deff 1), which, when substituted into Eq.(8), yields

KF;u ¼ 2 sin p 2 1 2Deff 1   : ð9Þ

Fig. 7shows the stable regions of the EWMA and double EWMA controller parameter space. The region above each curve is the stable region for the indicated value of n, the multiplicative model uncertainty parameter defined by:

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n¼ KP= bKP. Note that larger Deffand n values require lar-ger sfvalues (slower desired closed-loop responses) to en-sure stability. The implication is that for large effective delays and/or significant model uncertainty, closed-loop stability can only be achieved at the expense of closed-loop performance, which is in perfect keeping with practical experience (and robust control theory).

3. Control performance 3.1. Disturbance

In this study, we suppose that the process disturbance can be modeled as an integrated-moving-average (IMA) time series with a deterministic drift; i.e.

dt dt1 ¼ at h  at1þ k: ð10Þ

Here dtis the disturbance time series, t is the discrete time index, atis a gaussian white noise sequence with zero mean and variance r2

a; h is the moving average coefficient with 0 6 h 6 1; k is the slope of the drift. Many industrial pro-cess disturbance data are known to be well-represented by this model [22]. In transfer function form, Eq.(10)is

dtðq1Þ ¼1 hq1 1 q1 atþ

k

1 q1: ð11Þ

This disturbance can be treated as a combination of two signals: an IMA series and a ramp, as shown inFig. 8a. When h takes on the extreme value 0 or 1, the IMA series becomes, respectively, a random walk or a white noise process.

3.2. Effect of sampling intervals on disturbance model In Section2.2, we discussed the effect of sampling inter-vals on the effective delay in a control structure (Eq.(1)). Even though changing the sampling interval will not affect the nature of the disturbance, it is necessary to redefine the variables in the sampled-run domain (Box et. al.[1]). The relationship between the parameters in the original series (h and r2

aÞ and in the sampled series (h*and r 2 aÞ is: Nsð1  hÞ2 h ¼ ð1  hÞ2 h ; ð12Þ r2 a r2 a ¼ h h; ð13Þ

where Nsis the sampling interval r2

a and h*are the param-eters of the new sampled-run sequence as shown inFig. 8b dt  d t1¼ at  h  a t1þ Ns k; ð14Þ where a t is Nð0; r2aÞ.

Having now introduced the process, the controller, the disturbance as well as the concept of the effective delay, we are now in a position to consider the issue of controller performance.

3.3. Achievable performance

Under the condition that the setpoint is unchanged, the disturbance is the only input signal to the control system, so that the relationship between process output y*and dis-turbance d*inFig. 5a is yt ¼ 1 1þ GC GP qDeff  d  t: ð15Þ 1 2 3 4 5 6 7 8 0 2 4 6 8 10 τf Deff ξ=0.5 ξ=1 ξ=2 stable unstable 1 2 3 4 5 6 7 8 0 5 10 15 20 Deff τ f ξ=2 ξ=1 ξ=0.5 stable unstable

Fig. 7. Stable regions (above of each curve) for different nðKP= bKPÞ values

(a) for the EWMA controller and (b) for the double-EWMA controller.

1 1 1 1 q q θ − − − − 1 1 1 q− − 1 h − B * 1 1 1 1 q q θ − − − − 1 1 s N q− − at* k a k + + + + d* d

Fig. 8. The effect of sampling on an IMA time series (a) original sequence (Ns= 1) in production domain and (b) modified sequence in the

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For an IMA disturbance, substituting Eq. (11) into Eq. (15)gives yt ¼ 1 1þ GC GP qDeff 1 hq1 1 q1  a  t þ Nsk 1 q1   ¼ Saðq1Þ  at þ Skðq 1Þ  Nsk: ð16Þ

Note that the effect of the drift signal k for the EWMA controller will result in an offset of magnitude Nsk

KF from the target; with the double EWMA controller, this will not be the case because under these conditions, Skwill be zero at steady-state (from the final value theorem). By applying Parseval’s theorem to Eq. (16), we can compute the asymptotic variance as the control performance from the power spectrum

r2 y¼ 1 2p Z 2p 0 jSaðxÞj2dx r2 a: ð17Þ The ratio of r2

y, the sampled-run process output variance, to r2

a, the original variance of the white noise disturbance, can be obtained, from Eqs.(13) and (17)– a ratio we will use as our performance index. The controller parameter

may now be determined by minimizing this performance index, i.e. min sf r2 y r2 a ( ) : ð18Þ

To illustrate, consider a system using the EWMA control-ler for which

Saðq1Þ ¼ 1 h q1

1 q1þ KFqDeff: ð19Þ

The control performance with Deff= 1 can be calculated from Eq.(17)as r2 y¼ 2KFhþ ð1  hÞ2 ð2  KFÞKF r 2 a: ð20Þ

Eq. (20)is the same as that reported in[2,3], and optimal achievable control performance (achieved when r2

y¼ r2aÞ is obtained with KF= 1 h*. This result also indicates that the EWMA controller is a minimum variance controller when then the effective delay is 1.Fig. 9shows the achiev-able optimal control performance for different effective de-lays and sampling intervals. When the disturbance is a white noise sequence (h = 1), the achievable optimum con-trol performance is r2

a. This implies that a purely white noise disturbance cannot be eliminated with feedback con-trol; in fact any feedback control action taken will only am-plify this signal. This, of course, is the well-known result from classical SPC that the best way to deal with a white noise disturbance is to take no control action (KF= 0); alternatively, that the performance of a process subject to only white noise disturbance cannot be improved by taking control action of any sort.

The same analyses (Fig. 9) can also be applied to the double EWMA controller and, as will be shown later in Section4.2, it can also be extended for determining the tol-erable effective delay or sampling intervals.

4. Measurement strategy

Determining the appropriate number of metrology tools to use for a specific manufacturing problem is still a chal-lenging problem. On the one hand, by increasing the num-ber of metrology tools, one can reduce the loading of metrology tools and consequently achieve a higher sam-pling rate and throughput; however, the additional costs associated with such a decision can be significant. Box and Lucen˜o [3]and Box and Kramer[5] propose a dead-band control structure for determining the sampling inter-val, based on a minimum-cost scheme including adjustment cost, sampling cost, and quality lost. However, they do not consider the effects of sampling on stability – a very impor-tant issue in semiconductor manufacturing as we have shown already in this paper. Metrology strategies must be designed carefully and the sampling rate should be opti-mized to ensure acceptable control performance based on minimizing the cost of operation. However, defining and

1 2 3 4 5 6 7 8 0 2 4 6 8 10 12 Deff (σ2 y*/σ 2 a)opt θ=0 0.4 0.2 0.6 0.8 1 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Ns (σ2 y*/σ 2 a)opt 0.2 0.4 0.6 0.8 1 θ=0

Fig. 9. Achievable optimal control performance using EWMA controller: (a) for different effective delays with Ns= 1; (b) for different sampling

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quantifying the economic impact of such decisions is a very complicated issue in semiconductor manufacturing. In what follows we consider control performance as taking priority over economics. This is not a naı¨ve, unrealistic assumption; there are in fact critical steps in semiconductor manufacturing where the cost of quality lost is unquestion-ably more substantial than the cost of metrology tools. But we recognize that control performance will not always take priority over economics across the board.

4.1. Effect of sampling intervals and metrology delay Having characterized in Section3the achievable control performance for given sampling intervals and effective delays in general, we now present two specific examples here to illustrate how these results can be used to determine appropriate sampling strategies.

Example 4.1. Consider a stand-alone metrology tool that is shared by eight process tools. For simplicity, assume that for lot-to-lot control, the metrology delay due to queuing is eight runs; and that the effective delay is also eight because each lot is sampled. If a double-EWMA controller is implemented in the process tools, and the disturbance is identified as an IMA process with h = 0.6, we wish to investigate how the control performance is improved by introducing an additional metrology tool. If the new metrology unit shares half of the measurement loading, the metrology delay will be reduced to four.Fig. 10shows the effect of different Deff values on the control perfor-mance under these circumstances. It indicates that a 39% improvement in the variance can be achieved when Deff is reduced from 8 to 4. The margins of improvement is 29% when Deff is halved from 4 to 2 and 21% when Deff is further halved from 2 to 1. Note that such information can be used to quantify and justify the return on proposed investment on metrology tools.

As mentioned in Section 1, an alternative strategy for dealing with the effect of metrology delay is to implement virtual metrology (i.e., a soft sensor for quality). But since such soft sensor models are never perfect, the precision and accuracy of virtual metrology must be taken into consider-ation just as one would real sensors. If this soft sensor ‘‘measurement noise’’ is too large, (indicative of poor pre-cision of the virtual metrology component), the resulting performance may not be acceptable, despite the advantage of a much shorter effective delay.

Example 4.2. Consider a wafer-to-wafer control process tool with an integrated metrology tool in which measure-ment time is negligible (Nm= 1). The sampling interval is chosen to be four (Ns= 4) for a specific production rate, and the controller and disturbance are the same as in the previous example. If the same throughput is maintained, adding one more metrology unit to this equipment increases the sampling rate to Ns= 2. This strategy results in a 26% improvement in optimal performance as shown in Fig. 11.

Usually, control performance is more important than throughput; consequently the more interesting issue is how to increase throughput under the guarantee of perfor-mance. This is discussed next.

4.2. Maximum tolerable effective delay and sampling intervals

Before designing the measurement strategy, it is neces-sary to have available the following information about the process and control system:

• The process characteristics (e.g., gain).

• The disturbance characteristics (e.g., IMA or drift). • The appropriate controller (and controller tuning

param-eters).

• The desired specification limit.

0 10 20 30 40 1 2 3 4 5 6 7 τf σ2 y*/σ 2 a Deff = 1 Deff = 2 Deff = 4 Deff = 8

Fig. 10. The effect of Deff on the control performance inExample 4.1

(circles indicate optimal control performance).

1 2 3 4 5 6 7 8 9 1 1.5 2 2.5 3 3.5 4 4.5 τf σ2 y*/σ 2 a Ns = 1 Ns =2 Ns = 4 Ns = 8

Fig. 11. The effect of Ns on the control performance in Example 4.2

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We use a simple example to demonstrate how to deter-mine the tolerable sampling interval.

Example 4.3. Consider a CMP process with a perfect model (n = 1), subject to an IMA disturbance with parameters h = 0.4 and r2a¼ 24 A˚ . The specification limits are defined as 100 ± 25 A˚ . If the process is capable of achieving three sigma quality (only 1% of products out of spec under normal condition), so that 3rspec= 25 A˚ , the threshold value for the performance index is obtained as r2 spec r2 a ¼ð25=3Þ 2 24 ¼ 2:87: ð21Þ

If wafer-to-wafer control is implemented with an integrated metrology tool, by applying this threshold value toFig. 9, we reach two conclusions: (1) the effective delay cannot be greater than 4 if we sample every wafer; (2) the sampling intervals cannot be greater than 5 if Deff= 1.

The first conclusion not only guarantees control perfor-mance but also stability. If the tool cannot satisfy this con-dition, an additional metrology tool or wafer flow mechanism will have to be considered. The second conclu-sion could be applied to a CMP tool that can process two lots simultaneously, but with only one integrated metrol-ogy tool for quality measurement. In this tool, two process-ing equipments sharprocess-ing one metrology tool may result in a queue build up inside the tool. Due to the design, it is impossible to add more metrology tools within the tool. However, the preceding analysis of tolerable sampling interval suggests that one can make the sampling interval larger while achieving acceptable control performance. Applying this strategy will therefore substantially increase the throughput for this equipment.

Note that the optimal performance index inFig. 9is for optimal controller tuning obtained under the assumption of a perfect process model. In practice, the model will never be perfect, and we must take model uncertainty into

con-sideration. By considering plant/model mismatch in the range 0.8 < n < 1.2, is the maximum tolerable sampling interval still valid? Fig. 12 shows the achievable control performance as a function of the open-loop gain, KF, for different sampling intervals, Ns. The entire curve for Ns= 6 lies completely above the threshold value in Eq. (21), this strategy should therefore not be implemented because of specification violation. The circle on the curve marks the location of the optimal control performance and corresponding controller settings, KF,opt. Because KF is proportional to n, we use squares to locate the control performance at 0.8KF,opt and 1.2KF,opt, and the results indicate that robust performance can be maintained for ±20% steady-state gain variations. Actually, the curve crosses the threshold line at 0.61KF,optand 1.42KF,opt. This implies that, for Ns= 5, robust performance can be achieved for 0.61 < n < 1.42. The maximum tolerable effec-tive delay can also be determined in a similar fashion by analyzing control performance over different Deffvalues. 4.3. Measurement priority in queue for a metrology tool

Although integrated metrology tools can improve over-all process equipment productivity, capital investment con-siderations almost always dictate that stand-alone metrology tools are preferred because they can measure lots from several pieces of equipment, and one stand-alone metrology tool is less expensive than several integrated metrology tools. Especially in a stable process step, inte-grated metrology tools are usually not needed. When in a cluster of stand-alone metrology tools the loading of metrology tools increases suddenly, or one metrology tool shuts down, how should we prioritize the queue for measurements?

Let us define an index, Deff,i, to represent the current effective delay value for tool i. Deff,i should be updated every Ns runs and reset to zero when metrology data is fed back. From the previous section, it is possible to define the maximum tolerable effective delay for each process tool as Dmax,i, and then define the difference between these two values as a reference index

Ci¼ Dmax;i Deff;i: ð22Þ

We should sort the queue based on this reference index, Ci, and priority should be given to the lot with the smallest Ci. If this value reaches zero and the measurement still cannot be made, to ensure product quality, the equipment should be put on hold until the measurement is completed. 5. Conclusions

We have proposed a procedure for designing metrology strategies from the point of view of process control, includ-ing considerations for controller derivation, stability, and control performance. These strategies can help manufactur-ers determine the appropriate number of metrology units, if performance is the top priority. We also investigated

0.1 0.4 0.7 1 1.3 1.6 1 1.5 2 2.5 3 3.5 4 KF σ2 y*/σ 2 a NS=2 NS=3 NS=4 NS=5 NS=6

Fig. 12. Control performance as a function of forward loop gain KF¼ ðKP= bKPÞ=sf for different sampling intervals Ns (circle indicates

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strategies for determining measurement priority. These results are helpful in understanding the influence of sampling interval and metrology delay, and for determining appropri-ate sampling strappropri-ategies when implementing run-to-run control.

Nevertheless, economics is often more important for a fab manager who is usually more concerned with answer-ing such questions as: ‘‘should the fab purchase integrated metrology tools for each piece of equipment, or several stand-alone metrology tools to be shared by all of the equipment?’’ ‘‘Which will give a better return on invest-ment?’’ A larger number of metrology tools could provide a tighter process window and result in better yield, but it will also increase capital costs. Combining economics and process control to determine suitable metrology strategies will be the focus of future work.

Acknowledgements

We thank National Science Council for the support of the collaborated research. A.J.S. thanks TSMC for finan-cial support. Comments and suggestions from Jeffery Ward and David S.H. Wong are also appreciated. B.A.O. acknowledges the William L. Friend professorship for facilitating this collaboration.

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數據

Fig. 1. Multilayer configuration leading to repetitive characteristics in manufacturing
Fig. 2. Measurement complexity and sampling frequencies and data availability times at the recipe, metrology, and WAT levels.
Fig. 5. The control system in: (a) product domain and (b) sampled-run domain. To sampled-run domain N m =1Nm=2Nm=3Ns=3 System delay : q -1 D eff =1Wafers flowy*ty*t-1y*t-2y*t-3 N m =1Nm=2Nm=3Ns=3 :y*ty*t-1y*t-2y*t-3 12345678910 q -1 q -2 q -2
Fig. 8. The effect of sampling on an IMA time series (a) original sequence (N s = 1) in production domain and (b) modified sequence in the  sample-run domain, with sampling interval N s .
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