Microelectronic Engineering 10 (1991) 287-298

Elsevier

2x7

Use of the response surface method in IC manufacturing

Claudio Lombardi SGS-THOMSON Microe1ectrot~ic.s. Centrd R&D. TDMC‘ Sitrlultrtion Tools. l’irr C‘.

Olivetti 2. 20041 Agmte Brim:a, Mih. Itrrl)

Keywords. Optimization. statistical analysis. design ccntcring. target. input variable. per- turbations. sensitivity, parametric yield, experimental design, rcgrcssion analyGx. response surface, responses. screening, validation.

Claudio Lombardi was born in La Spczia. Italy in 1951. He received the

Dr. Eng. degree in Electronic Engineering from the University of Geneva.

I Italy in 1980. During 1981 he was special student at MIT. Cambridge. MA, USA where research activity on device simulation was carriccl out within Prof. D. Antoniadis’ group. From 1983 to 1986 he participated to the CVT (CAD-VLSI for telecommunication) Project. supported by EEC‘. Since 1986 he is participating in the ESPRIT project named EVEREST.

L

where he is presently acting as work package coordinator. Since 1989 he is participating in the ESPRIT project named STORM. where hc is prc- sently acting as work package coordinator.

1. Introduction

Process, device and circuit simulation is in wide and general use and its impact on process/circuit developments is certainly very high. Due to the in- creasing complexity of the process architectures and, moreover, to the increasing need of reducing the development cost and time, new methods and new tools are now necessary to successfully carry out optimization, statistical analysis and design centering tasks.

For both process and circuit design the problems to be faced are generally characterized by a high number of input variables and by a high number of constraints/targets to be simultaneously satisfied/achieved. Moreover, the de- sign sensitivity to process perturbations has to be minimized and taken into account in order to maximize the circuit parametric yield.

In order to efficiently manage problems of such a complexity. methods based on experimental design [l-3], and regression analysis [4] have been proposed (see for example [S-lo]). The central point of these methods is the minimization

0167-9317/91/$3.5&l 0 1991. Elsevier Science Publishers B.V

of the number of simulations needed to obtain a simple analytical expression describing the simulation outputs as function of the input variables. Once these expressions (macromodels or response surfaces) have been obtained they are used in place of numerical simulations. The CPU times spent to evaluate analyti- cal functions are orders of magnitude less than those needed for numerical simulation. This drastic cost reduction makes practical the application of all the available techniques for optimization and yield maximization which are generally not feasible when used with numerical simulation.

By using the response surface methods on different problems, experience has shown that there are several critical points that must be treated with care. This is specially true for process and device simulations where the involved CPU times are generally very high and the best compromise between the accuracy of the response surface and the number of required simulations has to be found. A precise sequence of steps does not exist but generally the application of the Response Surface Method (RSM) can be summarized as follows: _ analysis of the problem, identification of the responses. identification of the

input variables and their ranges: - choice of the model and choice of the appropriate E.vprrirnrr~fal Dcsigu plan: - Run of the experiments (simulations) for screerzir~g purpose and identification

of the most significant input variables: - new Experimental Design plan based on the selected variables and run of the

new simulations; - determination of the response surface by using regression analysis: - verification and validation of the obtained response surfaces: - application of the results.

In the present paper, first the RSM approach is shortly introduced and then one example will be presented where RSM has been applied to a process under development in order to estimate the spread (distribution) of the electrical indicators used to compute the spread of the main SPICE model parameters.

2. Overview of the basic RSM approach

As a first step the problem has to be analyzed. Generally there are a certain number r of responses Y = (y,, y2, . . . , y,) which are relevant for the problem we have to solve, like threshold voltages. current levels at the device simulation level or gain and propagation delay time at the circuit simulation level. The selected responses are functions of a number 17 of independent variables or input factors X = (x,. x2. . . . . .I-, ). The input factors. following [8], can be generall! expressed as a sum of two components: the cvmrollahlc component and the wcontrollable one, i.e. _y, = c, + 14,. When the input factor is completely under control or its random variation is negligible then II, = 0. When the input factor is not adjustable but its random variations are important then (I, = cwmmt. The 14, components are usually assumed to be Gaussian variables having mean zero and a standard deviation u, but other distributions can be assumed.

It is obvious that the choice of the input factors is very important and the validity of the results will depend very much on the appropriate choice of these variables. Also important, but less critical, is the choice of the range of variation for each factor xi. The range can be adjusted during the work but of course this will cost more simulations. The input factors s, are then normalized (linear transformation) over their range of variation

S; = X,.rni,x - Xj,min, Z, = ey, ’ 2, S,

where xi0 is the variable value at the center of the chosen range. The new input factors zi are now ranging between - 1 and + 1.

After the problem description each selected response y, is formulated as:

y, = fx(Z, 0) + E, (1)

where the function fx(Z, 0) is the assumed model. 0 = (0,. f&, . . . , H,,) is the array of the p model parameters and E is the error. As numerical simulators are deterministic the error here is only due to the lack of fit. The function

f&C 0) . is a o P lY nomial expression. The choice of the degree d of the polyno- mial expression that will be used is influenced by both the number (i.e. the cost) of the simulations to be performed and the desired accuracy. In practice, for real problems, it is not convenient to have a polynomial degree greater than 2 because of the explosion of the number of simulations.

Generally, the following polynomial expression is used:

,V = 0,) + ii OiX, + i i OjfXjXj. i= I ,=, i=;

The polynomial parameters are extracted by using regression techniques [ll]. A very convenient experimental design plan for the above model is the central composite one. But before running all the experiments which are needed in order to fit all the coefficients in (2), it is better to verify the decisions taken about the input factors and their range of variation. For example, during an optimization task the optimal point could be external to the region R defined by the ranges of the input factors. For this and for other similar reasons it is convenient to perform a first low-cost check to verify if the problem was well defined. Moreover, due to lack of knowledge of the problem or to its complexity, it may be necessary to select a relatively high number of input factors. Under these conditions it is mandatory to perform a first low-cost screening of the input factors in order to determine a restricted set of the most significant factors. Last, but not less important, it may happen that the linear model belonging in (2) (i.e. deleting the last polynomial sum) is a good approximation of the response and then there are no needs to use the full one. Whatever is the reason, we are often in the need of performing a first set of experiments and then a possible second phase where the number of experiments will be increased in order to improve the fitting.

290

Fig. 1. Central composite design for three variables. (-i centre point: 0 factorial centre point: (; star with added centre point.

with Xl&d

The central composite design [2,3, S] consists of two parts (see Fig. 1). The first one is a two-level (i.e. the input factors can take the two values + 1 and - 1) fractional factorial resolution V plan (requiring &(LJ + 1) + 1) experiments) which allows the estimation of all first order and two factors interaction polyno- mial terms. The second one (requiring 2v + 1 experiments), allowing the fit of the pure quadratic terms, consists of the center point (i.e. all Z, = 0) plus symmetrical star points arranged along the axes of the factors. This experimental design is an orthogonal one (i.e. it allows a better estimation of the model coefficients) and allows the generation of an increasing number of experiments (in order to fit polynomials of increasing complexity) without loosing all the already performed simulations.

When the number of selected input factors or when the cost of each experi- ment are very high, a screening procedure is very useful in order to minimize the number of runs to be performed by selecting those input variables having the greater effect on the response. This step is usually done by testing the main effect of the variables, assuming that the interaction effects are negligible. To this purpose a fractional factorial design of resolution III (or higher), which requires 2”‘l22(“+ ’ 1 experiments, can be used as this plan allows the estimation of all the main effects without confounding one main effect with another. A central point is normally added in order to help in testing the second order effects.

There are several methods that can be used to select the most significant factors [5, 111 like Stepwise Regression or the Best Subset Regression where statistical tests [ll] (like F-test. R2 test, adjusted Rz test and so on) are used to decide if a factor is important or not. These tests do not provide the user with definitive answers but threshold limits have to be decided.

Having selected the most important factors, a complete Central Composite experimental design can be planned in order to estimate the coefficient of the full polynomial expression in (2). The simulations done in the screening phase

C. Lornbardi I Response surface method in IC rnanufactwing 291

0 IDSC

- -. BISECTOR

can be mapped on this new plan and used again. Star points can be placed at +_cr (generally kO.5 or kO.75) dividing each factor in the composite plan in 5 levels: 0, +-a, +l. Regression analysis is applied again, like in the screening phase, in order to estimate the best model. The validation of the model can be done by using RcRESS [ll] or just by looking at the plot like the one shown in Fig 2.

Finally, the verification of the model can be only done by testing the model on points different from those used during the regression analysis. If the model is not adequate, the transformation [3,9] of both responses and input factors is a valuable method in order to improve the accuracy of the model.

3. An example

In this example RSM has been applied to a real problem using process- device simulation. The process under analysis is the CMOS part (minimum length 3.0 km) of a mixed, 60 volts BCD process (Bipolar-CMOS-DMOS). The main goal was to obtain statistical distributions of some meaningful process parameters and eventually, to perform an optimization of the devices under analysis. As simulators, SUPREM III (1-D) [12] for process, and MINIMOS-4 (2-D) [13] f or d evice simulation have been used. As regards the statistical part of this work (i.e. experimental design and regression analysis) RS/l Explore and RS/l Discovery [14] have been used.

292 C. Lombardi I Response surface method in IC rnanufacturiq

A set of RS/l procedures to automatically generate the input files for the simulators were written. Aiming to a characterization of the process the follow- ing responses were selected for the n-channel (NCH) and p-channel (PCH) devices:

VTL threshold of the long channel transistor (25 pm); VTB threshold of the long channel with body effect (V, = -5 V for NCH

and VB = 5 V for PCH); VTC threshold of the short channel transistor (3 km for NCH and 4 km for

PCH); IDSL drain current of the long channel transistors in the saturation region

(V,=V,=5V); IDSC drain current of the short channel transistors in the saturation region

(V, = VD = 5 V).

The choice of the input factors was based on the analysis of the process flow- chart, and on the knowledge of the process variations that might influence the selected performances of the transistors.

Among the several process parameters, the following small set of input factors was selected:

(1) NSUB

(2) QINN (3) QI& (4) TEMP (5) NSS (6) TOX

(7) DL (8) RES, (9) RESp

substrate concentration; dose of boron in p-well implant; dose of boron in PCH threshold shift implant; temperature of p-well diffusion; fixed silicon-silicon dioxide interface charge; gate oxide thickness; difference between nominal and effective length; source and drain serial resistances for NCH; source and drain serial resistances for PCH.

None of the above variables are fully controllable; in fact all of them are considered to have an uncontrollable component. Among them the substrate concentration, the oxide thickness, the channel length variation, the serial resis- tances and the fixed interface states have a constant controllable component.

It should be mentioned that this analysis was performed on a process whose development was already in progress and data resulting from a first process characterization were already available. This explains the choice of the above input factors. The ranges of the factors were then decided according to the knowledge of the process and of the production line were the process was developed.

A first screening analysis was planned in order to verify if all the variables were actually needed or if some of them could be neglected. So a first experi- mental design was set to find the most significant factors. By considering the number of involved variables (for each device there are 7 variables) and the cost of the simulations our choice resulted in a fractional factorial resolution IV plan with 16 experiments plus the central point (total of 17 experiments per

C. Lombardi I Response surface method in IC manufacturing 293

Table 1 Initial set of experiments

No. NSUB QIN TEMP TOX DL NSS RES

1 - 2 +

3 - 4 + 5 - 6 + 7 - 8 + 9 _

10 +

11 - 12 +

13 - 14 +

15 - 16 +

17 0

+ +

+ + _

_

+ + + + _ _

+ + _ _

+ + 0

_ _

_ _

+ _

+ + - + + + _ +

+ _ _ SL

+ +

+ _

+

+ +

+

+

+ _

+ _

+ _ + + + 0 0

+ 0

device). The complete list of experiments to be performed is shown in Table 1. The design scheme is the same for NCH and PCH, where QIN represents QINN and QINp and RES represents RE& and RESp respectively. For each experiment two SUPREM III and four MINIMOS-4 simulations were run in order to simulate all the desired responses.

The simulations of the 17 points of the design were performed and the effects of the input factors were analyed by using both the stepwise regression and the best subset regression methods. The same set of variables were selected by the two methods and the same values of R2 and Rz were obtained. In Table 2 the coefficients of the linear expressions of each response are reported.

A new set of experiments was then planned to determine a second-order polynomial function with interaction terms for all the responses by using the central composite design. The number of runs to be done, as a function of the number of input factors, is summarized in Table 3. Some of these experiments were coincident with some of the previous runs done for screening. The total number of new MINIMOS4 simulations was 90 for the n-channel and 56 for the p-channel.

Again the stepwise regression was used to estimate the model coefficients and the goodness of the fit was estimated by computing R2 and RgRESS. For all the responses high values of R2 and R2 PRESS (greater than 0.98) were found and the accuracy of the results were also visually inspected by plotting the fitted values against the simulated ones (one example is given in Fig. 2). In these plots the nearer the points are to the bisector line, the more accurate is the fitting.

Finally a validation control was performed on the analytical functions to

294 C. Lombardi I Response surface method irl IC marlufrrctwirlg

Table 2 Regression results for linear model

Constant Coefficients

NSUB QINP TEMP TOX DL NSS RSP

n-channel device VTL 0.797 VTB 2.813 VTC 0.752 IDSL 0.226 IDSC 1.865

p-channel device VTL -0.768 VTB -1.656 VTC -0.735 IDSL -0.091 IDSC -0.596

-0.037 0.133 -0.10 0.15 X -0.13 x -0.09 0.35 -0.22 0.47 X -0. I4 x -0.04 0.13 -0.10 0.15 X -0. 14 x

0.007 -0.026 0.018 -0.06 X 0.013 x

0.041 -0.155 0.111 -0.416 -0.273 0.086 x

-0.181 0.481 X

-0.254 0.445 X

-0.181 0.477 X

0.012 -0.024 0.012 0.061 -0.124 X

X

-0.122 X

X

0.075

X -0. 137 x X -0.133 X

X -0. 133 x

X 0.005 X

0.076 0.026 x

verify their prediction capabilities. New simulations were run by choosing input factors values different from those used for the fitting and the maximum differ- ence between these results and the predicted ones was found to be less than 5%. The ten second-order polynomial functions can be used, with confidence, in place of the simulation within the region defined by the input factor ranges.

2.1. Fit of N-channel measurements

For the first purpose an RS/l procedure that uses a grid search technique was written. A multidimensional grid is constructed in the variables space dividing each range into N equal intervals. An interval around the desired value is fixed for each response; the simulated values are considered to fit when they fall inside this interval. The first response is calculated for each point of this grid; the second only for the points for which the first is inside the constraints; the third is calculated only where the first two satisfy the constraints, and so on.

The grid search algorithm was used to verify the results obtained from the threshold corners lots: these corners were realized implanting three

Table 3 Number of runs required for analysis using second-order polynomial function

No. of variables No. of runs

3 15 4 25

5 27

6 4s

Table 4

C. Lombardi I Response surface method in IC manufacturing 295

QIN TOX NSS VTL VTB VTC

9.8 x 10” 6.2 x 10-h 5.4 x 10”’ 0.929 (0.950) 2.866 (2.88) 0.864 (0.84) 1.1 x lOI 6.0 x Wh 4.1 x 1oi” 1.024 (1.03) 3.046 (3.04) 0.960 (0.93) 1.3 x lOL3 6.2 x 10Ph 3.07 x 1o’O 1.213 (1.22) 3.505 (3.53) 1.144 (1.12)

different doses of QIN (nominal values: 0.9 X 10’” at/cm’, 1.1 X 10” at/cm*, 1.3 x 1013 at/cm*). The difference of VTL ,VTB, VTC values between typical and minimum case is lower than the difference between maximum and typical, while it must be the contrary: in fact it is well known Vthreshold CC m.

Imposing the constraint on VTL, VTB, VTC we effectively found three points corresponding to three different values of implant dose (see Table 4 where the measured threshold voltages are reported in parentheses). NSUB and TEMP were kept at their central values. It can be seen that, for the minimum, simulations fit measurements with a dose higher than the nominal one (9.8 x 101* vs. 9 x lo’*). A further control confirmed that the lot with the minimum dose was wrongly processed. In the same points the currents obtained

P-Channel Threshold Voltage for ~0”~ Channel

Fig. 3. Contour plot of the fitted p-channel threshold voltage as a function of implanted dose (QIN) and substrate concentration (NSUB).

296 C. Lomhardi I Response surface method in IC manufbctwirzg

were within the measured values k 10%. Another result is the rough estimation of the fixed interface charge to be considered.

2.2. Search of optimal implant dose for PCH

The second application of the fitted responses was the search of the optimal implant dose for threshold shift of PCH. This goal can be simply achieved because only the dose has to be changed and this has no influence on the other targets as there are no constraints on currents. By analyzing the contour-plot of the threshold voltage with QIN and NSUB, the most important input vari- ables for this response, varying in their whole range (shown in Fig. 3), the desired value (1 + 0.2 V) of VTL is rapidly reached for QIN = 4 x 10” at/cm’ (i.e. at the center of the NSUB range).

2.3. Statistical distribution of responses

Having the analytical expressions it is very simple to obtain the statistical distribution of responses with a Monte Carlo method applied to these functions.

Fig. 4. Histogram of the n-channel threshold voltage obtained from fitted response.

C. Lomhardi I Response surface method in IC manufacturing 297

Input variables are supposed to be statistically independent (and they are) with Gaussian distribution. Having no statistical data on the spread of these process variables, the ranges were set as follows: if existing, process tolerances were used (if TOX is out of TOX * 50 8, the wafer is rejected; the range 3 x 10” to 5 x 10’” for NSUB corresponds to the tolerance given by the manufacturer); in the other cases reasonable settings based on process knowledge were used, keeping in mind that on some variables were accumulated all possible variations existing in a real process.

One example of the simulated statistical distributions is shown in Fig. 4. The initial choice of the responses was done in order to build Worst Case Model Cards for Spice: in fact, elaborating statistical histograms of threshold and currents with the in-house developed software WCAP [15] it is possible to determine the spread of the most important Spice model parameters.

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[3] G. E. P. Box, W. G. Hunter and J. S. Hunter, Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building, Wiley, New York. 1978.

[4] A. R. Alvarez, B. L. Abdi, D. L. Young, H. D. Weed and E. R. Herald, Application of statistical and response surface methods to computer aided VLSI device design, IEEE Trans. CAD 7(2) (1988) 272-288.

[5] K. K. Low and S. W. Director, An efficient methodology for building macromodels of IC fabrication processes, IEEE Trans. CAD 8( 12) (December 1989).

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[lo] F. Severson and S. Simpkins, Hadamard analysis: An effective, systematic tool for worst circuit analysis, IEEE Custom Integrated Circuit Conf., 1987, pp. 114-118.

[ll] N. Draper and H. Smith, Jr., Applied Regression Analysis, Wiley, New York. 1981. [12] C. P. Ho, S. E. Hansen and P. M. Fahey, SUPREM III - A program for integrated circuits

process modeling and simulation, Stanford Electronics Labs. Stanford, CA, TR-No. SEL84- 001, 1984.

[13] S. Selberherr, Electrochem. J. Proc. 88(9) (1988) 43-86. [14] RSIDiscover Reference Manual, BBN Software Products Corporation, Cambridge. MA.

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