Device Robust-design by Using the Response Surface Methodology

Transcription

1 Devce Robust-desgn by Usng the Response Surface Methodology Xao-feng XIE, Yong LU, Wen-jun ZHANG, Zh-lan YANG (Insttute of Mcroelectroncs, Tsnghua Unversty, Bejng , P. R. Chna) Abstract: Devce robust-desgn s nherently a multple-objectve optmzaton problem. Usng desgn of experments (DoE) combned wth response surface methodology (RSM) can satsfy the great ncentve to reduce the number of TCAD smulatons that need to be performed. However, the errors of RSM models may large enough to dmnsh the valdty of the results for some nonlnear problems. To fnd the feasble desgn space, a new method wth objectves-orented desgn n generatons that takng the errors of RSM model nto account s presented. After the augment desgn of experments n promsng space accordng to the results of RSM model n current generaton, the feasble space wll be emergng as the model errors deceasng. The results on FIBMOS examples show that the methodology s effcently. Key words: response surface methodology; desgn of experments; devce robust-desgn 1. Introducton The technology CAD (TCAD) tools played a key role n the development of new technology generatons. For the deep sub-mcrometer devces, these tools provde a better nsght than any measurement technques and have become ndspensable n the new devce creaton [1]. Technology development, however, requres substantally more than a fundamental smulaton capablty: tools and methods to assst n exploraton of desgn trade-offs and to optmze a desgn are becomng ncreasngly mportant [2-4]. Devce robust-desgn s nherently a multple-objectve optmzaton problem because the desgners always want to attan more than one objectve at the same tme [5]. In order to obtan a robust devce, one should avod optmzng the desgn wth the consderaton of one sngle objectve only, because t usually leads to a devce that s not operable wth respect to other objectves. There s a great ncentve to reduce the number of TCAD smulatons that need to be performed. Vary one factor at a tme s a very neffcent procedure for optmzng and n many cases the best combnaton of desgn parameters may not be determned. Usng desgn of experments (DoE) [6,7] combned wth response surface methodology (RSM) [8, 9] s a strategy that can overcome these problems snce t can examne the whole parameter space whle at the same tme mnmzng the number of TCAD experments. The ntal step s to perform some computer based screenng experments that dentfy the key desgn parameters to reduce the number of desgn parameters to a more manageable number. Smulaton experments are then desgned and smulated, the results of whch are used to derve the response surfaces. There have some successful examples [10, 11] to desgn novel devces wth DoE/RSM approach by predctng the tendency among parameters and responses. And some frameworks, such as n DoE/Opt [3], VISTA [4], etc., have ntegrated the RSM and optmzaton capablty. However, the searched results by callng RSM model mght not satsfy the multple-objectves snce there always have errors between the results of the smulator and ts RSM model, even better model accuracy s acheved by a lot of methods [12-14]. In order to fnd the fully feasble desgn space that satsfes the multple-objectves, a new method wth objectvesorented desgn n generatons that takng the errors of RSM model nto account s presented. After the augment desgn n promsng space accordng to the RSM model n last generatons, the feasble space wll emerge as the model errors deceasng successvely. Then the method has been used to optmze the focused-on-beam (FIB) MOSFET [15] successfully. 2. Devce robust-desgn wth RSM

2 2.1 General representaton for devce desgn problems The devce desgn problems can be defned as fndng desgn ponts x S D such that f j ( x ) [c l, j, c u, j ], j = 1,, m (1) Where x =(x 1,, x,, x n ) T s a desgn pont, x represents a desgn parameter that specfy the topography and mpurty concentratons assocated wth a devce, such as effectve channel length, oxde thckness, dopng profle descrptors, S D s a desgn space defned as a Cartesan product of domans of desgn parameters x s (x [l, u ], 1 n). f j are real-valued functon of specfed devce performance on S D, whch defnes the response (.e. electrcal behavor) of the devce, such as on and off currents, threshold voltage, output resstance, etc. c l, j and c u, j specfes the desred objectve for the jth response. If c l, j = c u, j or only have small dfference, t represents an equalty constrant objectve. If c l, j = - or c u, j = +, t represents an nequalty constrant objectve. In addtonal, the optmum requrements should be transformed nto strong constrants objectve to obtan robust devces wth the process devatons for each devce parameter. Defnton 1: Each desgn ponts that satsfyng all of the m constrants s denoted as feasble pont. The set of feasble ponts s denoted as feasble space (S F ). Snce f j ( x ) are calculated by TCAD smulator at here, ts feasble space s also denoted as S F, SIM. 2.2 Desgn of experments (DoE) The choce of an approprate desgn of experments s extremely mportant n determnng the best model for multcharacterstcs. As TCAD codes become even more complex and computatonally ntensve, t becomes ncreasngly more mportant to reduce the number of TCAD smulatons requred. The value of statstcally based expermental desgns (the matrx of runs generated by specfc combnatons of desgn parameters n S D ) has been well establshed for automatc generaton of experments [6, 7]. The full-factoral desgn generates a unform grd wth user-specfed densty level m (totally m n ponts) coverng the nput parameter space. The Central composte desgns are useful to explore the parameter space S D wth a mnmum of requred experments. For example, a central composte crcumscrbed (CCC) desgn conssts of 2n axal ponts, 2 n cube ponts (full-factoral wth m=2) and one center pont. The rotatablty and the small number of necessary experments make central composte desgns very well suted for estmatng the coeffcents n a second-order model. The Latn hypercube samplng (LHS) [16] proposes the unform desgn concept. It provdes an orthogonal array that randomly samples the entre desgn space broken down nto r n equal-probablty regons (where r s the number of experments). LHS can be looked upon as a stratfed Monte Carlo samplng where the parwse correlatons can be mnmzed to a small value (whch s essental for uncorrelated parameter estmates) or else set to a desred value. LHS s especally useful n explorng the nteror of parameter space, and for lmtng the experment to a fxed (user specfed) number of smulatons. 2.3 Response surface methodology (RSM) Response surface method [8, 9] s a knd of methodology, whch generate mathematc model to descrbe the responses (devce characterstcs) n the space S D. An mportant role of response surface models s to mmc the more complex workngs of TCAD smulatons or experments. The most wdely used model functons are polynoms of second order [4] g( x ) = a 0 + n = 1 a x + n n = 1 j= a x x j j (2) The coeffcents for ths analytcal functon g( x ) can be calculated by a weghted least square estmaton. For both nputs ( x ) and responses (g), transformatons (e.g., log, exp, square, square root, nverse) can be specfed so as to ncludng the addtonal knowledge about the system behavor. The covarance of the estmates, a metrc of model stablty, s dependent on the nput desgn matrx and the lack of model ft. Choce of the nput desgn matrx s crtcal to determnng the model coeffcents, and mnmzng the covarance between the estmates of the model coeffcents. Scalng the nputs mnmzes the correlaton between the estmates of the coeffcents of the model [17]. In addtonal, weghted regresson s mportant: a common sequence s

3 to perform a broad expermental desgn, buld a model, optmze to desred regon wthn that model, and then refne the expermental desgn near that regon. In such cases, t s useful to weght the second set of smulatons more than the frst to ncrease model accuracy n the regon where the model wll be most used. 2.4 Objectves-orented augment desgn by consderng model errors Defnton 2: The feasble space of all the g j ( x ) [ c l, j + e l, j, c u, j + e u, j ] ( j = 1,, m) s denoted as maxmum potental space (S Fmax, RSM ). The feasble space of all the g j ( x ) [ c l, j, c u, j ] ( j = 1,, m) s denoted as feasble space of RSM model (S F, RSM ). The feasble space of all the g j ( x ) [ c l, j + e u, j, c u, j + e l, j ] ( j = 1,, m) s denoted as mnmum potental space (S Fmn, RSM ). When RSM model s utlzed to fnd feasble desgn space, the absolute errors δ j ( x ) = g j ( x )-f j ( x ) must be taken nto account snce S F, RSM s not a subset of S F, SIM, as shown n Fg. 1. For worse cases, the ntersecton of S F, RSM and S F, SIM s small, even s empty when the δ j ( x ) s large enough. SFmax, RSM SF, RSM SFmn, RSM SF, SIM g j ( x ) [ c l, j + e l, j, c u, j + e u, j ] g j ( x ) [ c l, j, c u, j ] g j ( x f j ( x ) [ c l, j + e u, j, c u, j + e l, j ] ) [c l, j, c u, j ] FIG. 1 A two-dmensonal scheme for the relatons of the feasble space n dfferent condtons Property 1: Suppose for any x, δ j ( x ) [e l, j, e u, j ], where e l, j and e u, j are the bound values for possble errors. Then for a gven x, t s easy to be proved that a) If g j ( x ) [ c l, j + e l, j, c u, j + e u, j ], then f j ( x ) [ c l, j, c u, j ]; b) If c l, j + e u, j < c u, j + e l, j, and f g j ( x ) [ c l, j + e u, j, c u, j + e l, j ], then f j ( x ) [ c l, j, c u, j ]. From property 1a), t s easy to know that S F, SIM S Fmax, RSM ; and from property 1b), t s easy to know that S Fmn, RSM S F, SIM. The propertes provde a methodology to fnd feasble ponts accordng to the response surface models,.e. to fnd S Fmn, RSM as feasble space nstead of S F, SIM. However, S Fmn, RSM may be empty as the range of errors [e l, j, e u, j ] s very large due to the hghly nonlnear response space. Hence, the errors, especally n promsng space for all the objectves, should be decreased. Ths wll be done by teratve generatons, just as n evolutonary computaton technology. For each generaton, the objectves-orented augment desgn n the S Fmax, RSM s performed for better accuracy of new RSM model, and potental spaces S Fmn, RSM and S Fmax, RSM are refreshed accordng to the new RSM model, as shown n Fg. 2. If S Fmn, RSM s stll empty, then perform next generaton. Augment desgn n SFmax, RSM for better model accuracy FIG. 2 Augment desgn n S Fmax, RSM for better model accuracy to enlarge S Fmn, RSM 2.5 Total desgn flow The total desgn flow can be descrbed as below: a) Defne desgn space S D and objectves; b) Prepare the nputs for the DoE modules (S D and the specfed desgn) and start the DoE program to create a DoE table; c) Calculate the responses for all the desgns n the DoE table by batched executons wth TCAD smulator and store nto a

4 persstent run database for all the experments; d) All responses that n specfed desgn space are collected from the fnshed runs and added to the experment table; e) Evaluate the data n the experment table to construct RSM models for all responses; f) Calculate all the errors for the experments and correspondng RSM calculaton results to fnd the range of [e l, j, e u, j ]; g) Fnd the maxmum potental space S Fmax, RSM and mnmum potental space S Fmn, RSM, accordng to the RSM model wth the range of errors [e l, j, e u, j ] and objectves n specfed desgn space; h) If S Fmn, RSM exsts, then output t as results; else screen the old desgn space wth S Fmax, RSM, return to b). 3. Desgn cases and dscusson The concepton of devce robust-desgn wth response surface methodology wll be demonstrated on a new-fashon focusedon-beam MOSFET [15], as shown n Fg. 4. Here most devce parameters are fxed. The effectve channel length s 0.35µm; the oxde thckness (T ox ) s 0.01µm. For source and dran, the juncton depth (X j ) s 0.1µm, dopng (N SD ) s 7.0E20cm -3. For the P + mplant n the channel, vertcal dstance (r p ) s µm, and vertcal devaton ( r p ) s µm. The desgn parameters nclude lateral mplantaton poston that start from source sde of channel (FIB-X), mplantaton dose (Dose), and substrate dopng concentraton (N sub ). The devce responses ncludes drve current (I on ) and dynamc output conductance (G out ), at V ds =1.5V and V gs =1.5V. For DoE n each generaton, the full-factoral desgn wth level=5 s used. A log transformaton of responses s used to ad model fttng. A numercal devce smulator PISCES-2ET [18] s used to calculate the devce responses. Tox SOURCE GATE DRAIN 0 N Xj N + P + FIB-X P-Substrate SUBSTRATE FIG. 3 Parameterzed representaton for FIBMOS devce Leff Ease case: Dose [1E12, 2E13] (cm -2 ), N sub [5E16, 1E18] (cm -3 ), when FIB-X=0.1µm; Fg 4 shows perspectve plots for a) I on and b) G out of smulated results and RSM model for dfferent Dose and N sub. Where the ponts are the smulaton results n the whole desgn space, and the surface s the RSM model that ftted to 25 data ponts. 1E-5 1E-6 1E-5 1E-6 I on (A/µm) 1E-7 1E-8 1E-9 G out (S) 1E-7 1E-10 1x x x x x10 17 N sub (cm -3 ) 1x x x x x10 17 N sub (cm -3 ) FIG. 4 (a) Perspectve plots for a) I on and b) G out of smulated results and RSM model for dfferent Dose and N sub (b)

5 It can be seen that the RSM model provdes consderable precson. Fg 5 shows the solnes of the RSM model for I on and G out. Where the sold lnes represent I on and dash lnes represent G out. Notce both the response results are transformed by a log functon. Snce the solnes for I on and G out are not parallel to each other, lower G out can be acheved at a fxed I on f hgher Dose and lower N sub are used. Fg 6 shows an optmzed result for that I on = 2E-4A/µm (.e. the solne that equal to 3.7 n Fg 5). Where the dash lne s a pont wth hgher N sub and lower Dose, and the sold lne s an optmzed soluton wth lower G out I ds (A/µm) 2.0x10-4 L eff =0.35µm FIB-X=0.1µm V gs =1.5V, V ds =1.5V 1.5x x x10-5 Case 1: Dose=1.00E12 cm -2, N sub =1.94E17 cm -3 Case 2: Dose=1.33E13 cm -2, N sub =5.00E16 cm x x x x x N sub (cm -3 ) V ds (V) FIG. 5 Response surfaces for log(i on ) and log (G out ) FIG. 6 An optmzaton example by RSM for larger G out Here the RSM model s utlzed to fnd feasble desgn space, whch the objectves for responses are set as I on >A/µm (.e. log(i on )>-4), G out <6.3E-6S(.e. log(g out )<-5.2). The range of response errors δ j ( x ) between the RSM model and the smulated results are [-, 0.12] for log(i on ) and [-0.12, 0.17] for log(g out ). Fg. 7 shows the feasble space n dfferent condtons, here the part of desgn space that N sub >5E17 cm -3 s not shown n order to demonstrate more clearly. The sold lnes,.e. the solnes for log(i on )= -4 and log(g out )= -5.2 of the smulaton results represent the boundary of S F, SIM, s used as reference. The dash lnes,.e. the solnes for log(i on )= -4 and log(g out )= -5.2 of the RSM model results represent the boundary of real S F, RSM. It can exactly to see that S F, RSM s not a subset of S F, SIM, whch due to the errors between RSM and smulaton results. Hence the model errors δ j ( x ) must be taken nto account. As n Fg. 7, the dash dot lnes,.e. the solnes for log(i on )= and log(g out )= of the RSM model represent the boundary of S Fmax, RSM, and the dot lnes,.e. the solnes for log(i on ) = and log(g out )= of the RSM model represent the boundary of S Fmn, RSM. Here S Fmn, RSM s a subset of S F, SIM, and then robust desgn ponts can be selected n ths regon. For example, as a desgn pont A n S Fmn, RSM, whch wth N sub =5E16cm -3 and Dose =1.7E13cm -2, then we have responses as I on =1.44E-4A/µm, G out =3.96E-6S. A S Fmn, RSM S F, RSM Legend for Fg 7 and Fg S Fmax,RSM Boundary of SF, SIM Boundary of SFmax, RSM S F, SIM Boundary of SF, RSM 1x x x x x10 17 N sub (cm -3 ) Boundary of SFmn, RSM FIG. 7 The feasble space n dfferent condtons for the objectve that I on >A/µm and G out <6.3E-6S Hard case: Dose [1E12, 2E13] (cm -2 ), FIB-X [, 0.30] (µm), when N sub =0.1cm -3 ; Fg 8 shows perspectve plots for a) I on and b) G out of smulated results and RSM model for dfferent Dose and FIB-X. Where the ponts are the smulaton results n the whole desgn space, and the surface s the RSM model that ftted to 25 data ponts.

6 I on (A/µm) 1E-5 G out (S) FIG. 8 (a) Perspectve plots for a) I on and b) G out of smulated results and RSM model l for dfferent Dose and FIB-X (b) S Fmax, RSM S Fmax, RSM S F, RSM S F, SIM S F, RSM -4.0 S Fmn, RSM B x x x x x10 13 FIG. 9 (a) The feasble space n dfferent condtons for the objectve that I on >A/µm and G out <3.16E-6S (b) Here the RSM model s utlzed to fnd feasble desgn space, whch the objectves for responses are set as I on >A/µm (.e. log(i on )>-4), G out <3.16E-6S(.e. log(g out )<-5.5). The range of response errors δ j ( x ) between the RSM model and the smulated results are [-0.09, 0.12] for log(i on ) and [-0.19, 0.23] for log(g out ). Fg. 9a) shows the feasble space of RSM n dfferent condtons that takes the model errors nto account. The dash lnes,.e. the solnes for log(i on )= -4 and log(g out )= -5.5 of the RSM model represent the boundary of real S F, RSM. The dash dot lnes,.e. the solnes for log(i on )= and log(g out )= of the RSM model represent the boundary of S Fmax, RSM. However, n ths case, the dot lnes,.e. the solnes for log(i on )= and log(g out )= of the RSM model cannot construct the set S Fmn, RSM wth feasble desgn ponts snce c l, j + e u, j > c u, j + e l, j, accordng to property 1b). In order to fnd feasble desgn space, the augment desgn should be performed n S Fmax, RSM for better model accuracy. Here the desgn space s decreased to Dose [1.2E13, 2E13] (cm -2 ) and FIB-X [, 0.225] (µm). Then new generaton s started wth a full-factoral desgn wth level=5. The range of errors δ j ( x ) between the RSM model and the smulated results are [-0.04, 0.03] for log(i on ) and [-0.09, 0.07] for log(g out ). Fg. 9b) shows the feasble space of new RSM model n ths desgn space n dfferent condtons. The sold lnes,.e. the solnes for log(i on )= -4 and log(g out )= -5.5 of the smulaton results represent the boundary of S F, SIM, and the dot lnes,.e. the solnes for log(i on )= and log(g out )= of the RSM model results represent the boundary of S Fmn, RSM. It can be found that the enhanced RSM model accuracy by the augment desgn n S Fmax, RSM nduces that (c l, j + e u, j ) to be less than (c u, j + e l, j ), and S Fmn, RSM whch s a subset of S F, SIM s not empty now. As a pont B n S Fmn, RSM, whch wth FIB-X=0.075µm and Dose =1.8E13cm -2, then we have responses as I on =1.12E-4A/µm, G out =2.32E-6S.

7 4. Concluson Ths paper has shown how response surface methodology combned wth TCAD smulaton can be used n the devce robustdesgn,.e. fnd the feasble space that satsfes mult-objectves, by consderng model errors. The fully feasble space S Fmn, RSM whch s a subset of real feasble space S F, SIM wll emerge by successvely accelerated enhancng the model accuracy n promsng space accordng to the objectves-orented augmented desgn n maxmum potental space S Fmax, RSM of last generaton. Future work s needed to employ new methods that enhance the accuracy of RSM model, snce better model accuracy makes for fndng S Fmn, RSM. References: [1] Dutton R W, Strojwas A J. Perspectves on technology and technology-drven CAD. IEEE Trans. on Computer-aded Desgn of Integrated Crcuts and Systems, 2000, 19(12): [2] Hosack H H, Mozumder P K, and Pollack G P. Recent advances n process synthess for sem-conductor devces. IEEE Trans. Electron Devces, 1998, 45 (3): [3] Bonng D S, Mozumder P K. DOE/Opt: A System for desgn of experments, response surface modelng, and optmzaton usng process and devce smulaton. IEEE Trans on Semconductor Manufacturng, 1994, 7(2): [4] Plasun R, Stocknger M, Selberherr S. Integrated optmzaton capabltes n the VISTA technology CAD framework. IEEE Trans. on Computeraded Desgn of Integrated Crcuts and Systems, 1998, 17(12): [5] Chen M-R, Chang P, Ln L. Devce robust-desgn usng multple-response optmzaton technque. IEEE 5th Internatonal Workshop on Statstcal Metrology, 2000: [6] Box GE P, Hunter W G, Hunter J S. Statstcs for expermenters. New York: John Wley, 1978 [7] Lorenzen T, Anderson V, Desgn of experments. Berln Germany: Marcel Dekker, 1991 [8] Alvarez A R, Abd B L, Young D L, et al. Applcaton of statstcal desgn and response surface methods to computer-aded VLSI devce desgn. IEEE Trans. on Computer-aded Desgn of Integrated Crcuts and Systems, 1988, 7(2): [9] Myers R, Khur A I, Carter W H. Response surface methodology. Technometrcs, 1989, 31(2): [10] Burenkov A, Tetzel K, Lorenz J. Optmzaton of 0.18µm CMOS devces by coupled process and devce smulaton. Sold-State Electroncs, 2000, 44: [11] P. Moens, M. Tack, H. Van hove, M. Vermandel, and D. Bolognes, Development of an optmsed 40V pdmos devce by use of a TCAD desgn of experment methodology, Conf. on Smulaton of Semconductor Processes and Devces, 2000: [12] Warng T G, Walton A J, Ferguson S, Sprevak D. Applcaton of covarance structures to mprove the ft of response surfaces to smulaton data. IEEE Trans on Semconductor Manufacturng, 1999, 12(3): [13] Josh S, Sheral H D, Tew J D. An enhanced response surface methodology (RSM) algorthm usng gradent-deflecton and second-order search strateges. Computer s & Operatonal Research, 1998, 25(7/8): [14] Xe H, Lee Y C, Mahajan R L, Su R. Process optmzaton usng a fuzzy logc response surface method. IEEE Trans. on Components, Packagng, and Manufacturng Technology Part A, 1994, 17(2): [15] Shen C-C, Murgua J, Goldsman N, et al. Use of focused-on-beam and modelng to optmze submcron MOSFET characterstcs. IEEE Trans. Electron Devces, 1998, 45(2): [16] McKay M D, Conover W J, Beckman R J. A comparson of three methods for selectng values of nput varables n the analyss of outputs from a computer code. Technometrcs, 1979, 21: [17] Draper N R, Smth H. Appled regresson analyss. New York: John Wley, 1981 [18] Yu Z, Chen D, So L, Dutton R W. PISCES-2ET manual. Integrated Crcuts Laboratory, Stanford Unversty, 1994