Measurement and Model Identification of Semiconductor Devices

Transcription

1 Measurement and Model Identfcaton of Semconductor evces JOSEF OBEŠ MARTIN GRÁBNER epartment of Rado Engneerng Faculty of Electrcal Engneerng zech Techncal Unversty n Prague Techncá Praha 6 zech Republc dobes@fel.cvut.cz http : // Abstract: urrent models of semconductor devces are very sophstcated especally ones for BJT and MOS- FET. The equatons of such models contan typcally one hundred parameters. Therefore a measurement and partcularly dentfcaton of full set of parameters s very dffcult. In the paper an optmzaton method s presented whch s usable for dentfcatons of even very complcated models wth a relatvely small number of teratons. The algorthm has been mplemented nto the orgnal software tool called.i.a. (rcut Interactve Analyzer) nto ts statc and dynamc analyss modes. Hence the dentfcaton s able to dentfy both and capactance models of semconductor devces. The process s demonstrated n the paper usng varous transstors. Key-Words: Modelng semconductor devce measurement parameter extracton optmzaton BJT MOSFET Introducton The.I.A. optmzaton algorthm sees to fnd up to 25 (n the current stable verson of the program) unnown parameters of the crcut for fulfllment of user-specfed requrements. The algorthm starts the analyses sequentally and changes these parameters after each of them to gradually fulfll the user s requrements. optmzed dependent varables R R 2 R 3 R 4 R 5 R 6 2 The. I. A. Optmzaton Algorthm Let us assume that some two crcut outputs are to be montored n three ponts as seen n Fg.. The crcles mar user-specfed requrements for the outputs and the squares mar values of the outputs obtaned after an analyss. The algorthm sees to mnmze the sum of squares of dfferences between them S (x...x n ) = m R 2 (x... x n ) n m () = where the unnown optmzed parameters of a crcut are mared by x...x n and R =... m are the dfferences. An extreme of the functon of n varables () can be found n the standard way. e. S = m 2R R =. (2) = ndependent control varable Fgure : A dagram of a typcal optmzaton tas. After a standard dervaton [] the generalzed leastsquares procedure s obtaned applyng the condton (2) J t J x (l) = J t r x (l+) = x (l) + x (l) where l s the teraton ndex and [ r = R x (l)] r = R [ x (l)] x x r r x x n J = l =... l max (3).. r m r m x x n =... m =... n.

2 ()( A) (dent) B Quassaturaton regon v E () ()( ) A (dent) (dent wth quassaturaton) B v E Fgure 2: Forward characterstcs of BJT K58. Fgure 4: Impact of quassaturaton model for BJT dentfcaton..2 5 ( )(A) (dent) B B ( )(A) (dent) B δ (%) v E E v BE Fgure 3: Reverse characterstcs of BJT K58. Fgure 5: Input characterstc of K58 (collector dsconnected). The generalzed least-squares procedure s very fast but sometmes nsuffcently stable. For ths reason the method s combned wth the gradent one x (l) = 2J t r l =... l max to the relable Levenberg-Marquardt modfcaton of (3) [ ] J t J + λ (l) x (l) = J t r x (l+) = x (l) + x (l) l =... l max (4) where s unt matrx and λ (l) s a scalar teratondependent factor. There are many ways to optmally determne that factor for each teraton the most sophstcated ones use an estmaton based on egenvalues of the Jacoban n (4) [2]. However smpler emprcal ways are mostly also successful []. The.I.A. program also contans a verson of the emprcal methods (however a way based on the egenvalues s also possble) whch sees to mnmze the λ (l) factor sequentally (. e. to mae the generalzed least-squares method more nfluental at the end of the process whch s natural): λ () = λ (l+) = λ(l) 5. (5) However ths monotone decay must be nterrupted (and therefore the gradent method must be sometmes made

3 ( ) ( )(pf) c c c c (dent) E (dent) E c E c v 2. v BE B (V) ( )( A) (dent) B v E Fgure 6: ollector and emtter juncton capactances of K58. Fgure 8: Forward characterstcs of mcrowave BJT KT E ( )(A) (dent) E E-6.4E-6.6E-6.8E-6 E-6 t(s) δ (%) v E (dent) (A) Fgure 7: Identfcaton of transt tme model parameters of K58. Fgure 9: Relatve dentfcaton errors n the selected stable area. more nfluental) when the method seems to dverge: f l > S (l) l mn j= S(j) then x (l) := x (l ) λ (l) := λ (l) 5 2 where the frst multplcaton by 5 compensates the dvson by 5 n (5) and the second multplcaton by 5 ncreases that scalar factor. Unfortunately the method descrbed above s nsuffcent for the majorty class of the crcut optmzaton problems. Thus an mproved method has been mplemented to the IA program. The mprovement conssts n the followng steps: The dfferences defned n (3) must be normalzed; These dfferences should also be weghted; The Jacoban J n (4) must be normalzed too; The Jacoban can qucly be evaluated by senstvtes; Evaluatng the Jacoban s not necessary n each teraton; Possble dvergence of teratons (4) can be damped.

4 ( )(A) (dent) v GS v S (V) ( )(A) where (nput) and (output) mar measured and optmzed values f the optmzaton s used for the den (dent) v S (V) v GS Fgure : Forward characterstcs of enh PMOSFET 2N368. ( )(A) (dent) v S (V) 3.85 v GS Fgure : Forward characterstcs of enh NMOSFET BUZ Fgure 2: Forward characterstcs of dep NMOSFET KF52. ( ) ( )(pf) c c c c (dent) S (dent) S c S c v v BS B (V) Fgure 3: ran and source juncton capactances of KF Normalzaton of the System of Equatons The models of BJT and MOSFET contan values of extreme orders (tny ones together wth the huge ones). For such systems the standard optmzaton algorthms are unstable. Therefore a normalzaton of dfferences s ncluded n the.i.a. program as a new feature (together wth ther weghtng of course) R [x (l)] y (output) [ x (l) ] y (nput) w y (nput) + y (null) =... m (6) tfcaton purposes. However many numercal experments have proved that a normalzaton of the Jacoban s also necessary: R [ x (l) ] x y (output) [ x (l) ] :=w x x (max) y (nput) x (mn) + y (null) =... m =... n (7) / where y (output) x s a result of senstvty analyss. The equaton (6) s a defnton. However the equaton (7) represents an assgnment. Therefore a soluton of the system (4) must be modfed by the assgnment [ ] x (l) := x (l) x (max) x (mn) =... n

5 after each teraton where x (mn) and x (max) represent mnmum and maxmum allowable values respectvely they are specfed by the user. The optmzaton s one of the most mportant advantages of the.i.a. program n comparson wth the SPIE ones. The total number of optmzed crcut parameters s lmted to 25. However there s no problem to ncrease that number arbtrarly. The optmzaton may be appled upon the operaton pont drect current transfer frequency and even transent analyses. 3 The Results of Model Identfcatons All the model equatons whch have been used for the model dentfcatons have been defned n the appendx of The SPIE boo [3]. A detaled physcal theory on modelng the semconductor devces s avalable n [4]. 3. BJT 3.. A Low Frequency Transstor The frst dentfed BJT was K58 whch s a zech equvalent of B8. The transstor has been frstly dentfed wthout the quassaturaton part of the model whch s smpler of course. The results of the dentfcaton are shown n Fgs. 2 and 3 the frst one (forward mode) wth the root mean square (rms) error 9.6 % and maxmum absolute value of relatve dfferences (δ max ) 43. % and the second one (reverse mode) wth these values rms = 4.85 % and δ max = 2. %. The optmzaton has gven the values of the model parameters I S = 7 3 A I SE = 2.98 A I S =.5 A β F = 974 β R = 5 n F =. n R =. n E = 2.6 n =.69 V AF = 4.9 V V AR = 4.9 V I KF =.2 A I KR =.28 ma and r = 3.2 Ω. As shown n Fg. 2 the saturaton part of the characterstcs s not modeled optmally. Therefore the equatons for modelng the quassaturaton must also be consdered. The results of such mproved dentfcaton are shown n Fg. 4 (they are drawn n natural lnear coordnates here n comparson wth the two prevous logarthmc ones). The optmzaton has gven the addtonal model parameters r O = Ω V O = V and γ = 7 [5]. Wth the ncluson of the quassaturaton the errors of the dentfcaton are lesser than those above rms = 3.5 % and δ max = 4.9 %. The parameters of the nonlnear base resstance model are dentfed usng the nput characterstc of the transstor as shown n Fg. 5. The nput characterstc has been dentfed wth the errors rms = 3.5 % and δ max = 35. % and the optmzaton has gven the model parameters r B = 26 Ω r BM = 37 mω I rb = 3.4 µa and r E =.53 Ω. The dynamc part of the model has also been dentfed. Frstly both junctons capactances have been determned as shown n Fg. 6. The dentfcaton has had the errors rms =.57 % (E).64 % () and δ max = 2.5 % (E) 2.73 % () and the optmzaton has gven the model parameters JE = 4.38 pf φ E =.65 V m E =.4 J = 3. pf φ =.4 V and m =.273. Secondly the transt tme model parameters have been dentfed as shown n Fg. 7. The optmzaton has gven the model parameters τ F =.249 ns I τf =.35 A V τf = 8.52 V and X τf =.33 wth the errors rms = 3.8 % and δ max = 94.4 %. The last ones seem to be large however the dfferences are determned usng the vertcal dstances whch are not optmal here of course (actually the dentfcaton can be consdered qute successful). The reverse transt tme has been dentfed n the same way wth the result τ R = 23 ns A Hgh Frequency Transstor The second dentfed BJT was the mcrowave one: Russan KT39. In Fg. 8 ts forward characterstcs are shown. The rregulartes are probably caused by oscllatons durng the measurement t s very dffcult to perform the measurements for the mcrowave transstors due to problematc stablty of such transstors. The optmzaton has gven the values of the model parameters I S = 8 A I SE = A I S = 7 A β F = 33 β R =.6 n F =.5 n R =.3 n E =.86 n =.75 V AF = 23 V V AR = 2 V I KF = 8 ma I KR = 86 ma r = 2 Ω r B = Ω r BM = Ω I rb = µa and r E =.6 Ω wth the dentfcaton errors rms = 6. % and δ max = 6.7 %. However f only the trangular stable regon s used as shown n Fgs. 8 and 9 then the errors are lesser: rms = 5.99 % and δ max = 22.2 % (and the mcrowave lnear transstors are manly used n such regons...). 3.2 MOSFET 3.2. Enhancement Mode Transstors Frstly let us dentfy the models of enhancement transstors. The frst one has been the low power standard P-channel 2N368 see Fg.. The dentfcaton procedure has gven the values of model parameters V TO = 4.77 V φ S =.657 V φ O =.86 V W = 37.9 µm L = 3.46 µm X J =.54 µm

6 X JL =.762 µm t ox = 98.7 nm N FS = 5 m 2 N A = m 3 v max = m/s µ O =.79 m 2 /(Vs) E P = 3.4 MV/m κ =.44 K P = A/V 2 γ =.294 V δ =.989 η =.3 θ =.334 V and ι =.34 (the last one s only present n the.i.a. program where serves as an addtonal fttng factor). The parameters of the model have been found wth a great precson rms = 2.8 % and δ max = 5.4 % only! The second one has been the hgh power standard N-channel VMOS BUZ345 see Fg.. The dentfcaton procedure has gven the values of model parameters V TO = 3.26 V φ S =.578 V φ O =.8 V W =.46 m L = 4.97 µm X J =.289 µm X JL =.79 µm t ox = 74.7 nm N FS = 5 m 2 N A =.73 2 m 3 v max = m/s µ O =.585 m 2 /(Vs) κ =.36 K P = A/V 2 γ =.366 V δ = θ =.384 V ι =.572 r =.249 Ω and r S =.435 Ω (for the power devces the dran and source resstances must be dentfed too; n the prevous example ther values have been fxed to the defaults Ω). The dentfcaton errors for that power devce have been greater than those for the prevous one (whch s natural): rms = 8.67 % and δ max = 28.8 %. Moreover the value of W s extreme but logcal power devces are composed of many sngle structures and therefore such value represents an ntegral A epleton Mode Transstor Secondly let us dentfy the model of a depleton-mode transstor whch was an N-channel KF52 see Fg. 2. The dentfcaton procedure has gven the values of the model parameters V TO =.48 V φ S =.334 V φ O =.789 V W = 443 µm L = 4.83 µm X J =.932 µm X JL =.827 µm t ox = 7.8 nm N FS = 5 m 2 N A = m 3 v max =.7 5 m/s µ O =.535 m 2 /(Vs) E P = 49 V/m κ =.4 K P = A/V 2 γ =.568 V δ = η =.8 θ =.2 V ι =.929 r =.8 Ω and r S = 5.7 Ω. Agan the dentfcaton has fnshed wth small errors rms = 4.6 % and δ max = 4.5 %. For the KF52 MOSFET ts juncton capactances have also been dentfed see Fg. 3. The dentfcaton procedure has gven the model parameters JO area S = 2.7 pf JO area =.57 pf JOsw permeter S =.26 pf JOsw permeter =.82 pf φ O =.789 V φ Osw =.789 V m S =.32 m Ssw =.83 m =.23 m sw =.286 agan the relatve errors of the dentfcaton are relatvely small: rms = 2.73 % (S) 3.5 % () δ max = 4.36 % (S) 6.9 % (). 4 oncluson An optmzaton algorthm has been presented whch s convenent for the robust and effectve dentfcatons of complcated tass. The algorthm has been mproved usng the equatons normalzaton whch s mportant for stablty of optmzatons wth BJT and MOSFET. The modfed algorthm has been mplemented to the.i.a. program and typcal measurements and dentfcatons of model parameters have been demonstrated. 5 Appendx The root mean square and maxmum devatons computed for the results n Fgs. 2 3 are defned naturally n p ( ) y (dent) y 2 = y rms = % n p δ max = max np y (dent) y = y % respectvely where y (dent) and y are the dentfed and measured values and n p s the number of all the measured ponts. Acnowledgement Ths paper has been supported by the grant of the European ommsson TARGET (Top Amplfer Research Groups n a European Team) by the Grant Agency of the zech Republc grant No. 2/5/277 and by the zech Techncal Unversty Research Project MSM References [] Fletcher R. Practcal Methods of Optmzaton. John Wley & Sons 978. [2] Fnsch L. An mplementaton of the Levenberg- Marquardt algorthm. Edgenősssche Technsche Hochschule Zűrch 996. [3] Vladmrescu A. The SPIE Boo. John Wley & Sons 994. [4] Massobro G. and Antognett P. Semconductor evces Modelng Wth SPIE. McGraw-Hll 993. [5] Hrušovč L. Quassaturaton Model of BJT. Master s thess zech Techncal Unversty 24.