# Small signal AC model for the velocity-saturated MODFET and the prediction of the microwave characteristics of MODFETs

код для вставкиСкачатьINFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may b e from any type of computer printer. T he quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate th e deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. H igher quality 6" x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact U M I directly to order. University M icrofilm s International A Bell & H ow ell Information C o m p a n y 3 0 0 North Z eeb R oad . A n n Arbor. Ml 4 8 1 0 6 -1 3 4 6 USA 3 1 3 /7 6 1 -4 7 0 0 8 0 0 /5 2 1 -0 6 0 0 O rder N um ber 9201684 Small sig n a l AC m odel fo r the velo city -satu rated M O D FET a n d the p re d ic tio n of th e m icrow ave characteristics o f M O DFETs Kang, Sung Choon, Ph.D. The Ohio State University, 1991 UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106 S m a l l S i g n a l AC M o d e l f o r t h e V e l o c i t y - S a t u r a t e d MODFET a n d T h e P r e d i c t i o n o f t h e m ic r o w a v e CHARACTERISTICS OF MODFETS D IS S E R T A T IO N Presented in Partial Fulfillment of th e Requirem ents for the D egree D octor of Philosophy in the G raduate School of T h e Ohio S ta te U niversity By Sung Choon Kang, B .S .E .E ., M .S.E.E., s|e s|c >|c s|c sfc The Ohio State U niversity 1991 D issertation Com mittee: Professor Patrick Roblin Professor Steve Bibyk Approved by %. L-.l Adviser Professor Furrukh Khan D epartm ent of Electrical Engineering A cknow ledgem ents I wish to express m y sincere appreciation and gratitude to m y advisor, Professor Patrick Roblin for his advice, guidance and financial support throught my M aster’s and P h.D . research at T he Ohio State University. Furthermore, I wish to thank Professor Steve B ibyk and Professor Furrukh S. Khan for reading m y dissertation and providing constructive criticism . I also express m y gratitude to Dr. Hardis Morko$ in University of Illinois at Urbana-Cham paign for providing the m easured data for m y research work. Finally, I wish to express m y gratitude to my wife, Kyungsook, m y son, Sangwoo, and m y daughter, Yousun, for their help and understanding. Especially, I owe to my parents and m y relatives in Korea who have supported m y Ph.D . studies. V it a June 18, 1955 ................................................... Born — ChoonChun, Korea 1973-1977 ...........................................................B .S .E .E ., The Seoul National University, Seoul, Korea 1977-1979 ...........................................................M ilitary Service 1980-1985 ...........................................................Manufacturing Engineer, G old Star Cable C o., Seoul, Korea 1985-1988 ...........................................................M .S .E .E ., The O hio State University, C olum bus, Ohio 1986-1991 ...........................................................G raduate Research Associate, D epartm ent of E lectrical Engineering, T h e Ohio State University, Colum bus, Ohio PUBLICATIO NS 1. P. Roblin, S.C . Kang, and H. Morkog, “Analytic Solution of th e VelocitySaturated M O S F E T /M O D F E T Wave Equation and Its Application to the Pre diction of th e Microwave Characteristics o f M O D FET’s ,” IEEE Trans. Electron D evices, vol. ED-37, No. 7, p p .1608-1622, 1990 2. P. Roblin, S.C . Kang, and H. Morkog, “Microwave Characteristics of the MODF E T and th e Velocity-Saturated M O SFET W ave-Equation,” Proceedings of the 1990 International Sym posium on C ircuit and System s, vol.2, p p .1501-1504, M ay 1990 3. P. Roblin, S.C . Kang, A. K etterson, and H. Morkog, “Analysis of M O D FET Microwave C haracteristics,” IEEE Trans. Electron D evices, vol. ED-34, N o.9, p p .1919-1928, 1987 FIELDS OF STU D Y M ajor Field : Electrical Engineering Studies in M icroeletronics : Professor Patrick Roblin Studies in Com puter Engineering : Professor Fusun Ozguner Studies in Physics : Professor Thom as Lemberger T able of C ontents AC K N O W LED G EM EN TS ................................................................................................. ii V I T A ............................................................................................................................................ iii LIST OF T A B L E S ................................................................................................................... viii LIST OF F I G U R E S ............................................................................................................... ix LIST OF S Y M B O L S ............................................................................................................... xv CHAPTER I I N T R O D U C T I O N ........................................................................................................ 1 B a c k g r o u n d .................................................................................................... ... . Problem S ta te m e n t............................................................................................ Structure of D isse r ta tio n .................................................................................. 1 3 6 TH E V ELO C ITY -SA TU R A TED M O D FET W AVE EQ UATIO N A N D ITS SO LUTION ........................................................................................................... 8 1.1 1.2 1.3 II 2.1 2.2 2.3 III PAGE Derivation of th e W a v e -E q u a tio n ................................................................ E xact Solution of the Velocity-Saturated M O D FET wave-equation Y-Param eters w ithin the Frequency Power-Series approxim ation . . 8 15 17 EQ UIVALENT CIRCUIT R E P R E S E N T A T IO N ............................................... 26 3.1 3.2 3.3 3.4 26 27 37 48 Introduction ....................................................................................................... T h e First Order Equivalent Circuit for th e GCA r e g io n ..................... T he O ptim al Second-Order Equivalent C ir c u it....................................... T he V elocity-Saturated M O D FET Equivalent C ir c u it ......................... IV PR ED IC TIO N OF T H E MICROWAVE CH AR AC TERISTIC S OF M ODF E T ’S ................................................................................................................................. 4.1 4.2 4.3 V 55 E xtraction of param eters for the ac m o d e l ............................................. Com parison of the m easured and calculated d a t a ............................... D is c u s sio n .............................................................................................................. 55 57 72 UNILATERAL P O W E R GAIN R ESO NAN CES AND f T-f\iAX O R D E R ING ..................................................................................................................................... 77 5.1 5.2 5.3 Introduction ...................................................................................................... Long and Short Channel M ode and the ac-current G a i n ................. Unilateral power gain of th e wave-equation m o d e l ................................. 77 79 82 V I C O N C L U S IO N ................................................................................................................. 88 6.1 6.2 C o n c l u s i o n .......................................................................................................... Future Work ....................................................................................................... 88 90 A P PE N D IC E S A Frequency Power-Series Solution for the Velocity-Saturated M O D FE T wave e q u a t io n ................................................................................................................. A .l A .2 A .3 Calculation of Vg c ( x = L ) .............................................................................. Calculation of Vq C{ X s ) ................................................................................. Power-Series Solution of W a v e -E q u a tio n ................................................. A .3.1 Calculation of v q ................................................................................... A .3.2 Calculation of v \ ................................................................................. A .3.3 Calculation of v 2 ................................................................................. A .3.4 Calculation of ig and i d ................................................................... A .3.5 Calculation of V12and K2 2 .................................................................. A .3 .6 Calculation of Yu and Y2 1 91 91 92 93 97 100 106 I ll 112 114 B Exact Solution for V elocity-Saturated M O D FET Wave E q u a tio n ......................116 C The Fourth Order Frequency Power-Series Solution in th e GCA R egion . vi 127 D D evelopm ent of Equivalent Circuit for the velocity-saturated M O D FET . 131 D .l Equivalent circuit for the saturation r e g io n ............................................... D .2 Calculation of th e Y-param eters for the two region m o d e l ............... 131 136 B I B L I O G R A P H Y ................................................................................................................... 139 vii L ist o f T a b l e s PAGE D evice parameters for calculating the intrinsic Y -p a r a m e te r s ............... 25 D evice parameters for the dc characteristics o f the A lG aA s/G aA s M O D F E T (2045) and th e G aA lA s/InG aA s/G aA s pseudom orphic M O D FET (2379) . . ............................................................................................................... 58 Microwave parasitics for the A lG aA s/G aA s M O D FET (2045) and the G aA lA s/In G aA s/G aA s pseudom orphic M O D FE T (2379) 69 D eviation of calculated S param eters from th e m easured data for all bias conditions for the device 2045 ( ds = 1500 A, C g d = 50 fF, and C o s = 50 f F ) ........................................................................................................... 70 D eviation of calculated S param eters from the m easured data for all bias conditions for the device 2379 ( da = 500 A, C g d = 70 fF, and C g s — 50 f F ) ............................................................................................................ 71 D eviation of calculated S param eters from th e measured data for device 2045 ............................................................................................................................. 74 D eviation of calculated S param eters from th e measured data for device 2379 ............................................................................................................................. 75 D evice parameters for the intrinsic short-channel M O D F E T ............... 81 L ist o f F ig u r e s FIG U R E PA G E 1 T he equivalent circuit of an extrinsic M O D F E T ......................................... 4 2 Equivalent circuit for th e velocity-saturated intrinsic M O D FE T pro posed by Rohdin [ 2 1 ] ............................................................................................. 5 3 Idealized representation of th e intrinsic M O D F E T ..................................... 9 4 E xact solution (solid lines) and frequency power-series solu tion (dotted lines) for m agnitude o f Y u .................................................................................. 21 E xact solution (solid lines) and frequency power-series solution (dotted lines) for phase of Y u ............................................................................................. 21 E xact solution (solid lines) and frequency power-series solu tion (dotted lines) for m agnitude o f Y u .................................................................................. 22 E xact solution (solid lines) and frequency power-series solution (dotted lines) for phase of Y u ............................................................................................. 22 E xact solution (solid lines) and frequency power-series solution (dotted lines) for m agnitude o f > 2 1 .................................................................................. 23 E xact solution (solid lines) and frequency power-series solution (dotted lines) for phase of Y n ............................................................................................. 23 E xact solution (solid lines) and frequency power-series solution (dotted lines) for m agnitude o f Y 2 2 .................................................................................. 24 E xact solution (solid lines) and frequency power-series solution (dotted lines) for phase of Y 2 2 ............................................................................................. 24 A pproxim ate sm all-signal equivalent-circuits for the intrinsic MOD F E T in GCA region.................................................................................................. 28 5 6 7 8 9 10 11 12 13 Comparison of the am plitude of Y \ \ / g o ............................................................ 30 14 Comparison of the phase o f Y n / g o ...................................................................... 30 15 Comparison of the am plitude of Y u / g o ............................................................ 31 16 Comparison of the phase o i Y u / g o ...................................................................... 31 17 Comparison of the am plitude of Y n / g o ............................................................ 32 18 Comparison of the phase of Y u / g o ...................................................................... 32 19 Comparison of the am plitude of Y i i / g o ............................................................ 33 20 Comparison of the phase of T^/flp..................................................................... 33 21 Plot of f 5 %(Yij)/fo for i i i asa function of the biasing param eter k. . 35 22 Plot of fs%(Yij)lfo for Y \ 2 asa function of the biasing param eter k. . 35 23 Plot of fs%(Yij)/fo for I 21 asa function of the biasing param eter k. . 36 24 Plot of fs%(Yij) / f 0 for Y 2 2 asa function of the biasing param eter k. . 36 25 Two different optimal second order sm all-signal equivalent circuits for the intrinsic M ODFET in GCA r e g i o n ......................................................... 38 26 Comparison of the m agnitude of Y n / g o for k = 0 . 6 5 ................................ 40 27 Comparison of the phase of Yn/go f ° r k = 0 .6 5 ........................................... 40 28 Comparison of the m agnitude of Y n / g o for k = 0 . 6 5 ................................ 41 29 Comparison of the phase of Yu/go for k = 0 .6 5 ........................................... 41 30 Comparison of the m agnitude of Y n / g o for k = 0 . 6 5 ................................ 42 31 Comparison of the phase o f F21/50 for & = 0 - 6 5 ........................................... 42 32 Comparison of the m agnitude of Y 2 2 /go for k = 0 . 6 5 ................................ 43 33 Comparison of the phase of 43 34 Plot of fs%{Yij)/fo for T ii as a function of the biasing param eter k. . 45 35 Plot of fs%{Yij)/fo for T12 as a function of the biasing param eter k. . 45 for k = 0.65 ............................................ x 36 Plot of h % ( Y i j ) l f 0 for Y21 as a function o f the biasing parameter k. . 46 37 Plot of fs%(Yij)lfo for F22 as a function o f the biasing parameter k. . 46 A pproxim ate second-order sm all-signal equivalent-circuits for th e in trinsic M O D F E T ..................................................................................................... 47 First-order non-quasi-static equivalent circuit for the velocity-saturated M O D FE T w ave-eq u ation ...................................................................................... 49 38 39 40 Comparison of th e am plitude o f Y\\ for Vd s 41 Comparison of the phase of Y u for = 3 F and 42 Comparison of the am plitude of F12 for 43 Comparison of the phase of Y 1 2 for Vd s Vd s Vd s = 3 F and Vg s — 3 F and = 3 F and 44 Comparison of th e am plitude of F2i for Vd s 45 Comparison of th e phase of Y<i\ for = 3 F and 46 Comparison of the am plitude o f >22 for Vd s 47 Comparison of th e phase of I 22 for = 3 F and 48 Vg s = OF. . . = OF................... 51 = OF.................. 51 52 = OF................... 52 — OF. . . 53 — OF................... 53 Measured (solid lines) and calculated (dotted lines) IV characteristics of the A lG aA s/G aA s M O D FE T 2045 ............................................................. 59 Measured (solid lines) and calculated (d otted lines) IV characteristics of the pseudom orphic G aA lA s/In G aA s/G aA s M O D FE T 2379 . . . . 59 50 Equivalent circuit for the extrinsic M O D FE T ............................................ 60 51 Measured (solid lines) and calculated (d otted lines) scattering param e ters for VGS = -08 V and VDS = 0.5 V for th e A lG aA s/G aA s M O D FE T 2045................................................................................................................................. 61 Measured (solid lines) and calculated (dotted lines) unilateral pow er gain for V g s = -08 V and V d s = 0.5 V for th e A lG aA s/G aA s M O D FE T 2045................................................................................................................................. 61 49 52 xi Vd s Vg s = 3 F and Vg s Vgs Vg s = OF. . 50 . Vg s = OF. 50 . . Vd s = 3 F and Vgs 53 54 55 56 57 58 59 60 61 62 Measured (solid lines) and calculated (dotted lines) scattering param eters for V g s = -08 V and V d s = 0.75 V for th e A lG aA s/G aA s MOD FE T 2045...................................................................................................................... 62 M easured (solid lines) and calculated (dotted lines) unilateral power gain for V g s = -08 V and V d s = 0.75 V for th e A lG aA s/G aA s MOD FE T 2045...................................................................................................................... 62 M easured (solid lines) and calculated (dotted lines) scattering parame ters for V g s = -08 V and V d s = 1.0 V for the A lG aA s/G aA s M ODFET 2045................................................................................................................................. 63 M easured (solid lines) and calculated (dotted lines) unilateral power gain for V g s = -08 V and V d s = 1.0 V for the A lG aA s/G aA s M ODFET 2045................................................................................................................................. 63 M easured (solid lines) and calculated (dotted lines) scattering parame ters for V q s = -08 V and V d s = 3.0 V for the A lG aA s/G aA s M ODFET 2045................................................................................................................................. 64 M easured (solid lines) and calculated (dotted lines) unilateral power gain for V g s = -08 V and V d s = 3.0 V for the A lG aA s/G aA s M ODFET 2045................................................................................................................................. 64 M easured (solid lines) and calculated (dotted lines) scattering parame ters for V d s = 3 V and V g s = - 0.15 V for th e G aA lA s/InG aA s/G aA s pseudom orphic M O D FE T 2379........................................................................... 65 Measured (solid lines) and calculated (dotted lines) unilateral power gain for V d s = 3 V and V g s = -0.15 V for th e G aA lA s/InG aA s/G aA s pseudom orphic M O D FE T 2379............................................................................ 65 Measured (solid lines) and calculated (dotted lines) scattering param eters for V d s = 3 V and V g s — -0.08 V for th e G aA lA s/InG aA s/G aA s pseudom orphic M O D FE T 2379........................................................................... 66 Measured (solid lines) and calculated (dotted lines) unilateral power gain for V d s = 3 V and Vgs = -0.08 V for th e G aA lA s/InG aA s/G aA s pseudom orphic M O D FE T 2379........................................................................... 66 xii 63 64 65 66 67 68 M easured (solid lines) and calculated (dotted lines) scattering param eters for Vd s = 3 V and Vgs = 0.25 V for th e G aA lA s/InG aA s/G aA s pseudom orphic M O D FET 2379............................................................................ 67 M easured (solid lines) and calculated (dotted lines) unilateral power gain for Vd s = 3 V and Vgs = 0.25 V for th e G aA lA s/In G aA s/G aA s pseudom orphic M O D FET 2379............................................................................ 67 Measured (solid lines) and calculated (dotted lines) scattering param eters for Vd s = 3 V and Vg s = 0.56 V for th e G aA lA s/InG aA s/G aA s pseudom orphic M O D FET 2379............................................................................ 68 Measured (solid lines) and calculated (dotted lines) unilateral power gain for Vd s = 3 V and Vgs = 0.56 V for the G aA lA s/In G aA s/G aA s pseudom orphic M O D FET 2379............................................................................ 68 Current and transconductance weight functions w \ (a ) (plain line) and iw2( a _1) (dashed line) plotted versus a and a -1 respectively................... 79 Variation of th e unity current gain cutoff frequency f j versus gate length Lg plotted versus a = E cLg/( V gs — Vt ) in log scale for an intrinsic M O D FE T with Vgs = 0, 0.1, and 0.2 V and Vd s = 1 V. . . 80 69 Exam ple of th e unilaterization of a two port device by loss-less feedback. 82 70 M agnitude of th e unilateral power gain versus frequency for an intrinsic M O D FET (V gs = 0 V and Vd s = 1 V) w ith a gate len gth of 3 /z (dashed-dotted line), 1 /z (dashed line), and 0.3 fi (plain lin e )................ 83 Equivalent circuit for the extrinsic M O D FET . C gs and C g d are the fringe capacitors of the gate.................................................................................. 85 Unilateral power gain versus frequency for a 0.3 /z extrinsic M O D FET w ith parasitics resistances R s = R g = R d = 0-01 ^ (plain lin e ), 0.1 D (dashed line), 1 fI (dotted dashed line), and 5 f! (dashed dashed line). 85 Unilateral power gain and short circuit current gain (plain line) ver sus frequency for a 0.3 /z extrinsic M O D FET using two different gate resistances R g = 5 fl (dashed line) and 25 D (d otted dashed line). . . 87 List of calvn p r o g r a m ............................................................................................... 128 71 72 73 74 xiii 75 List of calix p r o g r a m ............................................................................................... 129 76 Equivalent circuit for the saturation r e g io n .................................................... 135 xiv L ist o f S y m b o l s Cg : the gate capacitance per unit area Cgs '•the fringe capacitors between gate and source Cgd '■the fringe capacitors between gate and drain ds fMAX : the channel width in th e saturation region '• m axim um oscillation frequency E c : the critical electric field to attain th e peak velocity ei : the dielectric constant for the channel material 62 : the average dielectric constant for th e high-bandgap region I(x,t) : total current in the channel Idc(x) : the dc channel current i( x ) : the ac channel current q : electron charge Lg : the gate length I n s( x , t) : the len gth of the saturation region : two dim ensional electron density in the channel H : the channel m obility vs : saturation velocity o f electron ts : vs V t — tim e delay due to saturation region v3 : the saturation v elocity of electrons ■ the threshold voltage Vc s( x) : DC channel to source voltage at th e position x vcs(x) : total channel to source voltage at th e position x vg c {x ) : total gate to channel voltage at th e position x V g c (x ) •’ DC gate to channel voltage at th e position x VgC( x ) : AC gate to channel voltage at th e position x Vg s VgS Vd s • DC applied voltage between the g a te and the source : AC applied voltage between the g a te and the source ’ DC applied voltage between the drain and th e source xv Vds k '•AC applied voltage between th e drain and the source = Vd s / V gs — Vr xs for unsaturated device : instantaneous p osition of the G C A /satu ration boundary Xs : dc position of th e G C A /satu ration boundary xs : ac m otion of the G C A /satu ration boundary Wg : the gate width (Vgs ~ VT) u ° ■ t1 r 2 -------- xvi C H A PT E R I IN T R O D U C TIO N 1.1 B a c k g ro u n d T h e recently developed M O D FET is a prom ising device for both microwave and m illim eter-w ave applications and high-speed digital circuits. It has dem onstrated re m arkable high-speed performances [1]. Propagation delays as small as 12 ps and below 10 ps have been obtained in ring oscillator m easurem ents at 300° K and 11° K respec tiv e ly [2], [3]. Very low noise figures have been m easured at microwave frequencies (0.4 db with 14 db gain at 10GHz at 11°K [4]). Its low noise figure at high frequencies m akes it attractive in microwave applications. M O D FET s based on novel com pound sem iconductors are showing very prom ising results (see [5] for a review). Recently th e G E Electronics Laboratory reported 0.15 fi gate length InA lA s/In G aA s/In P latticem atched M O D FETs w ith m axim um frequency of oscillation as high as 405 GHz [6]. In support of the developm ent of the M O D FET technology , there has been a strong m odeling effort reported in th e literature. Indeed since the first dc model re ported by D elagebeaudeuf [7] a large number of dc m odels for the M O D FETs have been published. Som e authors have in addition derived sm all-signal m odels for th e M O D FE T using the quasi-static form ula Cgs = d Q / d V gs and Cgd = d Q / d V gd where 1 Q is the charge stored in th e channel. So far, however, there have been fewer re ported attem p ts to compare the microwave performance predicted from these models w ith the published m icrowave data available for the M O D FETs. Yeager and Dutton [8] reported a large-signal m odel which they compared with reasonable success to scattering parameters m easured at 4 GHz. A num erical m icrowave m odel based on a transm ission line circuit m odel was reported [9]. T h e sim ulation results obtained w ith their numerical m odel show a good agreement w ith the scattering parameters of a 0.3 m icron gate length M O D FE T m easured at a single bias point. An analytic microwave m odel including distributed effects for the unsaturated M O D FE T [10] was recently reported. It perm itted us to reasonably reproduce si m ultaneously the dc characteristics and microwave perform ance of an unsaturated M O D FE T using a unique set of device param eters. This analytic ac m odel was however lim ited to the linear regime up to the edge of saturation. Since the M O D FET wave equation has the same form as the three term inal MOSF E T ’s, it is useful to investigate the previous work on the M O SFE T wave equation. T he wave-equation for th e unsaturated M O SFET was derived independently by Can dler and Jordan [11], Geurst [12] and Hauser [13]. Geurst derived an exact solution of th e three term inal M O SFET wave-equation in term s of Stokes’ functions [12]. Burns [14] and Treleaven and Trofimenkoff [15] derived independently an exact solution in term s of B essel functions. These exact solutions are not however analytic per se, as th ey involve Stokes or B essel functions which much be num erically generated. Ap proxim ate equivalent circuits were derived by both Burns [14] and Treleaven and 3 TrofimenkofF [15] for the case of the M O SFET operated in pinch-off. For this m od e of operation th ey reported approxim ated analytic expressions for Y u and Y2\ (in pinch-off Y X 2 = Y 2 2 = 0). A n alternative procedure based on an iterative scheme was introduced b y Van Nielen [16] to obtain accurate approxim ate results of the M O S F E T wave equation. T his iterative solution was used by Bagheri and Tsividis [17] and Bagheri [18] for deriving the sm all-signal Y-param eters of th e long-channel four term inal M O SF E T and three-term inal M O D FET, respectively. M ore recently using a frequency pow er series first introduced by Van der Ziel and Ero [19] for the junction FE T , Van der Ziel and W u [20] solved the M O D FET w ave-equation and calculated Y\\ in term s o f a frequency power series for the unsaturated M O D FE T . Roblin and Kang [10] continued their calculation and derived th e remaining Y parameters Y 1 2 , Y2\ and Y22- T he frequency power series has the advantage of being analytic and holding up to high frequencies. To sum m arize there ex ist both an exact solution and an analytic frequency pow er series solution of the M O D FE T (and M O SFET) wave-equation. T h e M ODFET (and M OSFET) wave-equation applies however only to th e unsaturated M OSFET up to th e edge of saturation or to long-channel devices operated in pinch-off. However saturation in a M OSFET or M O DFET results from velocity saturation and not pinchoff for subm icron gate length. 1.2 P r o b le m S ta te m e n t In order to analyze the microwave characteristics of th e velocity-saturated M O D FET, an equivalent circuit is often used to fit the device perform ance m easured at various 4 Lg Rg C dg Cgs ^ Rd Ld V m in m — Source Figure 1: The equivalent circuit of an extrinsic M O D FET frequencies. A typical equivalent circuit is shown in Figure 1. This approach does not perm it to determ ine the elem ent values from the device parameters and to predict its bias dependence. The elem ent values of the equivalent circuit are generally valid only over the frequency range for which the parameter extraction is performed, so that attem p ts to extrapolate the response of the circuit beyond this frequency range can produce m isleading results. The sm all-signal ac m odels quoted in previous section are directly derived from their dc m odels using the quasi-static approximation. Since these m odels cannot account directly for the propagation delay across the channel and for distributed effects such as the effective channel charging resistances of the device capacitances, these m ight not successfully sim ulate the observed frequency and bias dependence of the M O D FE T characteristics. A high frequency ac m odel should account for the 5 g'dd C'dd G S S Figure 2: Equivalent circuit for the velocity-saturated intrinsic M O D FET proposed by Rohdin [21] propagation delay and th e distributed effects to predict the correct high frequency dependence o f the M O D F E T characteristics. Recently Rohdin [21] extended th e unsaturated m odel [10] to th e saturated MOD FET with th e aid of a drain resistor and capacitor in parallel to represent th e charac teristics of th e saturation region. T here is however no system atic way to predict the values of th e drain conductance and capacitance from the d ev ice parameters or the bias conditions. Consequently, their values are fitted so as to o b ta in a good agreement with the m easured data. In order to account for these saturation effects in the m icrowave characteristics of the velocity-saturated M O D FET , th e wave equation in saturation region should be derived and solved. T h e equivalent circuit based on wave equation w ill gives better representation of th e velocity-saturated M O D FET . It w ill give the basis for 6 th e developm ent of a large signal m odel for short channel M O D FETs. 1.3 S tr u c tu r e o f D is s e r ta tio n In Chapter II th e wave equation of velocity-saturated M O D FE T including both velocity-saturation and channel length m odulation effects will be derived, using a sim ple transport picture. T h e intrinsic M O D FET is divided into two regions sepa rated by a floating boundary. The w ave equation holding in these regions will be derived. B oth an exact solu tion using B essel function as reported in [14] and an ap proxim ate frequency power-series solution are derived. The comparison betw een two m ethod will b e m ade to estim ate the validity of th e analytic frequency power-series solution. In Chapter III an equivalent circuit representation of the velocity-saturated MOD F E T will be introduced. F irst a sim ple RC equivalent circuit for the GCA region (or long channel M O D FET ) w ill be derived based on th e frequency power-series solution of the GCA w ave equation. An optim al second-order equivalent circuit developed using fourth order frequency power-series solution w ill also be derived. An equivalent circuit for the saturation region based on the exact solution in th e saturation region, will also be developed. T h e total equivalent circuit for the velocity saturated MOD FE T will be then constructed by com bining the circuits for the GCA region and the saturation region. In Chapter IV we will discuss the integration of this ac-m odel with a dc-model. We then com pare the frequency and bias dependence o f the scattering param eters cal culated from this microwave m odel with the scattering parameters of a A lG aA s/G aA s 7 M O D FET and G aA lA s/In G aA s/G aA s pseudom orphic M O D FE T m easured for sev eral bias conditions. In order to introduce som e physical insights, a discussion on the significance of some of th e physical param eters will be given. In Chapter V the exact solution of the wave equation of velocity-saturated M OD FE T will b e used to analyze the m icrowave characteristics of intrinsic M O D FETs. The dependence of th e unilateral gain, / x and / m a x dependence upon gate length, parasitics resistance and capacitance w ill then be discussed. Chapter V I concludes this dissertation by discussing the future developm ent re quired in order to im prove the ac m odel and develop a large signal m odel. The appendices give th e detailed calculation of the frequency power-series solution (Appendix A ), exact solution (A ppendix B) and equivalent circuit for th e saturated M O D FET (A ppendix D ). Also included is M acsym a program to calculate the fourthorder frequency power-series solution of the wave equation. C H A PTER II THE VELOCITY-SATURATED M O D FET WAVE EQUATION AND IT S SOLUTION 2.1 D e r iv a tio n o f th e W a v e-E q u a tio n T h e distributed ac-m odel for the saturated M O D FE T is based o n a simple b u t well founded transport picture which assum es that transport is taking place in the 2 DEG channel and relies on th e following electron velocity-field relation ve = — fiE for E < E c v a = nEc for E > E c (2.1) where va is an effective saturation velocity, v, ty p ica lly corresponds to the p e a k ve locity of th e stationary velocity-field relation of the m aterial constituting the channel [22]. V elocity overshoot over the stationary velocity is partially im plied in t h e as sumption th a t the electron velocity rem ains the p ea k velocity v a for channel fields beyond th e critical field E c.Velocity overshoot above the effective saturation v elo city is neglected since it has a minimal im p a ct on the d c characteristics for m oderately sub-micron gate-length M ODFETs. Indeed it have recently b een shown in [22] for a 0.5 fi M O D F E T that a hydrodynam ic model developed by W id iger and H ess [23], allowing th e velocity to overshoot u p to three tim e s the peak stationary velocity, 9 Gate O Jl GCA Region V S ource O— Saturation Region d Drain ^ ds 2 DEG channel Xs —O Lg Figure 3: Idealized representation of th e intrinsic M O D FET predicted the same drain current and transconductance as a sim ple analytic m odel using th e peak stationary velocity for effective saturation velocity. The hydrodynam ic model predicts however a transport induced degradation in th e drain conductance not accounted for by th e analytic m odel. Therefore th e sim ple transport m odel selected here is m ore appropriate than th e stationary velocity-field relation used in [9] for micron and m oderately sub-micron gate length F E T ’s . For th e purpose o f analysis th e intrinsic M O D FE T is divided into two regions, the so-called gradual-channel approximation (G C A ) and saturation regions, as was done in a previously reported dc m odel for th e saturated M O D FET [24]. As shown in Figure 3, the G CA and saturation regions are located betw een the gate and th e 2DEG channel on th e source and drain side respectively. In the G CA region th e gradual channel approximation holds and the 2 D E G concentration ns is controlled by the gate to channel potential qns ( x , t) = Cg [v©s(<) - vc s ( x , t ) - VT] (2.2) where Ca is the 2DEG gate capacitance. In the satu ration region tw o dim ensional field effects dom inate and th e GCA approximation breaks down. T h e channel potential v c s can th en be approxim ately ob tain ed by solving the Poisson Equation a lo n g the 2DEG channel <PvCS( x , t ) — s ^ — q n s ( x , t) nrt_ ^ = - s ^ r = m x ’i) where /? = 1/ e i v sW gd a. ON ( 2 -3 ) This sim p le model has th e advantage over the G rebene Ghandhi m odel [24] o f predicting a larger drain conductance. However a detailed analysis of two-dim ensional field effects in the satu ration region [22] reveals th a t the charge distribution in th e channel o n ly partially account for th e d c drain conductance go (additional contributions to gD appear to be transport and traps related). As it will be seen in Chapter IV, it is preferable for th e microwave m o d e l to use a physical value for th e channel w idth da even though an artificial closer fit of the m easured dc drain conductance g o can be achieved with unphysically large channel openings ds. Following the pioneering work o f Grebene and Ghandhi it is assumed as in dc model [24] that the boundary betw een the GCA a n d saturation regions occurs when the channel field d v c s / d x reaches t h e critical field E c. Consequently the G C A region includes th e portion o f the channel where the electron velocity has not yet reached saturation and the saturation region includes the portion of th e channel in w h ich the velocity saturation is taking place. Let us now establish the wave equation w hich applies in each region. The wave equation for the GCA region was derived in [13], [12] and, [11]. The relationship between th e ac current and voltage in the GCA region is [20] *'(*) = \9 (Vac(x))vac{x)] (2.4) T he wave equation obtained for th e GCA region is [20] [9{v Gc{x))vgc(x)\ = j u C g WgVgc(x) (2.5) using the function g(VGc(x)) = fJtWgCg(VGc(x) — Vr)- The dc potential Vg c {%) — Vr is given by (see A ppendix B) Vg c { x ) - VT = (VGS - VT)^I 1 + (Ps - 2ks) ^ ~ (2.6) with ks = Vc s (X s )/(V g s —Vt ) and V cs(A s) the dc channel to source potential across the entire GCA region. The channel current in the saturation region can be expressed by I ( x , t ) = Idc + i ( x ) e jut = qWgn s (x, t)v a (2.7) and the continuity equation in the channel dijx^t) = dx w d v sn s ( x , t ) _ 9 dx dn s ( x , t ) _ 9 dt 1 dl{x,t) vs dt Extracting the ac part from Equation (2.8) and retaining the first order terms yields 12 In th e saturation region the ac current is related to th e ac voltage by the Poisson Equation (2.3). D ecom posing Equation (2.3) into dc and ac parts yields the following relationship between the ac voltage vgc(x) and current i = - m (2 .io ) Equations (2.9) and (2.10) m ake up th e wave equation for the saturation region. T h e solution of the w ave equation across the entire channel requires a set of boundary conditions to b e enforced at x = 0 and x = Lg and at th e boundary betw een the GCA and saturation region. The boundary conditions to be used at x = 0 and x = Lg for the com m on source configuration are Uflc(0) U^c(T^) = vgs — Vgs (2-11) Vds (2.12) The continuity of the 2D E G carrier concentration, channel electric field and channel potential, electron velocity and current are enforced at the G C A /satu ration boundary. These are naturally enforced by the continuity of the ac voltage vgc and ac current i at th e boundary. N ote that according to saturation picture, the channel electric field at the floating boundary between the G C A and saturation region is the dc (constant) critical field E c. T he ac channel field is therefore null at the boundary, and the GC A /satu ration boundary m ust m ove when ac voltages are applied at th e device term inals so as to m aintain a zero ac channel field. In th e sm all signal analysis the total (dc + ac) position of the GC A /satu ration boundary is written 13 x s { t ) = X s + x aeiwt (2.13) where X s is the dc position and x a th e ac m otion o f the boundary. Let us now derive th e relationship between th e ac m otion x a of th e G C A /satu ration boundary and the GCA ac field v'gc. The total (dc + ac) channel field at th e floating boundary xs is th e spatial derivative of th e total potential vgc at this boundary v ’ gc ( x s ) = V c c ( x s ) + v'gc(xs )e:,wt (2.14) T he ac electric field at th e floating boundary is th en , neglecting second order terms V ( * s ) = VZc ( X s ) x a + v'gc( X s ) (2.15) Setting th e ac electric field at th e floating boundary to zero yields th e boundary m otion x a as a function o f v'„ gc Xa~ V a d X s f 90^ (2 '16) where one can easily calculate Vq C{ X s ) to be given by: (see Appendix A .2 for detail calculation) k H i-\k .y v c s-v T Vcc(Xs) - - — S I J , X2— (217) T he solution of the wave equation across th e entire intrinsic M O D FET relies on the continuity of the ac voltage and ac current at th e floating boundary. It is therefore necessary to calculate th e ac voltage at the floating boundary and account for the m otion o f th e G C A /saturation boundary. Let us derive th e modified G C A channel potential obtained at th e floating boundary. T h e total (dc + ac) channel potential at the floating boundary is given by 14 vg c ( x s , t) = VGc { x s ) + vgc(xs )e3wt (2.18) w here vgc is th e ac potential obtained by solving Equation (2.5). Expanding Equation (2.18) with a Taylor series around the dc boundary position X s for small variations x s o f the boundary position yields the ac voltage vgc( x s ) at the floating boundary x s (second order term s are neglected) vflC(x s) = VqC(X s ) x 3 + vgc( X s ) (2.19) = —Ecxa + v gc( X s ) (2.20) T h e potential drop across the saturation region is also m odified by the m otion of th e boundary w hich m odulates the w idth of the saturation region. Integrating the Poisson equation f iVGC^ x } ) = _ j3I{x^ ax* = ^ + ,(x)eM ) (2>21) across the tim e varying saturation region yields th e ac potential at x = L g (see A ppendix A .l for detail calculation) vgc(Lg) = ( E c + Phcl) Xs + Vgc{xs ) + A vgc(l) (2.22) w here we introduced I = (L g — X s ) the dc width of th e saturation region and where A vgc(l) is the ac potential vgc(x) obtained by solving th e Poisson Equation (2.10) for a fixed saturation region w id th I, and zero ac potential vgc( X s ) = 0 and zero ac field v'gc( X s ) = 0 at X s - Substituting Equation (2.20) into Equation ( 2 .22) gives Vgc(Lg) = f ll d jx s + A Vgc(l) + vgc( X s ) (2.23) 15 One observes that th e contribution of the m otion x a of the G C A /saturation boundary is to add the ac potential term /3IdJxa. F in ally note th at the ac current at the floating boundary x s is to first order equal to the ac current at the fixed boundary X s - T his originates in th e fact that the dc current Idc is continuous (constant) along th e channel. 2.2 E x a ct S o lu tio n o f t h e V e lo c ity -S a tu r a te d M O D F E T w a v eeq u a tio n It has been shown by Burns [14] that the voltage-w ave solution of the wave-equation (2.5) can be expressed in term s of the m odified B essel functions I± 2 / 3 (Y ) (see A p pendix B for detail calculation) v ( x , u ) = C J 2 / 3 (Y ) + C 2 I . 2 / 3 ( Y ) (2.24) The current-wave is then derived from Equation (2.4) to be i(x,u>) = G'doaV S ' P ^ 4 [ C J . 1 / 3 ( Y ) + C 2 I 1/3( Y )] (2.25) where Y is a variable defined by Y = 4 /3 y / S ' ( P ) 3^4, P is a position variable defined by P = 1 — (2ks — k 3 ) x / X a, S' is the norm alized frequency S' = jui/ujok with u>ok = p{Vgs ~ Vt ){2ks - k l Y / X j , and G'dos = (2A:S - kl)Gdos with Gdos in Section 2.3. T h e wave-equation in th e saturation region can be readily derived. The current-wave is obtained by integrating Equation (2.9) *(*) = i ( X a)e~j ^ {x~x,) and th e voltage-wave by integrating Equation (2.10) (2.26) 16 A <W O = \ p 0 ) <(*.) 1 - l] (2.27) The unknown coefficients Ci and C 2 are obtained from th e boundary conditions (2.11) and (2 . 12), Ygc{Lg) = kgc(O) = ^ l lC l + A \ 2 C 2 = Vgs — Vds (2.28) A 2 \C\ + >122^2 = Vgs (2.29) where th e coefficients A,j are evaluated using Equations (2.22) and (2.24) An = h / z m + G ^ J S 'P 'J ' | g ( ^ ) 3 [e- i * ' - l ] + , - ^ l / ------- - p l d J ks An = x 2 [ i _ i / 3( x aj+ h / 3 ( Y ,) \ s ) ^ >u*' / - 2/3 (U ) + G i„ .V S ;P ' ' 1 ^ 0 ) ! [ e - > S ' - l ] + j / 3 ^ ; - p l d J — — 7------, r -------l-/i/3(Es) + i _ 5/3(r s)J k, (1 - \ k s) Kut A 21 = I 2/3 A 22 = I-2 /3 using A = j4iiA22 — ^ 12^ 21 , Ys — i / ^ ' / S ' ( P s)3^4, and Ps = (1 — ks)2. The unknown coefficients C 1 and C 2 are then derived from the system of equations (2.29) to be Cr O2 = A 22 — ^12 ~v gs ^22. ^ v ds A A 2i A n — A 2i — ^ Vgs + ^ Vds (2.30) (2.31) F in a lly the ac current flowing into the gate ig and th e ac current flowing into the drain id are given in terms of th e applied gate to source voltage Vds and drain to 17 source voltage vgs by id ig = i(L g) = i { X s) e - j % ‘ (2.32) = (2.33) G'doay/S' •P.1/ V ' £ , [C'1/ - 1/3(F .) + C 2 I 1 / 3 (YS)\ = i ( 0 ) - i ( L g) (2.34) = G'doaVS> [CaJ_1/3 Q x / S 7) + C 2 I 1 / 3 ( j ^ ) ] (2-35) - G'doaV S ' P ^ 4 e - j ^ l[ C , I . l / 3 (Ys) + C 2 h , 3 (Ya)} (2.36) T hese currents hold for arbitrary large frequencies. N ote that the m odified Bessel functions can be num erically calculated using the expansion [25] /y\ n ^ = ( 7 00 ( —) 2j ) Sm ^TTT) ^ The drain and gate currents obtained for th e saturated M O D FE T m odel reduce for / = 0 , X s = Lg and V c s(A s) = Vd s to the drain and gate currents of the unsaturated M O D FET s. 2.3 Y -P a r a m e te r s w ith in th e F req u en cy P o w er-S eries ap p ro x im a tio n The ex a ct solution derived in th e previous section was obtained in term s of modified B essel functions. Since these functions m ust be generated num erically it is convenient to use instead a frequency power series solution which usually holds up to very high frequencies. This frequency power series solution have directly derived from the waveequation using th e m ethod proposed by Ziel and Ero [19]. T he detailed calculation is described in A ppendix A. 18 The frequency power series yields th e ac current flowing into the gate ig and the ac current flow ing into the drain id in term s of th e applied gate to source voltage Vda and drain to source voltage vga. ig = Y \ \ v ga + Y\ 2 Vda (2.38) id = Y 2 1 vga 4- Y2 2 Vda (2.39) T h e Y coefficients calculated are the com m on source Y-param eters of the intrinsic M O D FET. Port 1 is defined between gate and source and port 2 between drain and source. Before givin g the calculated Y-param eters let us first define the following terms; 1 A ( k a) B ( k a) 6 (l C ( k a) 5-5*. D ( k a) *-**' (I-**.)* i _ l l 4. i p E ( k a) 6 6"'» ^ M J F ( k a) Lt 24 30 * * .) 3 J_p 30 a ' 180 * (i G(ka) H ( k a) i _ i t _1 {i - f a ) 4 i t I J_p 24 120 4 . -1-k 2 20 a ^ 80 * ~ 72 » 180 * (1 - P . ) 5 l_p 1440 * 19 Ri = Rgc = ( 1 - f c , ) 2 IdcPIXs k]{\ - \ k , f Vout ■'Os GdOs Ry = E cX s (1 - ka)Vout Ra = (1 — ka)R{ + -jjj-Gdoa( l — ka) Rb = 1 + (1 — ka)RiRy Rd = 1 Ra + Rb G dO s 1 + @^-GdOs(l — ka) + i? ,(l — ka)( 1 + R y) nC gW g(VGs - V T ) = Xs Vcs(Xs) ka = Vgs ~ Vt C os Cg W g X s = The Y u and Yu param eters for th e saturated M O D FE T are: ~ [~GgaTa + Cga + Ega] F ii = cu2 Y2 1 = Gga- u G ga = GdoaRd {ka + (1 — ka) R y R i ) C ga — R a c G a s D — H gs = R g c G g s E — R gcCoaB — R Ega — R gcG gaR d [ R a D + R b E ] — C o aR d [ R a A + R b C ] -------- — G gaG d a Fga — R g c R j ^ l l l - C G — a,T a + H ga + F ga T2 —Cgara -f- H,gt + ju> [—Ggara + Cga] where - <-gc'~’gs-L' C o s A — ■ E‘-'gs , \sOaJi- g CE g a D — Fga f3l2r [ R g c G ga ( R a F + R b H ) — C q s ( R a B + R b G ) E „ ( R AD + R BE)\ + ^ - G i , G t . - ^ - G i , C 3, 20 Gds is defined in the next section. T he Yi2 and for th e saturated M O D FET are: G dl JU) [ G dsr s 2 ~ 2 ^Ts ~~ CdsTs + Hda + Fds Y12 — W Y2 2 = Gds —w2 ^ Gds = G dOs R d (1 — ks) Cds = R g cG d sD — E ds H ds = R 2gcG d s F — R g c E d sD — Fds Eds = R gcR dG ds Fds — z l l - C d s T a + Hd, + ju> [—GdaTa + Cds "I” Eds\ Cds] where P I2- [Ra D + R b E) - ^ ~ G 2da [RycGj., (R^F s /-i2 24 ~ trj, Jds — R g H ) — E^t, ( R a D + R b E)\ P l\ G dsC ds N ote that th e Y-param eters derived for the saturated M O DFET reduce for / = 0, X s = L g and V cspC s) = V d s to the Y-param eters reported in [10] for th e unsaturated M O D FET , excep t, however, for the term f?22(^), w h ich as pointed out by [26] is incorrect. T h e correct term R 2 2 (k) is 9 R ( k ) 22( } = ^ ^(45 180^ 720 ^ (l-£*)6 1 6 0 + 14400 k4) (2.40) The correction introduced is small. It is necessary to assess the frequency range of valid ity of the frequency powerexpansion solution. For th is purpose th e frequency dependence of th e Y parameters 21 MAGNITUDE OF Yll xlOE-3 350 Y I N M H 0 POWER./ 250 150 1 10 100 FREQUENCY (GHz) Figure 4: Exact solution (solid lines) and frequency power-series solution (dotted lines) for m agnitude of Yn PHASE OF Y l l POWER 80 EXACT 70 60 1 10 FREQUENCY (GHz) 100 Figure 5: Exact solution (solid lines) and frequency power-series solution (dotted lines) for phase o f Yn 22 MAGNITUDE OF Y12 xlOE-3 3 .5 POWER / 1 .5 EXAC' 0 .5 1 10 100 FREQUENCY (GHz) Figure 6: Exact solution (solid lines) and frequency power-series solution (dotted lines) for m agnitude of Yn PHASE OF Y12 -110 D E G R E E -1 3 0 POWER -1 5 0 EXACT -1 7 0 FREQUENCY (GHz) 100 Figure 7: Exact solution (solid lines) and frequency power-series solution (dotted lines) for phase of Yu 23 MAGNITUDE OF Y21 xlOE-3 160 Y I N M H 0 140 POWER j 120 EXACT 100 1 100 10 FREQUENCY (GHz) Figure 8: E xact solution (solid lines) and frequency power-series solution (d otted lines) for m agnitude of Y2i PHASE OF Y21 -20 D E G R E E -4 0 POWER', -6 0 -8 0 -100 FREQUENCY (GHz) 100 Figure 9: E xact solution (solid lines) and frequency power-series solution (dotted lines) for phase of Y2i 24 MAGNITUDE OF Yll xlOE-3 350 POWER./ Y I N M H 0 250 150 1 100 10 FREQUENCY (GHz) Figure 10: E xact solution (solid lines) an d frequency power-series solution (dotted lines) for m agnitude of Y22 PHASE OF Y l l POWER 80 EXACT 70 60 1 10 FREQUENCY (GHz) 100 Figure 11: E xact solution (solid lines) an d frequency power-series solution (dotted lines) for phase of Y22 25 Table 1: D evice parameters for calculating the intrinsic Y-param eters Parameters l 3 W9 /* vs VT d ds Cl ^2 value gate length (fim) gate width (fim) mobility ( cm / V . s e c ) Saturation velocity ( m / s e c ) threshold voltage (V ) gate to channel spacing (A) channel width in saturation (A) channel dielectric constant gate dielectric constant 2 1 290 4400 3.45 x 105 -0.3 430 1500 13.1 e0 12.2 e0 are com pared for a fixed bias of both the analytic and exact solutions in Figure 4 11. T h e device param eters used for th is comparison are shown in Table 1. T he intrinsic Y-param eters are calculated for Vg s = OV and Vd s = 3V . One observes that the analytic solution com pare to the exact solution for frequencies up to 40 GHz. This frequency is near th e extrinsic f max as will be found in Chapter IV. C H A PT E R III EQUIVALENT CIRCUIT R EPRESENTATIO N 3.1 In tr o d u c tio n The M O D FE T wave-equation adm its an exact small-signal solution in the frequency dom ain in term s of th e m odified Bessel functions. However Bessel functions are difficult to generate and do not perm it the developm ent of a large-signal m odel. The approxim ate analytic solutions are preferred for CAD applications since they are fast and perm it the developm ent of equivalent circuits useful for tim e domain analysis. The equivalent circuit for the velocity-saturated M O DFET can be directly derived from the frequency power-series solution using a first-order RC topology for the entire M O D FET. However th is does not provide a physical representation of the circuit so that it is desirable to derive the equivalent circuits for each region and cascade them together to obtain a m ore physical equivalent circuit. Two m ethods, the iterative and power-series m ethods, have been used to obtain approxim ate solutions o f the M O D FET wave equation in th e G CA region. The smallsignal Y parameters obtained by an iteration [18] of order two adm it a frequency power-series expansion valid up to power two. These iterative Y-param eters hold for higher frequencies and have the advantage of providing a m ore graceful degradation 26 27 outside their frequency range of validity, compared to th e Y-param eters obtained by the frequency power-series of order two [18]. The iterative procedure yields for M O D FET up to first order, sm all-signal Y-param eters of th e following form Y ii = 9ij 1 + jujbij (3.1) This equation suggests the first-order RC topology for th e equivalent circuit m odel in GCA region. However the circuit directly derived from th e iterative m ethod does not im prove the total performance of the circuit, so that it is needed to derive the circuit from the second-order power-series solution using th e first-order RC topology. Based on the first-order RC topology, the second-order RC topology can be derived from the fourth-order frequency power-series expansion. In order to develop the equivalent circuit for the saturation region th e Poisson’s equation and wave equation have to be solved. The solution will account for the drain delay and th e potential drop in the drain region. A s seen in Chapter II this equation can be solved exactly and this is used to develop th e equivalent circuit of the saturation region. Combining the equivalent circuit o f th e GCA and saturation regions give the com plete equivalent circuit. 3.2 T h e F irst O rder E q u iv a len t C ircu it for th e G C A region A sim ple RC equivalent circuit m odel will be introduced to provide a graceful degra dation of th e Y-param eters for frequencies u> larger than u>o. T he RC m odel selected consists of th e DC (u; = 0) sm all-signal param eters gij shunted by a capacitor C,j in 28 g o- -O R11 © © j < o C 12 C11 1 + j C 0R12C 12 s ± Vds d R 22 9d 9m + C 22 j ( 0 C 2i 1 +jiaR2iC 2i O - Figure 12: A pproxim ate sm all-signal equivalent-circuits for th e intrinsic M O D FET in G CA region. series with a charging resistor i?,j. The resulting intrinsic Y-param eters are ju C n = Y12 = Y -*21 v -— _ 22 1 + juR nC n jwCii 1 + ju}Ri2Ci2 9a m 4+ , 9d + 1 + j w R 2\C2\ j u C 22 l + i a ; i ? 22C22 T h e associated equivalent-circuit for th e intrinsic M O D FET is shown in Figure 12. For frequencies to < < 1/ (RijC\j) these Y-param eters adm it the frequency powerseries Y j — 9ij + j u C i j + u>2 RijCfj (3.2) We can now readily identify th e resistors and capacitors to be ^ On n _ = _ 9o(Vg s )1'u(k) w0 -------------------- 9 o ( V Gs ) I i 2( k ) D_ tin = D _ Ru go(VGS)Ih(k) O i2 — ------------------------------------------- J1 1 2 = — w0 R 12 9 o ( V Gs ) ^ 2 ( k ) 29 n _ 9 o(V g s )^2 l(&) d Cl1 " ^ n, _ 9o(VGs)^22{k) C-22 — -------------------- _ 7^21 R n - ~g.(Vo s m * ) n II22 4122 — fl'0(V"G5)222( fe) where 11, i JL J . 2 ____ 7 _ i 3 6 ' 80 240 J 12 ^ n (A :) = Tx x{ k ) = IM5 (1-5*) 1 - k + \k2 (i n 12(k) (1 - fc)(l - I k) 2 i a(fc) = 2(1 - I*)* ^ 2l(*) = = 72.22 (&) = X22(*) = w (! ~ * ) ( £ - £ p k + jgfc2 - 3Igfe3) (1 - ! * ) « = J 2i ( *) i___ 1_ L4 ' 360 J 9a i n ~ 4an'v JL I _ 3L , U 2 9 (1 43 p _ I p . J L p ___ 1 Jf5 «ftA’ * iaaftf'' i«nnA' (1 - !2* ) • 1 L i.3 a "■ 20 (1 - I*)® - * )(& - if e * + - JL fc* + (1 - | * ) 6 ( ! - * ) ( ! - j * + ^ * 2) (1 - 12'*)* The time-constants r,j = RijCij appearing in the small-signal Y-parameters are then given by m - r i2 - 1 - t 2 r22 - D ^ 11 11 _ 1 60 — 120& + 81 A:2 —21 A:3 + 2k4 - ^ 15^2 _ k^ 6 _ 6k + k2j D ^ 2C 12 - - 21 21 22 22 1 30 - 41* + 16P - 2*3 ^ _ k) 1 600 — 1440* + 1290*2 — 540*3 + 110A:4 — 9 * 5 ^ 3 Q ^2 _ k^ 3Q _ A5k + 20fc2 _ ^ 1 320 - 560& + 340P - 90P + 9*4 - ^ 30^2 _ fc^ 2() _ l5k + 3jfc2^ MAGNITUDE OF Yll 12 POWER EXAC' 0> < v < 8 — B 4 0 ^= 0.1 1 F/FO 10 100 ^ igure 13: Comparison of the amplitude of iii/gro- PHASE OF Y l l MPWOKO 70 50 \ ^ \ POWER 30 10 L 0.1 1 F /F 0 10 100 Figure 14: Comparison o f the phase o f Yn /g0. 31 MAGNITUDE OF Y12 4 EXAC' ? POWER / 3 — B 2 1 O1^ 0. 1 1 F/FO 10 100 Figure 15: Comparison of the am plitude of Yu/go- PHASE OF Y12 -110 R -1 3 0 EXACT \ \ -1 5 0 -1 7 0 F/F0 p 8 wer 100 Figure 16: Com parison of the ph ase of Y^/go. 32 MAGNITUDE OF Y21 4 POWER 3 EXAC' I 2 1 0 100 F/FO Figure 17: Comparison o f the am plitude of Yix/go- PHASE OF Y21 -40 -80 -120 EXACT -160 0.1 w 1 F/FO 10 100 Figure 18: Comparison of the phase of Y2i/g0. MAGNITUDE OF Y22 4 POWER/ EXAC1 OK-^K 3 2 1 0 100 F/FO igure 19: Comparison of the amplitude of Yn/go. PHASE OF Y22 MMptJCDWO 45 EXACT 35 25 \ \\ 15 \ 1 F/FO 10 V POWE t 100 Figure 20: Comparison of the phase of Y^/go- 34 The m agnitude and phase of each Y-param eters are shown in Figure 13 - 20 for k = 0.65, obtained with the RC equivalent-circuit (dashed-dotted line, E Q ), the exact solution (plain line, E X A C T ), the frequency power-series (dashed line, P O W E R ), and the second-order iterative Y-param eters derived in [18] (dashed line, B ). A s can be seen in figures, the first-order equivalent shows th e graceful degradation. In order to establish the range of validity of the RC circuit representation for all bias conditions, the frequency fs%(Yij) for each parameter Yij is calculated. An error Err(Y{j) of 5 % is obtained between the exact B essel solution and the approxim ate results for th e frequency fs%(Yij). T he error E r r (Y ij) is \Yij(exact) — Yij(approximate)\ . \Yij(exact)\ For the sake o f comparison /s% (Y j)//o for each Yij param eter are plotted in Figure 21-24 as a function of th e biasing param eter k for the frequency power-series model (dashed line, P O W E R ),the second-order iterative results [18] (dashed line, B2), the first-order iterative results [18] (dashed line, B l ) and the sim p le RC circuit represen tation of th e frequency power-series m odel (dashed-dotted line, EQ). O ne observes that the sim ple RC representation of the frequency power series holds for all bias conditions up to a higher frequency than both th e frequency power series and the it erative results. On the sam e curve we have also plotted the u n ity current gain cut-off frequency fr /fo (dashed line, FT) and the m axim um frequency of oscillation f max/fo (plain line, F M A X ), (frequency at which the unilateral gain is one [27]). B oth f x and /max are calculated using th e exact Bessel solution. All approxim ate sm all-signal m odels except th e first-order iterative m odel hold 35 Yll ERROR 5 PERCENT FMAX 6 4 2 POWER FT B1 0 0 0.2 0 .4 0.6 0.8 1 Figure 21: Plot of fs%{Yij)/fo for Y u as a function of the biasing param eter k. Y12 ERROR 5 PERCENT FMAX 6 4 2 POWER FT B1 0 0 0. 2 0 .4 0.6 0.8 1 Figure 22: Plot of f 5 %(Yij)/fo for Yu as a function o f the biasing param eter k. 36 Y21 ERROR 5 PERCENT 10 FMAX 8 F 6 ( 0 4 2 U . POWER B1 FT 0 Figure 23: P lot of fs%(Yij)/fo for Y2 1 as a function o f the biasing param eter k. Y22 ERROR 5 PERCENT 6 FMAX 4 2 POWER FT 0 B1 Figure 24: P lot of fs%{Yij)/fo for Y22 as a function o f the biasing param eter k. 37 for frequencies larger than the cut-off frequency f o for all bias conditions. The RC circu it representation holds for frequencies larger than the m axim um frequency of oscillation f max for k sm aller than ~ 0.9. For k larger than ~ 0.9, / 5% is however sm aller than fmax• Note th a t both th e exact and the approxim ate m odels predict an in fin ite m axim um frequency of oscillation at k = 1. O bviously in the extrinsic device th e unavoidable source, drain and gate resistances and drain output conductance w ill limit fmax to a finite value. The infinite f max predicted for the intrinsic FET is nonetheless an indication o f the lim ited validity of the long-channel m odel. Indeed e v e n in long channel devices the drain current saturation ultim ately results from v elo city saturation and not pinch-off so that we always have k < 1 in the unsaturated p a rt of the channel. To conclude note that th e norm alization frequency f 0 is bias dependent. For gate voltages approaching the threshold voltage, the norm alization frequency f 0 is small a n d none of th e se so-called high-frequency approxim ate m odels can account for the distributed effects arising even at low frequencies. 3 .3 T h e O p tim a l S eco n d -O rd er E q u iv a len t C ircu it T h e simple R C equivalent circuit shown in Figure 12 is valid when the frequency considered is sm all enough so that the unsaturated M O D FE T behaves like a lumped d ev ice. At high-frequencies transm ission line-effects becom e im portant and a secondord er equivalent circuit becom es desirable. T he topology of the optim al second-order equivalent circuit will be based on th e second-order RC topology obtained by rewriting the second-order iterative Y- 38 Ri o------- A V -------- ------ A V ------^ - Ci Yij ‘ 9|g =- c2 O (a) Ci ° C2 + lf" Yij - 9ig Ri (b) Figure 25: Tw o different optim al second order sm all-signal equivalent circuits for the intrinsic M O D FET in GCA region param eters under the form juciij -|- ( j u f b i j Yij — 9ij d- (3.4) 1 + jujcij + ( j u f d i j A ctually tw o different second-order RC equivalent circuits can be used to im plem ent this equation, as is seen in Figure 25. (a) is preferable over (b) as its topology physically im plem ents the distributed effects of the channel. Let us now evaluate each elem ent in th e equivalent circuit (a). First Equation (3.4) is rew ritten in term s of the tim e constants Tuj, r 2 ij, and T3,j. Yij — 9ij d" jw C ij 1+ j ^ i j (3.5) d- ( i^ r ) 2r3<j where Cij TU j — fljj — C \ij "f" Cij2 Cli jC 2 i j R 2 ij , = U}0 b ij = U o ~ W ij T C^2ij T2ij = WoCij T3ij = ^ Q ^ ij = W0 = = U0(C 2ijR 2 ij W qC \ij + C iijR iij + C 2i j R l i j ) Cj2ij R l i j R2ij 0 r (Vgs - V j) 27T/o = p yx-----Ll In order to extract th e value o f t u j , r 2 ij, and Th Equation (3.5) is expanded in a fourth-order frequency power series. The denom inator of Equation (3.5) is obtained - T V,- „ N2-2 = 1 - b “ r20- + l + 3 —0TKj + \3—0) T3ij “o + [ j ^ r 2ij + ( ; ^ ) 2r32t, ] 2 - \ j ^ r 2ij + ( A Wo Wo Wo o 27& ]3 (3.6) Wo Substituting Equation (3.6) in Equation (3.5) and neglecting the fifth- and higher order term s in w gives the fourth-order frequency-power series. Yu = Sij + i - F i j - U — f S i j + 0 - ) % - - ( j - ) ' D h Wo Wo where th e coefficients, tu j, r 2 ij, and Wo (3.7) Wo are given in term s of Fij, Sij, T,j, and Dij by MAGNITUDE OF Yll ITER POWER 12 8 4 0 10 0 20 F/FO 30 40 Figure 26: Comparison of the m agnitude of F n/<7o for k = 0.65 PHASE OF Y l l ITER POWER -10 F/FO F igu re 27: Comparison of th e phase of Yn /g0 for k = 0.65 MAGNITUDE OF Y12 5 ITER 4 POWER 3 2 1 0 0 10 20 F/FO 30 40 Figure 28: Comparison o f th e m agnitude o f Y u / g o for k — 0.65 PHASE OF Y12 D E G R E E POWER -4 0 -8 0 -120 EXACT -1 6 0 ITER -200 F/FO Figure 29: Comparison o f th e phase of Y^/go for k = 0.65 MAGNITUDE OF Y21 5 ITER 4 POWER EXACT 3 2 1 0 0 10 20 F/FO 30 40 Figure 30: Comparison of th e m agnitude of F21/50 for k = 0.65 PHASE OF Y21 D E G R E E -4 0 -8 0 -1 2 0 ITER -1 6 0 -2 0 0 F/FO Figure 31: Comparison o f the phase o f Y2i/go for k = 0.65 MAGNITUDE OF Y22 5 IT E R POWER 4 EXACT 3 2 1 0 0 20 10 F /F O 30 40 Figure 32: Comparison of the m agnitude of Y 2 2 / 9 0 for k = 0.65 PHASE OF Y22 140 100 D E G R E E EXACT IT E R 0WER -2 0 -6 0 -100 20 F /F O Figure 33: Comparison of th e phase of Y2 2 /go for k = 0.65 44 T h e coefficients F,j, Sij, Tij and D{j can be obtained from the M O D FET waveequation using th e m ethod developed by Ziel [20]. The procedure used and the obtained F,j, Sij , TtJ- and Dij coefficients for each Y param eters Yij are given in A ppendix C. N ote that these parameters are all dependent on the norm alized bias param eter k = v v*‘V t ■ The elem ents of the optim al second-order RC circuit can now be obtained by inverting the system of Equations (3.5) so as to express R u j , R 2 ij, Cuj, and C 2 ij in term s of rltj, r 2 ij, and r3tj. R \ i j — ik T\ij _________ (Cj jl ~3j j ~ T \ j j T 2i j ) 2_______ r lij(CfjTlj - CijTujTxj + Tfc) C U j k = C i j T 3 ij C 2 ij = C j j T3 ij ~ ~ T l i j T2 ij C i j T l i j T 2 ij + ^ • j T3i j ~ T U j T 2ij T h e m agnitude and phase of each Y-param eters are p lotted in Figure 26 - 33 to dem onstrate th e graceful degradation. The Y-param eters are obtained w ith the second-order equivalent circuit (dashed-dotted line, EQ ), th e exact solution (plain line, E X A C T ), the frequency power series (dashed line, P O W E R ), and fourth order iterative solution (dashed dashed line, ITE R ). A s can be seen in figures, th e secondorder equivalent circuit exhibits a more graceful degradation compared to th e fourthorder iterative solution and power series solution. The frequency range of validity of this equivalent circuit can be evaluated by calculating the frequency /s%(Y'j) for each param eter Yij for which a 5% error is obtained. Figure 34 - 37 show /s%(Tij) for each param eter Yij. O ne observes that the circuit holds to a much higher frequency than 45 Yll ERROR 5 PERCENT 60 50 40 30 20 IT E R 4tH 10 0 IT E R 2nd l****"-**"T * ^ —-fc-» -i—AU 1— POWER K F igure 34: Plot o f / 5% (Kj)//o for Y\\ as a fu n ction of the biasing param eter k. Y12 ERROR 5 PERCENT EQ IT E R 4 th _ITER 2nd ~*1 ' 1 — •*— POWER K Figure 35: Plot o f /s%(VIj)//o for Yu as a function of the biasing param eter k. 46 Y21 ERROR 5 PERCENT 80 60 F / F 0 40 ITER 4 th 20 IT E R 2nd 0 POWER Figure 36: Plot of /s% (Y j) /fo for Y2\ as a function of the biasing param eter k. Y22 ERROR 5 PERCENT 40 30 F / 0 F 20 ITER 4 th 10 0 POWER 4 th 0 0.2 0 .4 0.6 0.8 1 Figure 37: Plot of / 5 % (Y j)//o for Y22 as a function of the biasing param eter k. 47 Gate Drain 111 R 211 ^T ' Cm Source 122 Source Figure 38: Approxim ate second-order sm all-signal equivalent-circuits for the intrinsic M O D FET any other m odel for all k values except for >2i- /s%(F2i ) / fo for Y 2 1 is the sam e as th a t of first-order RC equivalent circuits at k = 0.9. As m entioned above tw o different topologies for th e second-order equivalent circuit are possible (see Figure 2 5 ). The transfer functions of both circuits for sm all-signal analysis in th e frequency dom ain are th e sam e, even though different RC elem ents are used. For th e same b ias condition, th e values o f R iy and C iy in circuit (b) are th e same as th at of Ry a n d Cy in th e optim al first-order RC circuit. This m eans th a t circuit (b ) extends th e frequency range by adding the R2ij, C2y circuit to the in itial RC circuit. However in circuit (a) th e sum of capacitors, Cu j + C^y, give the capacitor C y o f the optim al first-order R C circuit. T h e small-signal analysis does n ot differentiate between (a ) and (b), how ever circuit (a) will be preferable to circuit (b) since circuit (a) is a m ore physical representation o f the distributed channel for large-signal analysis. T he resulting equivalent circuit for long gate length device is 48 show n in Figure 38. 3 .4 T h e V e lo c ity -S a tu r a te d M O D F E T E q u iv a len t C ircu it T h e sm all-signal model presented above for th e intrinsic M O D FET holds only for th e region of th e channel for w h ich the gradual channel approxim ation (G C A) holds. H owever in saturation it b eco m es necessary to account for the contribution of the b u ilt-in potential. A more co m p le x equivalent circuit results in which the equivalent circu it introduced for the M O D F E T wave-equation is now just a subcircuit. L et us dem onstrate this approach for th e velocity-saturated M O D FE T waveequation. In th is conventional M O DFET m odel the F E T channel is divided into th e GCA and saturation regions of length X s = Lg — £ and £ respectively. In the saturation region the electron velocity is assum ed to saturate (to a value v s) while th e GCA is failing. The channel potential in the saturation region is then assumed to b e supported uniquely by t h e electron distribution in th e channel. T h e derivation of the equivalent circuit for saturation region is given in Appendix D , which is based on the exact solution. B y combining tw o equivalent circuit, one of G C A region and the other o f saturation region, one can obtain Figure 39. T h e total Y-parameters Y i j( s ) in terms o f the Y-param eters of the GCA region Yij(g) of reduced gate length X s = Lg — £, are obtained as follows Yn (sat) = Yn (s ) + V M S , + Ynisat) = Yn{g)l, + 1 + r » (j)& (« ) ~ (J )(1 - ~ U ^)Yn(g)) - * < « ) * ■ ( ,) ) 49 >d I Jy. ——o—'V'vAr—— -°d Rdd R od CD CD X go YsVd. * v . © Y^)Vd6 Cgg Y,a(®) Cdd 6 6 >d6 ■JOT, icoCnd 1 +jtORg(JCg(j Y2i(co) = g m + foCda 1 +jmRdgCdg Figure 39: First-order non-quasi-static equivalent circuit for the velocity-saturated M O D F E T wave-equation * 22(sat) = -, v e"Jur' 1 -f Y2 2 {g)Zs {ui) where t , = v , /£ is th e transit tim e of the saturation region, Z,(u) an im pedance specified below and 7. = 1 - S , = 6 , and y s tw o constants given by x + p lDc(A w ith A = B = 2X 5(1 - ks) ( 2 k, - k 2 )(Vas - VT) 4 X 5(1 - k , f G dosi^k, N ote th a t k, = — k f ) 2(VGs ~ Yt ) Vc s ( X s ) /( V g s — Vt ) and Gdos = ^ C gWg(VGs — V t ) / X s are values used for k and the drain conductance gd respectively, in th e G CA Y-Param eters Yij(g) given in section 3.2. 50 MAGNITUDE OF Yll xlOE-3 600 I EQUI POWER Y I N 400 M H 0 200 F/FO Figure 40: Comparison of the am plitude of Yu for Vd s = 3 F and Vg s = 0V. PHASE OF Y l l 90 70 50 EXACT EQUI 30 10 POWER 0 2 4 F/FO 6 8 Figure 41: Comparison of the phase of F n for Vds = 3 F and Vgs = OF. 51 MAGNITUDE OF Y12 xlOE-3 20 / POWER 16 EXACT EQUI 12 8 4 0 0 2 4 F/FO 8 6 Figure 42: Comparison of the am plitude of Yi2 for Vds = 10 3 V an d Vg s = O V . PHASE OF Y12 300 200 D E G R E E POWER 100 EXACT -100 F/FO Figure 43: Comparison of the phase of Y12 for Vds = 3F and Vgs = 0V . 52 MAGNITUDE OF Y21 200 160 I POWER Y N 120 M H 0 EXACT EQUI F/FO Figure 44: Comparison o f the am plitude of F2i for Vd s = 3F and Vg s = OF. PHASE OF Y21 400 300 D E G R E E POWER 200 100 EQUI EXACT -100 F/FO Figure 45: Comparison o f the phase o f Y2 1 for Vds = 3 F and Vgs = OF. 53 xlOE-3 MAGNITUDE OF Y22 2U / POWER EXACT EQUI F/FO Figure 46: Comparison o f the am plitude of K22 for Vd s = 3 V and Vgs = OV. PHASE OF Y22 400 POWER 300 G R EXACT 200 EQUI 100 F/FO Figure 47: Com parison of the phase of F22 for Vbs = 3V and Vgs = OV. 54 T h e im pedance Z a(u) is approxim ated by a first order RC network providing th e correct second-order frequency power-series expansion = £.1 + i1 r+ ?j u rC aRDa 2 (3-9) with p i DCt B - \ p e R si = R s2 = „ _ 3(1 + P I d c U ) 3 (1 + p i p p i A ) ~ 8 1 + PI dc ^A 2pe2 9 Ta pi2 using P = 1l t xv aWgda. T h e resulting equivalent-circuit provides an optim al first-order non-quasi-static equivalent-circuit adm itting th e correct second-order frequency power expansion as well as a graceful degradation. T his is dem onstrated in Figure 40 - 47 for an intrinsic M O D FE T with th e parameters given in Table 1 in Chapter II and for an intrinsic bias o f Vd s = 3 V and Vg s = OV. The phase and am plitude of Y-param eters versus frequency calculated using this first-order RC equivalent-circuit (dashed-dotted line, E Q U I), the exact solution (plain line, E X A C T ), and th e frequency power-series ap proxim ation (dashed line, PO W E R ) are compared in figures. The optim al first-order RC m odel (EQ U I) is seen to hold to a m uch higher frequency than th e frequency power-series approximation (P O W E R ). C H A PTER IV PR E D IC T IO N OF THE MICROWAVE C H ARACTERISTICS OF M O D FET’S 4.1 E x tr a c tio n o f p a ra m eters for t h e ac m o d e l The intrinsic ac model developed relies on the m aterial, device and bias param eters ei, L g, W g , (i, v„, Cg, V t, V g c {X s), I or X s , VG , I d c , which sh ou ld all be obtained directly from the dc-m odel. As was m entioned in Chapter II, it is necessary to use an accurate dc m odel to m o d e l the m icrow ave performance. The dc m odel used is th e dc m odel recently reported [24], except for th e saturation voltage 14 = Vd s —Vcs(Xs) which in accordance w ith the proposed saturation picture is derived from Equation (2.3) to be 14 = I d c X l H ^ d ' W g V , ) - ECX S (4.1) This dc m od el provides tw o features: a field dependent m obility (see Equation (4.3) below) and a non-linear charge control Tis = nso [(1 (1 — - a )) + atanh Vq c - V gm' 14 which perm its one to o b ta in an im proved fit of th e dc-characteristics (nso, (4.2) Vg m , a , and 14 are used as fittin g parameters). Since the ac-m odel introduced in C hapter II 55 56 relies on a constant m obility /z (in th e GCA region), constant gate capacitance Cg, and constant threshold voltage Vt an extraction theory is required. Following the approach described in reference [10] th e m obility is given by, ** = 1 + E ( 0 )/E ! where Ei = E cf (/j E c/ v a — 1). N ote th at the m obility used by th e ac-m odel is the chordal m obility and not th e differential m obility (see Gunn [28]). In reference [10] the threshold voltage was calculated using VT = V g c ( X s ) x 14 ' ^ L(l-°) ( V c c ( . X s ) - Vc m V cosh2 ^ G c i X s ^ G M ^ (4.4) and the 2D E G capacitance using = „n,„(.1 - ° ) ^ Vi F g c (0) — Vgm V! (4.5) T hese expressions are only valid for sm all gate-to-channel voltages. At large gate-tochannel voltages the high-bandgap m aterial between the gate and th e 2DEG channel is no longer fully depleted. This leads to the saturation of th e 2DEG concentra tion in Equation (4.2), an effect reflected in turn in Equations (4.4) and (4.5) by th e reduction of both the threshold voltage and th e 2DEG capacitance. However due to the large RC constant of the depleted parasitic M ESFET channel, th e charge distribution (electrons and ionized donors) in the high-bandgap m aterial does have tim e to respond at high-frequencies. T h e 2DEG capacitance at large gate-to-channel voltages is then lim ited by the m axim um 2DEG capacitance Cgmax = + Ad) , 57 where d is the gate to channel spacing and A d is a constant arising solely from the variation of th e Fermi level w ith the 2DEG concentration [29]. T he gate-to-channel voltage Vgco a t which this ta k e s place is sim ply obtained from C*3(Vgco) = C a m ax- For gate-to-channel voltage larger than Vgco the threshold voltage selected is then given by Vrmax = Vt(Vgco). T h e use of th e m axim um 2DEG capacitance and m ax im um threshold voltage at m icrow ave frequencies for large gate-to-channel voltages supports the n otion that the R F transconductance gxf (R F) can be larger than the dc transconductance gM{dc). A greatly improved fit of the scattering param eters results from this choice. 4 .2 C o m p arison o f t h e m ea su red and c a lc u la te d d a ta In order to te s t this m icrowave model, th e theory is applied to tw o one-micron g a te length devices; an nAlo. 2 sGaO' 7 5 A s / iA lo . 2 sGao. 7 5 A s / G a A s (35 0 /3 0 /1 0 ,0 0 0 A ) M O D FET (d ev ice 2045) and an nAl.i 5 Gao,s 5 A s /iA lo .i 5 Gao,s 5 A s / I n o . 2 Gao,&A s / G a A s (3 5 0 /3 0 /1 5 0 /1 0 ,0 0 0 A) pseudom orphic M O D FE T (device 2379). T h e data are ob tained from U niversity of Illinois at Urbana-Champaign. The device parameters u sed for fitting th e IV characteristics of th e devices are shown in Table 2. The com parison of the calculated and m easured IV characteristics are shown in Figures 48 and 49. The device parameters for device 2045 are the sam e as reported in [10] except for ds , R a, and Rd- A smaller value for d„ (closer to the equilibrium channel width, see [22]) is used. T h e nonlinear source and drain resistance m odel reported in [30] is used for R s and RdThe equivalent circuit used for the extrinsic m odel is shown in Figure 50. Dis- 58 Table 2: Device param eters for th e dc characteristics of th e A lG aA s/G aA s M O D FE T (2045) a n d th e G aA lA s/In G aA s/G aA s pseudom orphic M O D FE T (2379) Parameters Lg wg V va Ec n so Vj V gm gate length (fim) gate width (fim) m obility (cm / V . s e c ) Saturation velocity ( m / s e c ) Critical Field ( K V/cm ) ( cm -2 ) 2 (V ) (V ) a d Ad ds Rs Rd C\ C2 gate to channel spacing (A) see [29] (A) channel width in saturation (A) low field source resistance (fl) low field drain resistance (fl) channel dielectric constant channel dielectric constant 2045 2379 1 1 290 290 4400 5600 3.45 x 105 3.5 x 105 10.9 31.2 1.02 x 1012 1.0 x 1012 0.3 0.19 0.15 -0.12 0.5 0.5 380 380 50 50 1500 500 3 2.7 6 2.7 13.1 eo 12.44 e0 12.2 eo 12.65 e0 59 I V CHARACTERISTICS x lO E -3 VG=0.7 VG=0.5 30 — VG=0.3 " VG=0.1 VOLTAGE Figure 48: Measured (solid lines) and calculated (dotted lines) IV characteristics of th e A lG aA s/G aA s M O D FET 2045 IV CHARACTERISTICS x lO E -3 VG=0.7 60 VG=0.5 VG=0•3 40 VG =0.1 20 V G = -0 .1 V G = -0 .3 0 VOLTAGE Figure 49: Measured (solid lines) and calculated (dotted lines) IV characteristics of the pseudom orphic G aA lA s/In G aA s/G aA s M O D FET 2379 60 CGd Gate Drain GS Source Figure 50: Equivalent circuit for th e extrinsic M O D FET tributed effects along th e gate w idth are not included, as they were found to be sm all for the gate w idth and frequencies considered. T h e value of the parasitics used for calculating th e extrinsic Y parameters are shown in Table 3. The gate to source and gate to drain Cgd Cgs fringe capacitances were estim ated to be 50-70 fF (see [21]). T h e bond inductance at the gate, source and drain term inals and gate resistance R g are obtained from deem bedding. T he source and drain resistances used are obtained from the dc model. The scattering param eters were m easured from 2 GHz to 18.4 GHz for 5 bias conditions for device 2045 and 13 bias conditions for device 2379. The extrinsic Y param eters, including parasitics, were calculated from the ac- 61 BIAS CONDITION VGS = 0.08 VDS = 0 . 5 S12 x 10 Figure 51: M easured (solid lin es) and calculated (dotted lines) scattering param eters for Vqs = -08 V and Y d s — 0-5 V for the A lG aA s/G aA s M O D FE T 2045. UNILATERAL GAIN 30 VGS = 0 .0 8 VDS = 0 .5 A I N (dB) 10 0 1 - 10 FREQUENCY (GHz) 100 Figure 52: M easured (solid lin es) and calculated (dotted lines) unilateral power gain for VGS = .08 V and Yds = 0.5 V for the A lG aA s/G aA s M O D FET 2045. 62 BIAS CONDITION Figure 53: Measured (solid lines) and calculated (d otted lines) scattering param eters for VGS = .08 V and VDS = 0.75 V for the A lG aA s/G aA s M O D FE T 2045. u n il a t e r a l g a in 30 VGS = 0 .0 8 VDS = 0 .7 5 20 10 0 1 10 FREQUENCY (GHz ) 100 Figure 54: Measured (solid lines) and calculated (d otted lines) unilateral power gain for V g s = -08 V and Vbs = 0.75 V for the A lG aA s/G aA s M O D FE T 2045. 63 90 b ia s condition VGS = 0.08 -]50 F ig u r e 55: M easured (solid lin e s ) and calculated (dotted lin e s ) scattering parameters for VGS = .0 8 V and VDs = 1-0 V for t h e A lG aA s/G aA s M O D FET 2045. u n il a t e r a l g a in 30 VGs = 0 .0 8 VDS = 1 20 10 0 1 10 FREQUENCY (GHz ) 100 F ig u r e 56: M easured (solid lin e s ) and calcu lated (dotted lin e s ) unilateral power gain for Vgs = .0 8 V and Vps — 1-0 V for t h e A lG aA s/G aA s M O D FET 2045. 64 BIAS CONDITION Figure 57: M easured (solid lines) and calculated (dotted lines) scattering parameters for Vgs = -08 V and VDS = 3.0 V for the A lG aA s/G aA s M O D FE T 2045. UNILATERAL GAIN 30 VGS = 0 . 0 8 VDS = 3 20 10 0 1 10 FREQUENCY (GHz) 100 Figure 58: M easured (solid lines) and calculated (dotted lines) unilateral power gain for V g s = -08 V and V d s = 3.0 V for the A lG aA s/G aA s M O D FE T 2045. 65 BIAS CONDITION Figure 59: M easured (solid lines) and calculated (dotted lines) scattering param eters for V d s — 3 V and V g s = - 0.15 V for th e G aA lA s/InG aA s/G aA s pseudom orphic M O D FE T 2379. UNILATERAL GAIN 30 VGS = - 0 . 1 5 VDS = 3 G A I N 20 <dB) 0 1 10 FREQUENCY (GHz) 100 Figure 60: Measured (solid lines) and calculated (dotted lines) unilateral power gain for V d s = 3 V and V g s — -0.15 V for th e G aA lA s/In G aA s/G aA s pseudom orphic M O D FE T 2379. 66 BIAS CONDITION Figure 61: M easured (solid lines) and calculated (d o tte d lines) scattering param eters for V d s = 3 V and V g s = -0.08 V for th e G aA lA s/InG aA s/G aA s pseudomorphic M O D FE T 2379. UNILATERAL GAIN 30 VGS = - 0 .0 8 G A I N (dB) VDS = 3 20 10 0 1 10 FREQUENCY (GHz) 100 Figure 62: M easured (solid lines) and calculated (dotted lin es) unilateral power gain for V d s = 3 V and V g s = -0.08 V for th e G aA lA s/In G aA s/G aA s pseudomorphic M O DFET 2379. 67 BIAS CONDITION Figure 63: M easured (solid lines) and calculated (d o tted lines) scattering param eters for V d s = 3 V and Vgs = 0.25 V for th e G aA lA s/InG aA s/G aA s pseudom orphic M ODFET 2379. UNILATERAL GAIN 30 VGS = 0 .2 5 G A I N (dB) VDS = 3 20 10 0 1 10 FREQUENCY (GHz) 100 Figure 64: Measured (solid lines) and calculated (dotted lines) unilateral power gain for V d s = 3 V and V g s = 0.25 V for th e G aA lA s/InG aA s/G aA s pseudom orphic M O D F E T 2379. 68 BIAS CONDITION F igure 65: M easured (solid lines) and calculated (d otted lines) scattering param eters for V d s = 3 V and V g s = 0.56 V for the G aA lA s/In G aA s/G aA s pseudom orphic M O D FE T 2379. UNILATERAL GAIN 30 VGS = 0 . 5 6 VDS = 3 20 G A I N (dB) 10 0 1 10 FREQUENCY (GHz) Figure 66: Measured (solid lines) and calculated (dotted lin es) unilateral power gain for V d s = 3 V and V g s = 0.56 V for the G aA lA s/In G aA s/G aA s pseudom orphic M O D FET 2379. 69 T able 3: M icrowave parasitics for the A lG aA s/G aA s M O D FE T (2045) and the G aA lA s/In G aA s/G aA s pseudom orphic M O D F E T (2379) 2045 Parameters 2379 C GS Gate to source parasitic capacitance (fF ) 50 70 C Gate to drain parasitic capacitance (fF ) 50 50 RG Gate resistance (Q) 4.16 5.4 LG Gate bond inductance (nH ) 0.35 0.3 L5 Source bond inductance (nH ) 0.08 0.07 LD Drain bond inductance (nH ) 0.33 0.36 gd m odel for the sam e bias conditions and then converted to scattering parameters for comparison with th e measured data. T h e scattering param eters and unilateral power gain versus frequency for four bias conditions are show n in Figures 51 - 58 for device 2045 and Figures 59 - 66 for device 2379. The scattering parameters for Vg s = -08 V and Vd s = 0.5 V, 0.75 V , 1 V, and 3 V are shown in Figures 51, 53, 55, 57 respectively for the A lG aA s/G aA s M ODFET. The scattering param eters for Vds = 3 V and Vgs = -0.15 V , -0.08 V , 0.25 V, and 0.56 V are shown in Figures 59, 61, 63, 65 respectively for th e G aA lA s/InG aA s/G aA s pseudomorphic M O D FE T . In th e polar and Sm ith chart p lots, the solid line is used to represent the m easured data and the d otted line is used for the calculated data. Figures 52, 54, 56, 58, 60, 62, 64, and 66 show the unilateral power gain given by [31] 70 Table 4: D eviation of calculated S param eters from th e m easured d a ta for all bias conditions for the device 2045 ( d a = 1500 A , C g d = 50 fF, and C g s = 50 fF) Vi. AS„ <1 0.08 0.25 0.09709 0.08 0.5 0.08 ^ Vg. a s 22 A f7(d B ) 0.12828 0.04286 0.08892 2.41553 0.12094 0.41329 0.03103 0.12960 0.60579 0.75 0.14212 0.61521 0.03263 0.21576 0.61067 0.08 1.0 0.12278 0.61916 0.04603 0.25717 1.45705 0.08 3.0 0.30447 1.32780 0.07536 0.28605 1.16160 rH cs A 5 i2 __________IF21 —F12 1_________ , . gx “ 4 (R e (y n )R e(F22) - R e ( y 12)R e(F 21)) 1 ' In these figures, the solid lin e represents the value o f U obtained from the m easured data and th e dotted line th e value of U calculated from the m icrowave model. To estim a te the perform ance of th e m odel the error obtained for all m easured bias conditions is sum m arized in Table 4 and Table 5 for devices 2045 and 2379 respectively. Since scattering parameters are norm alized quantities, th e error used is 1 N A Sij = "T7 ) ^ | Sijdaia (cj,') -'V i=l Sijcalculatedi^i) | (4.7) As can be seen in Figures 52, 54, 56, 58, 60, 62, 64, and 66 th e calculated and measured unilateral gain plotted on a log scale differ by a few dB over the entire frequency range. The error used for th e unilateral power gain is therefore 1 N AUdB = I l0log(Udata(Ui)) ~ 10log(Ucalculated^)) \ (4.8) 71 Table 5: D eviation of calculated S param eters from th e measured d a ta for all bias conditions for the device 2379 ( d s = 500 A, C g d = 70 fF, and C g s = 50 fF) vg. Vu A 5U <1 0.07 0.25 0.10488 0.07 0.5 0.07 A522 AC7(dB) 0.09403 0.06810 0.12519 3.25594 0.13381 1.01989 0.06220 0.54647 3.10049 0.75 0.14001 0.93923 0.03592 0.43403 1.10848 0.07 1.0 0.14074 0.67644 0.03820 0.32584 2.06435 0.07 2.0 0.15359 0.62508 0.05199 0.26670 3.08550 -0.15 0.5 0.14178 0.70650 0.03433 0.34632 1.24863 -0.08 0.5 0.12351 0.83907 0.04749 0.43257 0.99085 0.25 0.5 0.11836 0.16675 0.05722 0.10181 4.08520 0.5 0.5 0.13181 0.06170 0.06089 0.15007 0.87395 -0.08 3.0 0.14849 0.67419 0.04942 0.26523 2.96939 -0.15 3.0 0.15725 0.72022 0.04232 0.27461 2.60278 0.25 3.0 0.12164 0.89009 0.06550 0.24644 1.71173 0.56 3.0 0.09435 1.42184 0.07672 0.27800 1.19325 i-M a s 12 72 The error averaged o v er all bias con d ition s is given in Tables 4 and 5 for devices 2045 and 2379, respectively. A minimum average error on the unilateral gain of 1.25 dB and 2.17 dB is obtained for all bias conditions for devices 2045 and 2379 respec tively. N ote that an error o f + 2 dB in th e unilateral gain corresponds to an error in the calculated f max given by f m a x { d a t a ) / f max(theory) = 1.26. This error is reasonable considering that apart fro m ds and C q S an d C g d , all o f the m odel param eters are extracted from the dc m easurem ent or th e microwave deem bedding. A m ain source of error originates from the d c drain conductance and transconductance, which are only approxim ately fitted w ith th e present extraction techniques. This em phasizes the im portance of an accurate d c model. It is believed th a t the error obtain ed could be reduced w ith an improved d c parameter extraction technique. A dditional substantial error m ight also originate from calibration and deem bedding. 4.3 D iscu ssio n It is now relevant to elab orate on the physical insights gained from th e ac-model reported for the saturated M ODFET. F ir st let us ju stify the need for an ac-m odel for th e saturated MODFET. As m entioned the p reviou s solutions for the wave-equation were only valid up to the edge of saturation which occurs w hen the channel field is th e critical field E c. It is important to n o te that one ca n n o t infer th e performance of the device in saturation from the perform ance of the d e v ic e at the ed g e of saturation. Indeed once th e device is driven deeper in saturation, one observes a substantial increase of the transconductance, a stron g reduction o f th e drain conductance and a reduction of 73 th e effective channel length. These effects drastically modify, and in fact im prove the device performance. In a recent paper [21], Rohdin reported an ac-m odel for the saturated M O D FE T (shown in Figure 2 in Chapter I). In his m odel the GCA region is described b y an equivalent circuit based on the Y-param eters derived for the unsaturated M O D F E T [10]. To account for the saturated region, Rohdin introduces in series in the drain term inal, a conductance gdd> lim iting th e output conductance. A capacitance C d d ' in parallel with gdd' is introduced to account for the charge m odulation in the saturation region. U sing his model Rohdin develops a m ethod perm itting one to extract from th e measured Y-param eters the source resistance, fringe capacitances, gate len gth and effective saturation velocity. The physical equivalent circuit m odel proposed by Rohdin em phasizes the im portance of th e saturation region. Interestingly the GCA region in a saturated M O D FE T does not behave, at highfrequencies, as the GCA region of th e unsaturated M O D FET. This is verified by developing a m odel relying on the Y-param eters derived for the unsaturated M O D F E T (using for gate length: X s ) and th e exact solution of the wave-equation in the saturation region. The continuity of th e total current and voltage was enforced at th e floating boundary. T his approach failed com pletely, and th e S-parameters cal culated were uncorrelated to the m easured ones, for one cannot use the unsaturated Y-param eters in the GCA region of th e saturated M O D FET. Indeed these u n sa tu rated Y-param eters are derived with th e boundary conditions Vgco(Xs-) = vgc0( X s + , t ) (4.9) 74 Table 6: D eviation of calculated S parameters from the m easured data for device 2045 <ASu> <AS21) (A S 12) (A S 22) (A f/)(d B ) d s = 500 0 .1 5 2 2 0 .7 4 1 8 0 .0 4 8 4 8 0 .2 5 7 4 1 .8 8 9 8 ds = 0 .1 5 7 4 0 .6 2 0 7 0 .0 4 5 5 8 0 .1 9 5 5 1 .2 5 0 1 0 .1 6 7 3 0 .5 8 0 9 0 .0 4 4 2 9 0 .1 8 9 7 1 .5 2 6 0 0 .3 5 4 9 1 .2 7 3 8 0 .0 6 1 9 9 0 .3 7 8 3 3 .0 4 2 6 ds = A 1500 A 2500 A Cgs — C g d , v g c i ( X s —) = — 0 V g C2 ( X s —) = 0 ( 4 .1 0 ) which are incorrect for the saturated M O D FET. Instead it m ust be assumed, as is done in A ppendix A , that each voltage com ponent is continuous: wflCo,i,2(-Xs—) = u3co,i,2(-^5 + ) . Note th a t these correct boundary conditions are naturally enforced by a circuit im plem entation such as th e one proposed by Rohdin [21]. Let us now consider in more d etail the im pact of ds upon th e scattering parameters. ds is in ten d ed to represent the effective channel w idth in the saturation region. T he best fit of th e IV characteristics at all bias conditions was obtained for d„ = 2 5 0 0 A and ds = 1 5 0 0 A on devices 2 0 4 5 and 2 3 7 9 respectively. In Tables 6 and 7 it is compared th e error on the scattering parameters and unilateral gain averaged over all bias p oin ts obtained for different values of ds. It is noted from Tables 6 and 7 that a reduced error results from using a value of da = 1 5 0 0 A and da = 5 0 0 A which is closer t o t h e equilibrium channel w idth (see [2 2 ]). This supports the concept that th e RF extrinsic drain conductance g o ( R F ) can be sm aller than the dc extrinsic drain 75 Table 7: D eviation of calculated S param eters from the m easured data for device 2379 A da — 500 A ds = 1000 A ds = 1500 A ds = 300 C g S = C g d = 0 <ASU > (A S 21) <ASia) (A S22) (A C /)(dB ) 0.1323 0.7153 0.05459 0.3133 2.3031 0.1316 0.6796 0.05310 0.2918 2.1762 0.1315 0.6114 0.05080 0.2577 2.206 0.1324 0.5654 0.04975 0.2390 2.3746 0.2531 1.8118 0.09147 0.6197 5.2966 conductance gD{dc). T h e difference betw een the dc drain conductance <jfn(dc) and RF drain conductance g o ( R F ) is thought to originate in som e devices from traps present in the buffer [32]. It is p oin ted out in S ection 4.1 th at the RF transconductance g \ i { R F ) was larger in this device than the dc transconductance gM{dc). This ac-m odel does not account in itself for either the large charging tim e constant of th e buffer trap s or the large RC tim e constant in the w ide bandgap m aterial. These effects must therefore be reflected in the choice m ade for th e parameters o f the ac-m odel. For exam p le for a device with negligible g a te leakage (perfect insulated gate) the intrinsic drain conductance Gds and th e intrinsic transconductance G ga in th e ac-model should be related to the extrinsic R F drain conductance g n ( R F ) and extrinsic RF transconductance gM(RF) by Chou 76 and Antoniadis relationships inverted [33] 1 + Gds ( R a 4 - R d ) + G gaR „ = 9 d ^R F ) (4 ,U ) 1 + G da{ R a + R d) (4 '12) + G gaR s = 9 m ^R F ) Finally let us n o te the im pact o f the parasitics on the microwave characteristics. Tables 6 and 7 show that the error in the unilateral power gain increases by 2-3 dB when we set C g s = C g d = 0. Therefore parasitics play an im portant role prediction of the d ev ice performance. in the C H A PTE R V UNILATERAL PO W ER G AIN R ESO NANCES A N D j T “ f m a x O R D ER IN G 5.1 In tr o d u c tio n In the velocity-saturated model th e FE T is divided into two regions [34], a gradualchannel region (GCA) and a saturation region. In the gradual-channel region [35] the carrier concentration is determ ined by 0 q n a( V G c ) = for V g c ^ V r C 9 {V gc — V r) q n SM A x for for V t < V g c V g sm a x < < V g sm a x V sc Constant mobility fi, g a te capacitance Cg and threshold voltage Vj are assumed in this region. In the saturation region, the Poisson equation is solved in th e direction parallel to the channel, assuming a constant channel width d3. T he velocity of the electron is assumed to have saturated to vB in this region. T h e boundary between the GCA region and the saturation region is the position in the channel where th e channel electric field reaches th e critical field E c = This sim ple m odel can handle both long and short channel FE T s. In th e long-channel m ode the transconductance varies linearly w ith gate voltage. In the short-channel m ode the transconductance saturates 78 to a constant value for large gate voltages. The sw itch from the lon g channel m ode to the short channel mode is determ ined b y the ratio a = ( E cLg) / ( V a s — Vr). Indeed one can easily verify for th e velocity-saturated M O D FET m odel that th e ratio o f th e drain current at the onset o f saturation Idc(sat) (when th e channel field at the drain is equal to th e critical field E c) by th e drain current in pinch-off I dc(pinch) for th e sam e V g s voltage is given b y Idc(sat) I / ■ = 2a h s ----------w - Idcipmch) '' 1 + 4 2- 1 * = Wl(a ) (5 J ) Sim ilarly the ratio of the transconductance gm(sat) at th e onset of saturation by th e m axim um transconductance gm,MAx{sai) is 9m( s a t ) _ a-> _ = 9m,MAx(sat) ^ 1 + (a -l)2 (5 .2) The weight functions W i ( a ) (plain lin e) and w2(a - 1 ) (dashed lin e) are plotted in Figure 67 versus a and a -1 respectively. From the tangential dashed-dotted lines shown in Figure 67 it is seen that the transition from long- to short-channel m od e occurs for a = 1 / 2 for the saturation current ratio and a -1 = 1 for th e saturation transconductance ratio W2 - N ote that th is criterion is o n ly applicable in the range o f validity of th e GCA approximation. A G C A region is always expected in the case of high-aspect ratio (Lg/ d ) F E T where d is the gate to channel spacing. Small aspect ratio FETs, where two-dim ensional field effects are im portant over th e entire gated channel are not considered here. The G C A approximation will also fail in conventional high-aspect ratio FET when th e Vgs voltage reaches V g s m a x and th e channel charge n s saturates to tism a x - In a M O D FET th is occurs w hen th e parasitic M ESFET turns on [36]. The ratio a is therefore more correctly defined by 79 1 0.8 N E I G H T 0.6 0 .4 0. 2 0 0 2I 4 6 ALPHA AND 1 /ALPHA 8 10 F igu re 67: Current and transconductance w eight functions tw i(a) (plain lin e) and w2(a _1) (dashed line) p lo tte d versus a and a -1 respectively. a E c^g CL (5.3) m i n \ y GSM A X , I'gs] - V t ' As a consequence it is not possible i n practice to turn on th e short-channel m o d e in a long gate-length F E T (e.g., lo ^ ). 5.2 Long a n d Short C h a n n e l M o d e and th e ac-cu rren t G a in The high-frequency small-signal characteristics o f the velocity-saturated M O D F E T model w ill be stu d ied in both th e short and lo n g channel m ode. O bviously the transport picture u p o n which this ac-m odel is based will in practice break dow n for frequencies corresponding to the e n e r g y relaxation tim e ( ~ 1 T H z) and the m om entum relaxation time ( ~ 10 THz). 80 ALPHA VS FT 1000 100 (GHz) L g**2 0.01 0.001 0. 01 JU1L ALPHA 100 1000 Figure 68: Variation of the unity current gain cutoff frequency / y versus g a te length Lg plotted versus a = E cL 9/( V g s — Vr) in log scale for an intrinsic M O D FE T with V g s = 0, 0.1, and 0.2 V and V d s = 1 V . T he exact solution derived in Chapter II is used to calculate th e unity current-gain cutoff frequency /y ( m t ) of the intrinsic M O D FE T versus gate length Lg for th e gate to source voltages Vg s = 0 , 0.1, and 0.2 V and a drain to source voltage of Vos = 1 V. T h e M O D FET param eters used are given in Table 8 . These unilateral current-gain cutoff frequencies f r i i n t ) are plotted versus a in log scale in Figure 68. / r is defined here as the frequency at which we have , . \ 21 T |y 2 i(<*r)| _ M ^ r ) | |!/ii(w t)| \z-n{u}T)\ . f r i i n t ) increases with shrinking gate length ( a ). The increase of /y (in t ) switches from the 1/ L 2g law of the long channel FE T to th e 1 / L g law expected forth e short channel FE T. This results from the saturation o f the transconductance du e to the 81 Table 8 : Device parameters for th e intrinsic short-channel M O DFET value Parameters gate width (/zm) threshold voltage (F ) mobility ( cm 2jW .sec) saturation velocity ( m / s e c ) gate to channel spacing (A) channel width in saturation (A) channel dielectric constant gate dielectric constant w , VT /* Vs d ds ^1 ^2 290 -0.3 5600 1.85 x 10s 430 500 13.1 c0 12.2 e0 velocity saturation in the FE T channel. N ote that the switch from the long to short channel m ode occurs for a betw een 1 and 2 as predicted in the previous section. For the saturation velocity and m obility of Table 1 and V d s = 1 V the corner point a = 1 corresponds to a gate length of 1 and 1.66 /z for Vg s = 0 and 0.2 V respectively. If the effective saturation velocity v s were to increase with decreasing gate length L g one would have in th e sub-micron regim e an 1 / l aw with 1 < 7 < 2. Rohdin [21] has dem onstrated that despite the expected occurrence of velocity overshoot th e effective saturation velocity is essentially independent of gate length for M O D FE T s with gate length varying from 0.9 to 0.3 /z. H is analysis is based on the system atic reverse modeling of large number of FETs on different wafers. A 82 o — [jx N = -^ 2 - _ X ]2 p 22 X 22P12 Figure 69: Exam ple of th e unilaterization of a two port device b y loss-less feedback. constant saturation velocity of 1.8510s m /se c is used in this analysis. 5.3 U n ila te r a l p o w e r ga in o f th e w a v e -e q u a tio n m odel T h e analysis of the high-frequency performance of a device is typ ically done using th e unilateral power gain U derived by Mason [27]. if _ ___________ I gai ~ I2___________ 4 [Re(j/n)Re(?/22) - R e(y i2)R e(y2i)] I^21- zu (k ^ I2 4 [Re(2n )R e (222) - R e(z12)R e(22i)] U is th e m axim um available power gain (MAG) of a device once it has been unilaterized (?/i2 = z 12 = 0) using feedback techniques. Figure 69 shows a possible feedback circuit (proposed by Mason him self) to unilaterize a three-term inal device. T h e m axim um frequency of oscillation / max is defined as the frequency at w hich U is unity. / m a x is often referred to as the frequency at which a three-port device switches from active to passive. The importance o f U and / m a x for characterizing a device hinges on their invariance upon loss-less coupling (feedback and loading). 83 GAIN VS FREQUENCY 40 20 ■"s: V P;: G A I N is 0 (dB) -2 0 -4 0 -6 0 10 100 FREQUENCY (GHz) 1000 10000 Figure 70: M agnitude o f the unilateral power gain versus frequency for an intrinsic M O D F E T ( V g s = 0 V and V o s = 1 V ) with a gate length of 3 /x (dashed-dotted lin e), 1 n (dashed line), and 0.3 // (plain line). However because the feedback network required to unilaterize a device could only be achieved at a single frequency with loss-less passive com ponents, / m a x is a narrow band figure of merit. A narrow-band figure of m erit is useful in classifying transistors for the design of tuned amplifiers and oscillators. / m a x is therefore used as a R F or m icrowave figure of m erit. This is in contrast w ith f r which is a broad-band figure of m erit and is therefore more relevant for classifying transistors for the design of broad-band and large-signal circuits. Figure 70 shows th e m agnitude o f th e intrinsic unilateral gain calculated using the solution o f the velocity-saturated wave-equation for a 3, 1 and 0.3 fi gate length FET with Vg s — 0V and Vd s = IV . For sm all frequencies one observes th e usual 20 dB per 84 decade decrease of th e intrinsic Unilateral power gain. In all th ese FETs, on e observes at large-frequencies a periodic divergence of th e intrinsic \U\ and alternate regions of positive U and negative U. /„ = (n + T h e resonant frequencies are approxim ately given by for positive integer n where t s ( V g s , V d s ) = £ / v a is th e transit-tim e through the saturation region o f bias-dependent length £ ( V g s , V d s ) - Clearly these resonances are associated with the saturation region and occur for frequencies for which th e phase of the drain current phaser e x p ( —ju>Ta) is approxim ately ± 7r. The appearance of the negative U regions are also correlated w ith th e periodic development of a negative output ac-resistance. T h e unilateral power gain resonances could be an artifact of the FET m odel which assum es the existence of a constant velocity saturation region indirectly controlled by the gate. The possibility of steady-state gain at frequencies above the 20 dB /decade extrapolated f M A x {i n t ) is not however in contradiction w ith th e principle o f operation of an F E T . An F E T is a transit tim e device, and its switching speed is therefore limited by th e length of its gate. However the am plification of a steady-state signal does not convey any inform ation. For such a steady-state application, an F E T is therefore not necessarily transit-tim e lim ited. Note th at for the 0.3 fi intrinsic M O D F E T , the first resonant frequency f 0 occurs before the 20 dB /decade extrapolated fMAx{int). A resonance can also be predicted by the approxim ate solu tion of the wave-equation based upon the frequency-power series. In practical devices, lossy parasitics (see Figure 71) w ill prevent the observation of th e unilateral power gain resonances. Figure 72 show th e im pact of a source, gate 85 C gd Drain Gate GS Source Figure 71: Equivalent circuit for th e extrinsic M O D FET. C g s and C g d are the fringe capacitors o f the gate. GAIN VS FREQUENCY A (dB) -2 0 -4 0 .0 100 FREQUENCY (GHz) 1000 10000 Figure 72: Unilateral power gain versus frequency for a 0.3 fi extrinsic M O D FE T w ith parasitics resistances R s = R a = R D = 0.01 fi (plain lin e), 0.1 fi (dashed line), 1 0 (dotted dashed line), and 5 fi (dashed dashed line). 86 and drain resistances power gain of the 0.3 Rs = Rg = [/, M O D FET Rd = 0.01, 0 .1, 1 and 5 fl upon the Unilateral in the presence of th e parasitics capacitors Cgd = C g s = 50 fF (see Figure 71) and w ith bias Vgs = 0 V and V o s = 1 V. For large enough resistances the unilateral power gain exhibits a switch from the 20 dB to the 40 dB drop per decade approxim ately at the resonant frequency f 0. This is not w ithout resem blance with the perform ance of the equivalent circuit reported by Steer and Trew [37]. N ote however that here the corner frequency f 0 is above the extrinsic f M A x ( e x t)- Up to now a 40 dB per decade decrease of the unilateral power gain has not been experim entally observed/reported in M O D FE T s (or M O SFET s). It is noted that f a can be smaller than the intrinsic fMAx(int) (extrapolated w ith a 20 dB per decade slope) in a 0.3 fi M O D FET. However for realistic lossy parasitics the extrinsic / m a x (ext) is much sm aller than f 0 (see Figure72 ) and the unilateral power gain calculated from the extrinsic velocity-saturated M O D FE T wave-equation will exh ib it a 20 dB per decade decrease in the entire active range. Let us now address the issue of th e ordering of f o and / m a x • It is shown in Figure 73 th at by the use of a sufficiently large gate resistance in presence of the parasitics resistance R s = R d = 2 0 and parasitics capacitance C g s = C g d = 50 fF w ith bias V g s = 0 V and V d s = 1 V, it is possible to reduce the extrinsic unilateral power gain U while m aintaining a constant extrinsic /x ( e x f ) until / m a x (ext ) is smaller than f x ( e x t ) . Large gate resistances are indeed a problem in subm icron gate F E T ’s and m ushroom T- and L- shape gates are used to circum vent it. A s it was seen, lossy parasitics (gate, source and drain resistances combined w ith 87 GAIN VS FREQUENCY 40 20 G A I N 0 (dB) -20 -4 0 1 10 100 FREQUENCY (GHz) 1000 10000 Figure 73: U nilateral pow er gain and short circuit current gain (plain line) versus frequency for a 0.3 fi extrinsic M O D FE T using two different gate resistances R q = 5 fi (dashed line) and 25 ft (dotted dashed line). th e parasitics capacitors C g s and C g d ) play a dom inant role in shaping the highfrequency characteristics of a short-channel device (e.g ., 12 dB drop of U per octave). C H A PT E R VI CONCLUSION 6.1 C o n c lu sio n The velocity-saturated M O D FET wave-equation was derived. This ideal model is based on a piece-w ise linear charge-control m o d el and velo city field relation and ac counts for velocity saturation, and channel le n g th narrowing. T h e exact solution was obtained in term s o f Bessel functions and a n a ly tic expressions for the Y parameters in term s of a frequency power series. A sim ple RC equivalent circuit developed from the frequency power-series Yparam eters was presented for th e unsaturated intrinsic M O D FE T . T h is first-order RC equivalent circuit was found to hold to higher frequencies than th e frequency power-series from w hich it is derived or the m ore com plicated second-order iterative Y-param eters reported by [18]. Like the iterative Y-param eters this RC equivalent cir cuit features a graceful degradation of the sm all-signal param eters at high frequencies. A lthough quite sim ple the RC equivalent c ircu it selected departs from conventional equivalent circuit m odels which usually rely o n a transm ission line or RC delay for the drain transconductance and a C or series R C feedback elem en t betw een the drain and g a te and an inductor in series with the drain conductance. 88 89 In order to increase th e frequency range of validity an optim al second-order RC equivalent circuit which ad m its a fourth-order frequency power-series solution was developed. It was dem onstrated that th is equivalent circuit exhibits a graceful degra dation and holds to much higher frequencies than the first-order RC equivalent circuit, and even fourth-order iterative Y param eters of the M O D FE T w ave equation. This non-quasi-static sm all-signal equivalent circuit m odel was th en extended to th e short-channel velocity-saturated M O D FE T wave-equation. T h e resulting equiva lent circuit provided a graceful degradation of the sm all-signal Y-param eters at high frequencies. To apply th is ideal M O D FE T ac-m odel to real M O D FE T d evices a param eter extraction technique was proposed. T h e resulting m icrowave m od el perm itted one to reasonably predict the microwave characteristics (scattering param eters and u n i lateral power gain versus frequency for different bias) of a one m icron gate len gth A lG aA s/G aA s M O D FET and G aA lA s/InG aA s/G aA s M O D FET. T he param eters used by the intrinsic ac-m odel were all obtained from th e fit of th e IV characteristics alone. The m icrowave parasitic elem ents were either measured ( R g, Lq , Ls, L d ) or estim ated ( C g s , C g d )- T h e merit of th is analytic ac m odel is therefore its capacity to predict th e microwave performance from the dc characteristics. Using th e exact solution o f M O D FET wave equation, the high-frequency charac teristics of th e saturated-velocity M O D F E T wave-equation was studied. This nonquasi-static sm all-signal m od el presents som e novel features (e.g., unilateral pow er gain resonances). The observation or u se of these non-quasi-static effects in th e e x 90 trinsic M O D FE T seems how ever to require unrealistically sm all source and drain resistances assum ing this canonic ac-m odel is applicable to real devices. 6 .2 F u tu re W ork N ew m odels are required to evaluate the device parasitics and im prove the extraction techniques for the gate capacitance Cg and the threshold voltage Vr- Finally it m ight be possible to develop an extraction techniques for the m obility y and the effective saturation velocity v s, perm itting to apply this sim ple ac model to sub m icron M O D FE T s for w hich velocity overshoot and undershoot have a strong effect on device performance. The large-signal analysis is beyond th e purpose of the dissertation. However a large-signal m odel based on th e proposed first-order equivalent circuit was reported [38] recently for the long channel M O D FE T . However a large-signal m odel for the velocity-saturated M O D FET has not yet been developed. Even though the analysis for the saturation region is n o t as sim ple as for the GCA region, it should be possible to develop a large signal m odel from the sm all signal equivalent circuit m odel developed in this dissertation for th e velocity-saturated M O D FET. A ppendix A Frequency Power-Series Solution for the V elocity-Saturated M ODFET wave equation A .l C a l c u l a t i o n o f Vg c ( x = L g) T h e Poisson’s equation in the channel is given in (2.3). For convenience the equation is repeated below d?V G C dx 2 = —f3I(x,t) = —f3(Idc + (A .l) For sim plicity a new variable x' — x — X s is introduced, which make ( A .l) as d2VGC = - / ? / ( * ' , t) = - f 3 ( I dc + iejut) dx12 (A .2) T h e boundary conditions are 0) = i>Gc(a:s) v'Gc ( x ' = 0) = - E c vgc{x' = L et’s start w ith integrating (A .2) regarding to x'. d v Gc = a [ X'(T , .NJ , dvGc < ^ + ‘) ^ + - S r f x' —Phcx' — E c — (3 J i(x)dx 91 (A .3) v g c (Lg)is obtain ed by integration from 0 to £ VGc{Lg) = rLg—XS fLg—XS rx1 u g c (z s ) — E c(Lg — x s ) — ft I Idcx'dx' — ft / i(x) dx = v g c (x s — ft J Jo ) — E c(Lg —x s ) Jo Jo — f3-Idc(L g — x s ) 2 fLg-xs rx' J i(x)dxdx. (A .4) Retaining th e first order term s gives v Gc ( L g ) « Vg c ( X s ) - E J - f t l- I dce + E cx aejwt + vgc(xs ) e ^ + A v gJ e jwi + f tl dc£xaejujt (A .5) where I — Lg — X s and x' n i{x)dxdx' . The ac poten tial at x = L g is then: vge(Lg) = E cx aejut + vgc( x s )ejujt + A v gc(£)ejut + f t l dc£xaejut A .2 (A .6) C a lc u la tio n o f Vq C( X s ) T he dc poten tial Vgc ~ Vr is given in Equation (2.6). For convenience the equation is rewritten as below Vgc ( x ) - V t = (Vgs - VT)^1 + (k*a - 2 ka)^ ~ Differentiating both side of the above equation gives (A.7) 93 d2 x x=xs V out j — dx x=Xg d Vout — M i i _ i M dx M 1 - |fca) Xs (i + ( t ; _ 2 t.) -£ -)-■ /* A5 A5 x=Xs Vout * ? (i-& )2 x-=Xs Voutl 1 + fcs2 - 2fcs] - 3/2 Vou(^ ( l - ifc s) 2 (A.8) ^ s ( l ~ k3)3 where w e use Vout = v° c(X s) = “ A .3 Vg s — hr- x j ( i - k, Y (A .9) P o w e r -S e r ie s S o lu tio n o f W a v e-E q u a tio n The w ave equation in th e GCA an d saturation regions can be solved using the fre quency power series an alysis which w as first introduced by Ziel and Wu [20]. Since we are interested in the ran ge of frequency up to f max w e will use an expansion up to the second order terms. T h e frequency power-series solu tion up to th e fourth order will be shown in Appendix C without detailed calculation. The ac voltage and current, VgC(x), i ( x ) , and x, are expanded in Taylor series in powers of ju> up to the second order i(x) = iQ + juji-L + (j u ) 2i 2 (A .10a) 94 xs = x a0 + j u x ai + ( j u ) 2x a2 (A .10c) Substituting vgc( x ) and i(x) in to Equations (2.5), (2.9) and (2.10) and equating th e power of ju> yields for the G CA region - j ^ [ g ( V Gc ( x ) ) v gc0(x)} = 0 (A.11a) - ^ [ g ( V G c ( x ) ) v gcl(x)] = WgCgvgd0(x) ( A .lib ) ■ ^ \ 9 (v o c ( x ) ) v gc2(x)] = WgCgvgci ( x ) (A .11c) and for the saturation region, di0{x) dx dx di2(x) = 0 v.s 1 . _ dx (A .12) vs and d2A vsc0(a;) dx2 = - 0 i o{x) (A .13) Introducing th e new variable x' = x — X s in Equation (A . 12) gives | - v d%\ 2o dx1 di2 vs ii dx1 v2 (A .14) 95 and f^AUgcO = Z g r — 0 K <A-i5) iP&Vgcl „.- d ^ ~ = - ^ ’2 Solving th e Equation (A .14) yields io = Ci i\ — — - x -+- C 2 Vs (A .16) — *0 — ^ 72: <r/2 — X <r' -L n ?2 — + C3 2 u2 us where C \ and C2 are arbitrary co n sta n t. Substituting Equation (A .16) in to Equation (A. 15) a n d integrating from 0 to i g iv es the p o ten tia l across th e saturation boundary. Augco = A u g Cl = xn ^ * ' 3 - ^ 2 x '2 §Vs 2 (A.17a) (A.17b) (AJ7c» where w e used the boundary con d ition s AUgcO,l,2(AS) = 0 = 0 NoW th e boundary conditions are needed in order to solve th e above wave equation. The boundary conditions used at x = 0 and x = L g are Ugco(O) ~ Ugs (A.18a) 96 VgcO ^Lg) wf l d ( 0 ) — Vgg ~ ^ 302 ( 0 ) — v g c l { L g ) — v gc2 { L g ) — 0 Vds (A.18b) (A.18c) And th e following boundary conditions are used at the G C A /satu ration boundary: • Each com ponent of th e ac current is continuous at the boundary [5(V fcc(A s))usc0,i,2(As)] = io,i,2( A s + ) (A.19) • Each com ponent of th e ac channel electric field in the saturation region is set to zero at th e boundary Av'gdax2 = 0. • Each com ponent of th e boundary m otion is calculated using x,0,l,2 v „ ^ x ^ V gc0ih2( X s ) (A.20) • Each com ponent of th e ac voltage is continuous at the boundary so th a t we have ^flcO,l,2(Ls ) = PIdJXsO,l,2 + AvgcO, 1,2(0 + UscO,1,2(As) (A .21) Before th e calculation of the channel voltage for entire region, let us define the following simpler notation. Vout — Vgs — Vr L = lg V = VGc ( x ) V Vs = Vq c {x ) = Vgc ( A s ) 97 And it will be helpful for understanding the calculation that the following term s are calculated first. g(V) = G doa( l - y ) X s (A.22) SW = G doa( l - k3) X s (A .23) g'(Va) = - Gd o a (A .24) We are interested in th e ac current in terms o f vga and v da in order to calculate the intrinsic Y param eters. However th e boundary conditions are given in form of ac voltages so that the ac channel voltages have to be calculated before th e ac current can be obtained. A .3.1 C a lc u la tio n o f t>o T he first order ac voltage vq for th e entire channel can be obtained by solving the Equation (A .11a) for th e GCA region and Equation (A .17a) for the saturation region w ith the boundary conditions (A .18a), (A .18b) and G C A /saturation boundary con ditions. Integrating Equation (A. 11a) gives th e channel voltage in th e GCA region for Vo in term s of x g ( V ) v0{ x ) = d \ x + d2 where di and d2 are arbitrary constant. (A .25) d2 can be obtained from th e boundary condition (A.18a) w ith Equation (A .25) = vgaG doaX s Substituting Equation (A .26) into Equation (A .25) yields (A .26) 98 X *-f- VggGd OSX s «■<*) - ; (v ) <A -27> d\ will b e determ ined later on w h en the wave equation is solved in the saturation region. The calculation o f A v 0 in the saturation region (Equation (A .17a)) is obtained by solving th e Poisson equation (A .13) for a fixed saturation region width £, and zero ac potential vgc( X s ) = 0 and zero ac field v'gc( X s ) = 0. The relation between di and C\ can be obtained by settin g the ac current continuous at G C A /saturation boundary. The ac current at th e GC A /satu ration boundary is * o P k ) = ~ -j^[g(V)vo(x)]\x=xs = ~di and th e ac current in the saturation region is C i so that w e can find C\ = —d\. v0(x = Lg) can be found in E quation (A.21) from the zero order term w hich is u0(L3) = fHdJx o + A v 0(l) + wo(As ) (A .28) The value o f Ci (= —d \ ) can be found from this equation using th e boundary condition (A.18b). In order to obtain C \ , we have to express th e above equation in terms of C\ . Let us start w ith the calculation ^ ( A s - ) at the GC A /satu ration boundary v0( a :5 ) = d ! X s + V g y G ,J0 S X S Gdoa{ 1 —ks) X s ~ C \ X s + VgsGdosXs G d o s ( l - k a) X s . 1 yj a:0 can b e obtained by substituting Equation (A .9) into Equation (A .20) and extract ing the zero order com ponent 99 *o = X J(1 ,/ v , -k .)3 (A .30) 7oT, -----r r ^ vo ( * s ) V ^ A f tl - ifcfl)2 „ Vq( X s ) is obtained by differentiating Equation (A .27) at G C A /saturation boundary and expressing it in term s o f d\. • to ) = "j" VgsQdos^-S 5 s(v) X d = x=Xs d 'T x [g{v)\ VggGdosXs .9(V)\ x= Xs g ( V ) - x g ' jV ) d - * ~ d' T x . 4 ~ VgSGdosXs 92(V) x=Xs x=Xs g 'iv ) 9 2( V ) (A .31) x=Xs Substituting Equation (A .24) into Equation (A .31) yields Ci tiX s) = - Gdos{ 1 — ks) X s 1+ E CX S (1 ka^V0Ut + v'gs { E cX s (i - k s y x s vout (A .32) From Equation (A .30) and Equation ( A .32) we obtain (1 - K ) 2Ci { ^ f ) G doak ] { \ - \ k sf x0 = (1 - k a) 2v'g s ( ^ E CX S 1+ (1 - ks )V0Ut E CX S (A .33) ( i - I M 2 v out( i - ka) The next step is to calculate Xvo(£) in Equation (A.17a). Auo(^) = E ie (A .34) Substituting Equation (A .33), (A .34), (A .29) in to Equation (A .28) gives vo(Lg) = (1 - h f C r f h J - i+ { ^ f ) G d0s k i { \ - \ k ay + (1 - k a) 2Vga/3I dc£ (3 ? )* 2 0 Ci E CX S (1 - ka)Vout PCi E cX s Vout(l — ka) V,'gs ' G doa( l - k a) + ( I - k 3) — Vga Vda (A .35) 100 B y solving E quation (A .35) C \ will be known in term s o f vgs and Vds - For sim plicity we will introduce the new sym bols i2, and R y which are defined in Section 2.3. < L g) = - ^ [ i + R y] + R iR vvg. - f ¥ - C 1 ^ I* dot G doa{Cl 1 - ks) + h (1 - ks ) = vso ~ v *s (A .36) M anipulating th e above equation gives Gdos\.(<k a "I" 1 + ~ k a) ) v ga ^G doai 1 — k a) G dosRd[(ka + R i R y ( + 1 - (1 — “ I" ( 1 ^ s )U (is ] k a) R i ( l k a) ) v ga + -f (1 - Ry) fc s ) u d s ] where 8£2 R d = 1 + ~2~Gdoa( 1 — ka) + (1 — k a) R i ( 1 + R y) C a lc u la tio n o f v\ A .3.2 We shall now integrate the wave equation ( A .lib ) in order to get i>i in the G C A region i] = C , w sv„(x) = C , W ~ C ' X Jrg V" ° * “ X s ( A .37) Before to do so let us introduce some new variables which will sim plify th e integration. T he channel current is given by Ido = fxC g W g [F g s - F t - Vcs] Using Equation (A .38) dx can be expressed in terms o f V c s(z) as follow s ( A .3 8 ) 101 dx = [VG5 - (A.39) - VC 5( X )] rfVc s (a:) Id c Integrating Equation (A .39) from 0 to x yields x = » C gWg [(V o s -V iJ V o s W -i^ * )] Ii C , W , ( V g s -V Vcs(x) t? V g s — Vt Id c V -V t) Vc7s( s(x) 2 \V g s Let us introduce the new variable y Vcs(x) ^ Vg s — Vt so that we can write x = fiCgW g (VG S - V T)2 (A.40) Integrating Equation (A .38) from 0 to X s gives he = ^ ^ [ ( V gs- V t )V c s ( X s ) - \ v 3 s ( X s )\ fiCgWg (VG S - V T ) 2 Vcs(Xs) 1 ( Vcs(Xs)\ Xs VGs — V t 2 \ VGs — V t fiCgWg (VG S - V t )2 Xs (ka 2 kfj ) (A.41) where we introduced the new variable . Vcs(Xs ) 5 Vos - Vt Replacing Equation (A .41) into Equation (A .40) we obtain an expression related x to y x = xX s ( 1 2\ (l- it ,) !* " ? ') 102 Let us differentiate the above equation by y in order to obtain the relation between dx and dy — Xs dy k3 ( l - M_ 1 so that dx can be expressed in term s o f dy ( A ' 4 2 ) L et’s start integration o f the wave equation (A .37) using these new variables = fx C a W a Jo C\X -J- VgSGfi()S^Cs 1 ; ? r au3“ J dx W ) '9 ” 9 L ___ 1 X sC l - y) fy r X s ( y - h I 2) fvr As[y~ *y ) 1 X s ^ ~ y) .h. 0 U 1l ks ( l1 - I * ,) Gd0, ( l - y ) X s ks( 1 - \ k 3) ay - - r w 1 + GdOsXsVg n tnr ffV y GdOsXsVgs j , ° ’ W ' Lo G M, ( l -—y y) )X s dy + e ' G^osCi CosCi Ci fpy ( 11 2\ i G d o s k K 1 -- = \rk r3yy /Jo0 V I* G q 3G \ Gd0skj(l - \ k 3y Coav C0.i>„, ~ 22 y J) dy + G g d om sU ;v j - ±i fkc .)) I 1 - dy + e' Co3Vg3 ! G o 3V g 3 + 7 ;— TTi— ^ y + ei y j \ 2 y ~ 66 V GdoM 1 - \rkr 3) /I 2 1 3^ Integrating once more yields g(V)vi(x) G q3C i________ t x _ _____ / I 2 1 3\ G«k>.*2(1 - i M 2 A) \2^ ~Gdosk23( l - l k 3y l ,) £ y^ + k fi% = - o J ^ o - L y ( 2^ " 6 y3) dy+ f j - 6^ J Co3Vg3 fX J 1 1 + Jb,(l - iJb.) X V + ClX+ 62 * .( / - \ k 3) dy +e> ( ^ 2“ F 3+ k ) dy - U a)2 .) Jo ' C 0aX ssCx C1 G doak 3( 1 VgSC oaX s ' f l 1 4 Qy 3 [6y 1 3l I r1 2 + k i ( i - \ K Y l2y “ 5*1+ 62 . 1 sl 30 . , = 0 because Uo(0) = 0 from boundary condition (A .18c). ui(x) g(Y) C oaX s C C\\ fl 3 G dosk3( 1 - \Uk s) s ) 3 V6y VgaCoaX s f 1 2 *2U - 5*-) 1 4 6 j_ 5) 30 V 1 3^ . - F J + 6' 1 J (A .43) (A .44) AC potential at X 5 , where y = ka, is « i(* s ) = r ^-s) coac1 E + vgsC oaC + G do Ci (A .45) Unknown variable e\ will be obtain by applying the rem aining boundary condition to Equation (A . 2 1 ) for the first order term. For convenience Equation ( A .21) will be rew ritten for th e first order term . ux(Ls ) = f3Idclxi + Aux(^) + ni(A's) (A.46) T he other boundary condition v \ { L g) = 0 w ill be used to obtain ex w ith Equation (A .46). Equation (A .46) m ust b e rewritten in terms of ex through th e following steps. First the relation betw een ex and C 2 , which is used in saturation region, will be established through GC A /satu ration boundary condition, ac current continuity, the ac current in G CA region at X s is iiC Xs) = —^ [ 9 ( V ) v i ( x ) ] \ x=xs 104 And ac current in saturation region at X s { x ' = 0) is i i( x ' = 0) = C 2 C i can be expressed in term s of C2 = Cos“D G dlos l C \ Vg g C Q g A from the above equations (A .47) ei where A and D are defined in Chapter II section 3. N ext step is to rew rite Aui(£) in term s o f e\. From Equation (A. 17b) a „ iM = E x e - E i p 6va (A .48) 2 Sub stitu ting Equation (A .47) into Equation (A .48) yields C o D C \ V g g C o s A ■G dog e\ (A.49) Last step is calculation x\ in term s of ei. Substituting Equation (A .9) into Equation (A.20) and extracting the first order term yields Xl * 1(1 ~ k* f _ ' , v , v outk]{ 1- \ k ay l( 5) (A .50) In order to get xi in term s of e i, we m ust calculate ac electric field, tq (X s ), in term s of ei- ac electric field in GCA region is differentiating Equation (A .44) by x, 1 » .( * » ) = C o sC l D + vgsCosA + ei Gdc C oaX s C1 g '( V s ) E + vgsC 0SX s C + e \ X s 9 2{Vs) G (los 1 CoaC x D + + e\ U -Y vgaCoaA VgS GdosXs{ 1 — ka) ^dos g(V.) Q E CX S ( J d o a V out Gl,X}( C ,„ X sC , 1 - *.)2 Gd CoaC x G dosXs( 1 — ka) + E 4* VgSC oaX s C + e i X s ( D + E R y) G doa vgaC oa(A + C R y) + e i ( l -f f?y)] 105 where A , C , D , and E are defined in Chapter II. Now x x can be expressed in term s of ei (1 - K f Xi C oaC x , = ( D + vgaC oa( A + C R + e i(l + y ) + E R y ) Gdos Gjo, f e ) *J(1 - 1kay (A .51) R y ) ] R eplacing Equation (A .51), (A .49), (A .45) into Equation (A .46) and applying boundary condition (A .18c), v x(Lg) = 0. 0 Ci '^ ■ ( D = Gdos{ 1 ka) V a a C o a ■ [ R i ( C’do*(l ei ka) GdosiX P C i* 6u. ka) 1 - ka) (A + \cDD C \ R i ( D GL(1 - M vgaC 03 R i ( ks) ei Gdosi, 1 C R y ) + C ] vgaCoaA Ci ■Gdo cxc0 + '-r dos [1 + iZ«(l + R y ) { 1 — &s)] 2 C^dos(l + E R , ) ( 1 - k .) + ^ - E 'Jrdos + E R y ) { \ 1 - ka)(A + PI2 — ka) •+• E H— — G<i0» ( l — ka)D C R y ) + C 1 + /? ,(! + i?y)(l — ka) + ^ 3£2 + ^ - G doa{ 1 - ka)A 8£2 Gdosi)- + ka) ^ PC X£2 H 7 T'a ) where i?, and i?y are defined in Chapter II. M ultiplying Gdos(1 — ks) to both sides to make calculation easy gives 0 = _ C i C £a 3£2 Ri( 1 - ka) D + (1 + 7 ^ ( 1 - ka) ) E + ^ - G d o s { 1 - ka)D Gdos ~\'VgaC oa + e i Rd + Ri{ 1 - k2)A + (1 + J f c i ^ l - ka) ) C + — Ts G d 0 J( l — k a) 3£2 Gdoa( 1 - h ) A 106 e\ can be obtained by solving the above equation, which is ei = Rd 3£2 ClCos R i { 1 - k . ) D + R b E + ^ - G doa( 1 - ka) D Gdos f3£2 vgsCoa {JCxl 2 6 — GosRd R i ( 1 — ka)A -f- R b C + — ^ 05(1 - ka)A 'TaJ£dGdos{t1 Ci kg'] [ R a D + R g E ] — v ga[ R A A + R b C ] G do n r “ n As 7 s-ti'cl'C* dos\*- (A .52) J where RA and R g are g iven in Chapter II. Equation (A .47) and (A .52) gives Ci J3 (A .53) Vgs A ~ Ci ■Gdoa Now ui is known for en tir e gate len gth. A .3.3 C a lc u la tio n o f t;2 The wave equation for th e second order ac potential in GCA region is <P dx2 [g(v)v2] = c aw avi (A .54) Substituting Equation (A .4 4 ) into Equation (A .54) and integrating both side gives _ " r c‘w’ Jo () 9 0 f c°-c 'Xs - \k,y _CosVgaXs__ ( \ 2 _ 1 3^ i k 2a (l - \ k a)2 \ 2 y 3 / IK* ViM 6s T 30*J + fl C l , C^ 1i ________ fX 1 3 ^ 4 _L ^ A G d0akf( 1 - ±fcs)3 Jo g { V ) U 22 “ 6 y + 30y J J X 107 + i m ^ l l w ) & - \ & x + c ‘w‘ C w ) i x + h C l C11 i ______ _____ 1____ l y 3 _ 1 4 + *\ x s ( l y) , \ k a)3 Jo Vo GU>.XS 6 y + 3300 y )/ ka( U 11 - ±ka) U a) y G d0ak3( l - ±ka)3 G dos Xs (1 {l - «) y) \ 6 r . 4. ( ± - Gqsvga ry 1 f - v 2 _ i , / 3^ ~ y ) j„ Zy ) ka( \ - \ k a) y k * { l - \ k ay" J Jo o Gd0aX s ( 1l - y) \ 2 y r* w fv 1 X s { y - \ y 2) X s ( l - y ) fl V o G W M l - y ) M l - W M l - W ^ _______ G qsC i_____ / 4 1^ 5 1 6A G i)A 4( l - | M 4 V24y . 30y GftgVgs_____ (1 3 Gdo.fcJU - !&s )3 U y 180y J 1 4\ _________Cps6!_____ / I 2 12y ) G d0sk*(l - |fcs)2 V2y ^ 3^ , r 6y J + * Integrating again gives g ( V ) v 2{x) C 20aC " lx r 1= " *^* .)/ o Gjo.*J(1 - \ k , ) * L + G ^ { C - \k,Y 4 ! s + Coaex fy G - \ k a)3 Jo ” d0ak3(l ................. ( e y3 G jo4 ,4(1 - J*,)' CosXset G d0ak3( l - \ k a) 6 ^ 12y>) dx (?3 ft* + h fy ( 1«<3 ^ + m ye) k J h w ) dy ' - , ( i - 1 * .) ' f 1v 2 \2 C l C xX s rv 1 G 2d0M 1 - \ k a)3 J 0 \24 C%aVgaX S * * - 3 0 1' + l 8 0 I' ) ‘fe ±ka)* J 0 V24 y , - + + m f Vf l Gfogs fy G " d0sk3(l - \ k a)3 Jo ' + 1 ( f (f2- ?3)<>*+ CICX G 2d0sk<a(l - / "" . ' W i - 1W 108 C^XsCr ——y 5 — —y G H— — y 7 ------ — y 8] .120 80* 180 1440 J G l s k K i - l k ay + + v9s C l X s 1 777 y Gd0sK{ 1 - \ks)A L24 4 - 1 20 1 5 77772/ + 7^ ri 1 .4 , 1 G doak*( 1 - \ k a)3 * v ~ 6 V + 3 0 y C psXse i + f\x + f 2 Since boundary condition (A. 18c) gives v<i(x = 0) = 0, f 2 is zero. v 2(x) C tX sd, = 9 ( V ) [ G 2doak l ( l - 1 ksy + vaaC l X s [— y 4 G dosk i( 1 - \| kks) sY [24 + CoaX s £ 1 G doak*( 1 + .120V - — 8 0 V + 180y y5 + — 20V 72 1440y y 6l J The second order ac potential at x = X s , where y = ka, is given as v 2( X s ) = + C t X s C , „ . vaaC lX s H + 9^ os'4 - G G d0 do3 G doa( 1 — ka) X s C oaX s e 1 E + fiX s G dos 1 + CoseiE C oalCi H + G j oa “ 1 G ,os ~ 1 G d0 + ^ (A .55) where H , G , and £ are given in Chapter II. The other boundary condition gives ac potential at drain side zero, so that we can calculate the f \ . T h e second order ac potential at drain side can be obtained from Equation (A .21) by extracting second order term s v 2(\jg) = /3Idclx 2 + A v 2(£) + v2( X s ) (A .56) A ^ (^ ) is given in Equation (A. 17c) A v 2(£) = 24ug 6u, (A .57) 109 ac current is continuous at G C A /satu ration boundary so that i2(X s ) = - f a \ g ( V ) v a(ar)]|*=jrff c20A r v9sClD G doa = C3 = Coa&1 G doa G d, D -h (A .58) ( ' = 0) 12 2 w here jF, B, and D are given in Chapter II. From Equation (2.16) and Equation (A .9) *2 = * 1 ( 1 - k a)3 ToTi i T ^ 2( * s ) v outk i ( i - \ k ay (A .59) 77 1 *2 ( Xs ) C2 £ ± F + G ldos 9 ( V S) 9'{Vs) 9 ( V a) XJ —rn l^dos + /x Gdo I Gdos ^ S ^ g s^ o s I + Gdos( l — fcj)A s G d oVso u^t ^ = T ^ T T -T I-*dos B + Gdoa dos I f \r h + h Xs ^ 1 D + /, G doa [C 2, A s d1 / f + ^ , X s G l sX U l - k3)2 + n ^Jdoa rc|,d x „ . uflsC 2s + ^ S ^ O S ^ l jp H + — n ------- G + G G doa doa l X s E + /x A s “ ’ yy- \ l ¥ 1 ( F + n » H ) + ' ^ ( B + R llG) fc ajJiS I ^d o s doa + £ z * ( D + R yE) + f 1( l + R y)] &doa J (A .60) Substituting E quation (A .60) into Equation (A .59) yields (1 - ka)2 x2 = ^ ( F + RyH) doa + v„*c l ( B + HyG) + G ^ . ( D + R , E ) + M 1 + R , ) G doa G doa (A .61) 110 Substitu ting Equation (A .55), (A .57), and (A .61) into Equation (A .56) 0 = V2(Lg ) = + R i ___ C-s£ ± ( F + R . H ) + + R,G ) Gdoa &' doa {* d o a dos P C 1 / A i P C 2/!3 P C 3i 1 C osC i j j Gdoai^ 1 ^s) _ ^ VgSCos q G^a CcoSe i ^ ^ ^ Gdoa (A .62) Gdoa From Equation (A .58) we can obtain C 2 and introduce it in Equation (A .62) 0 = V2(L g) = R i _________ G doa + + Goa^l VgaCl, ( F + R SH ) + '- 'd os r doa (D + R y E ) + / l ( l + Ry) | — / ^ 2r 2Cx 24 Gdoa p e c 2Ts ^ c r F+^ c i B+c ^ D+fi p t2 r + 2 6 G doa l + G doa (1 & s) + RyG) U dos Gdo dos G doa G do G do pi2T2C\ pec2ra 4- 24 1 6 \Cldx 2-7dos(l *.) p e i? i(l — ka) ( F + R y H ) 4— — Gd04( l — ks) F 4* H dos pi2 i? ,(l — ks) ( B + R y G ) 4— — Gdos(l — ks) B 4- G 4G doa 4- Gos^l R i ( l - ks) ( D + R yE ) 4- ^ - G dos{ 1 - ks)D 4- E G dos + /l P i2 ■R,(l — &s) ( l 4- Ry) 4— — Gdoa{ 1 — ks) 4-1 = 0 p£2r2Ci pe2c2r3 4- fi = 6 24 1 C ld, G dos \ s-v2 [Ra F + R b H] + GdoaiP ks) 4- [Ra D 4- R b E) 4- f i / R d ] = 0 Rd Gd, t-*doa \ P t 2r 2C x 24 Gdo [Ra B 4- RbG] J G d o s il - ks) - G doai 1- ks) Ill [r a F + R b H] ^dos Ra B + R b G\ ^dos [Ra D + R BE\ {^dos ~ 2 4 ~ Gdos(1 “ ks)Rd " ^ 6 ^ ^ Gdos(1 “ *s) + [CosCl [Ra F + R b H] - vgaCos[RA B + R BG\ - E 1[RAD + R BE] LG,dos '-3 Cl,Cl — -3=^— r Wos A .3 .4 v„Cl Cm 33— i f - -3;— i f - i i <-*(*05 'GTrfoa C a lcu la tio n o f ig and id The ac current flow ing into the g a te can be obtained by subtracting i( x = Lg) from i(x = 0). We assum e that the ac channel current is flowing from drain to source. The ac current flow ing into the drain is i(x = L g). i(x = 0) can be obtained from the current equation o f the G CA region and i ( x = Lg) from the current equation of the saturation region. The relation between ac current and ac p otential in GCA region is given by E quation (2.4). Rewriting th is equation in term s of th e frequency com ponents at x = 0 yields *o(0) = ~ j ^ [ g ( V ) v 0]\x=0 = *i(0) = ~ - ^ l 9 ( V ) v i]U=o = - e i *2(0 ) = ~ ^ 9 { V ) v2]\x= o = —di = Ci (A .63) (A .64) -/l (A .65) The ac current in th e saturation region is given by Equation (A . 16) so th a t i(x = Lg) can b e obtained by substituting x' = 1 into E quation (A .16). *0 {Eg) = Ci i i{ L g) = ~~CiTa + C2 (A .66) (A.67) 112 i 2( La) = \C it2 .- C 2t. + C s (A .68) where rs = l / v 3. As shown in Equation (A. 10a) th e ac current consists of th e zero, first and second order term s. In order to get the to ta l gate and drain current we have to combine them using to Equation (A.10a). T h e ac current will be calculated up to second order term s *(0) = Ci - j u e i - (ju>)2/ i (A .69) i ( L g) = C 1 - M C 1ra - C 2) + (ju:)2 ( ^ C 1T2 - C 2Ta + C 3^ (A .70) The gate current is then given by ig = jw [-e 1 + CiTa - C 2] + ( j u ) 2 - f i - ~ ^ Ta + C 2Ta - C 3 (A .71) and th e drain current by Equation (A .70). A .3.5 C a lcu la tio n o f Yi2 a n d Y22 Fj2 and Y22 are obtained from ig/vda and id/vda respectively w ith vga = 0. For vga = 0 we m ust introduce th e new constants C[, C'2, C'3, e^, and f [ calculated in Equation (A .69), ( A .70), and (A .71). C\ — G dos (1 n' _ G2 ^3 t — _ ks)RdVda GoaC\ n D I* dos ei ^ — y — U —h CoaR d C i fr> n ! D C,1 ^ 4" -Rb-®] $ C xt 2_ D ^ /i a TsRdGdos{ 1 0 f N &s) 113 f'l = + - 2£ Cl Gdos( 1 - ka) R d - ^ I l R dG dos{ 1 - ka) 14 0 GoaRd [Ra F + R b H ] — Eda[RAD + R b E ] Gdoa Gdo \coacx ■ Substituting th e above equations in to Equation (A .71) and Equation (A .70) and dividing it by v ds gives Yx2 and Y22 Y12 = y 22 ig j u [ - e x' + C [ t s - C ' 2] + (jus)2 { - f ' x - l C ' r f + C'2t s - C ' 3] -2 - = Vds Vds — j w [ —Eds GdaTs ~ C da] = U) = id C[ + M - C ' x T a + C'2) + (.j u ) 2[\C [r * - C'2t s + C'3] — = + Fds + ^ + ( jw ) 2 [— T, ~ CdsTs + H ds Vds F ds ~ G d a / ^ T 2 ju)[Eda + C dsTs Gds^a "h Cds] Vds — Gds + j b j ( —GdsTs + Cda) + ( i w)2[G?ds/ 2 r 2 — Cd„rs + H ds] = Gds - a;2 [ ^ r 2 - C d. r . + Hds] - jw[G'lJ.r . - C ds\ where 9 1 — G dos{\ Vds C"2 _ CosGds Gds Cda Vds r dos ^S^Ed - Eds t e. Eds Vds C 0s E d C di Ordos [Ra D + R b E] - ^ 0 tsG 2 ds Cs _ C l G ds F _ CosEda D _ Vds Gdoa Gdoa Hds Fds I L —- Pn al s sr ids - S f/•>- rTs '~>da'Jdi r r 24 CoaRd C 0aGda (R a F + R b H ) — Eda{R AD + i?B ^)j Gdoa L Gdo Vds + — Hda\ 114 C a lc u la tio n o f Yn and Y21 A .3.6 F n and i 2i is the ac gate and drain current divided by vga w hen Vda = 0. Cj = GdosRd(ka “i Ry (1 ^gs = cDLG,Ci - D dos c; = C ‘-C \ F _ 0 1 , — Go$Rd _ 9 ^ 1 D _ fl Gdos Gdos cx [Ra D 4- R b E] — v93[R a A + -KbC]! r G dos pe C \ T s R d G dos ( 1 - fl = + ^ f C oaR d G dos — ' C d o s il - r c osG i . Gdo k .) k a)R d - ^ ^ T a R d G dos{ 1 - K ) [Ra F + R b H] — vgaC oa[RAB + R b G] E\ [R a D + R b E]] Substituting the above equations into Equation (A .71) and Equation (A .70) and dividing it b y vgs gives I n and I 21 Y11 = ia _ M ~ e i + Cx-T. - <%] + ( j uQ2[ - / ; - \ C l r t + (% t . - C l] — = ugs ug s — j u [ —E ga + GgaTs — Cga] + (juj)2[—Fga — Gas/ 2 r 2 + CgaTa —H g, s i y ja G gaTa -f- C g3] -- u> = id c ; + M - c y . + c i ) + c h 2[ ± c ; v 2 - c ; r . + c i \ — = ------------------------------------------------------------------------------ - C ,.r . + j i o [ E ga H ,gs Jgs ug s = Gga + j u j ( - G gaTa + Cga) + C?w)2[ - G sar 2 - CgaTa + Htgsi = G ga-u>> ~ C , . t. + U „ ju>[GgaTa C ga] where Ggs C —- = GdosRd{ks + R i R y ( l — ks)) = v gs c gs = = D - C oaA - E as Vgs Egs = ^ & dos = ^ J £ R d[RAD + R BE ] - C 0SR d[RAA + R BC] Vgs U gs ~ ~ ^ d03 Cl C lG g. ft, ~ Vgs r i2 CjTd o s Cl f l _ pPTg 93 24 C0SEgS r i {-*doS n dos 9S f3PTS - vgs ~ 93 ds + ^ ^ R d[RAF + R b H ] - ^ - R d[RAB + R BG} ^dos C 0SR dEgS Gdoa 6 33ds '- 'd o s [Ra D + R b E] A p p en d ix B E xact S olution for V elo city -S a tu ra ted M O D F E T W ave E q u a tio n The exact solution of the M O D FE T wave equation is based on the original calculation of Burn [14] for the M O SFE T in pinch off (k = 1). His wave equation can be m odified to hold for th e unsaturated regime. Before proceeding, we need to derive the channel voltage. The dc channel potential Vc(x ) in th e GCA region is obtained from the current equation. T he dc current in GCA region is IDC(x) = fiC(Vas - V T - V c s ( x ) ) dVcdSx{x) Since th e dc current Idc is independent of x, we have (VGs - V t - V c s ( x ) ) d ^G J ^ = constant Integrating both sides from 0 to a: and m anipulating with boundary conditions, Fcs(O) = 0 and V cs(A 5 ) = V 'D , yields 117 It is convenient to introduce the dc gate to channel voltage, V a c ( x ) = Vg s - V = t -V V g c ( x ), c s (x ) defined as (B .l) (Vos - VT ) J l - ( 2 k . - k * ) ± - (B.2) w here ks = vJ j i VTThe equation is derived for total voltage from continuity equation and current equation. The tim e dependent current equation can then be rewritten I ( x , t ) = - f i C v G c ( x , t ) - V^ * , t ^ (B.3) Vac(x, t ) = (B.4) w here V g c {x ) + vgc{ x , t) T h e continuity equation becom es d l j x , t) dx ^ d vG cjx, Q dt /g D ifferentiating Equation (B .3 ) on both sides with respect to x and substitutin g in E quation (B .5) yields an equation for t ¥ / 2 x ^ vgc (x ^ ) : 2 d v GC( x ,t ) = ~ a t ^ (B.6) For small signal analysis, th e above equation will be decomposed into a dc part a n d a sm all-signal ac part. It is assumed th at the second order term (such as VgC( x , t )) o f sm all-signal ac part is negligible. Using th is assum ption VQC( x ,t) can be rewritten approxim ately v G c ( X >0 » VG c ( x ) + 2 V G c ( x ) V g e( x , t ) 118 Substituting Equation (B .2) in the above equation and sim plifying yields vh o M * (Vos - VT )2 ( l - (2fcs - Ar2) | - ) + 2{Vgs - VT) ^ 1 - (2ka - k j ) j - v ge( x , t ) = (Vgs - Vt )2P + 2{Vo s - VT) y / P v gc( x , t) (B .7) where P = 1 - (2fca - A:2) X , The new variable P is introduced to sim plify calculation. The relation betw een dx and d P has to be calculated before differentiating Equation (B .7). X5 dP 2 ka — k2 dx = The procedure to differentiate Equation (B .7) is as follows d2 dx2 dP2 2 (2k, ^ ? ( V e S - V T ) ^ 2 = 2 = 2 = d2 r (2 * . - * . 2)i v 2GC( x ,t ) [l/Gcfa.O] p V ic M ] d v 9C , (2 ka - k2)2(VGS - V t ) d X2 dP 2y / P v + y / P d P (2ks - k2)2(VGS - VT) X2 (2 ks - k2)2(VGS - VT) d2v, Vp dP2 X 2 1 V+ 4 /3 3 /2 dv gc 2y / P d P + 1 dvgc y /P d P + 1 dv gc 2 y /P d P + VP 1 4 p 3 /2 V9c d2i gc dP2 (B .8) Replacing Equation (B .8) in Equation (B .6) yields d v ge( x , t ) n(VG S - V T)(2k3 - k 2s ) 2 dt X2 /~pd2Vgc(x i P) . 1 d V g / x , t ) _____ 1 dP2 y/P dP 4 P 3 /2 ^ C \ r p d?Vgc(x,t) 1 dvgc( x , t ) V dP2 + y /P dP 1 4 p 3 /2 ' ’ ' (B.9) 119 w here OJ0k li{VGs - V T ) { 2 k a - k l f ^ Laplace transform of Equation (B.9) is + p d ^ t ) _ (1 + $ ,p , /a)v(S i4) = „ (B 10) w h ere S' = S/to0kEquation (B .1 0 ) is the s space representation of the M O D F E T wave equation and can b e verified t o be equivalent to Equation (2.5). Equation (B.10) can be derived from Equation (2 .5 ) by substituting g (V o c (x )) = gW gCg (VGc(x ) ~ V r ) into Equation (2 .5 ) and differentiating. This is a modified B essel’s differential equation so that one can find an a n alytic solution [25]. The com p lete solution is w ritten as v,c(P,S) = ( ^ ( P ) 3' 4) + C2/_ 2/3 ( ] V ? ( P ) 3/4) (B .ll) w h ere C\ and C2 are arbitrary constants. T h e boundary conditions w ill define C\ and C 2 . T h e ac current will be obtained decom posing Equation (B .3) into its dc and ac com ponents w hile neglecting second order terms: i(P .S ) where = - ^ C ^ ( t , « ( P , 5 ) V b c (P )) = - /* C ( - (2* X * ; ) ) = G 'doa - ^ ) ^ ( V V ' P ) = ffdQ, - ± ; ( v V P ) (B.12) 120 r , y C gW a{VGS - V t ) (2 k a - k j ) ^dos E quation (B.12) has to be expanded in term s of C\ and C 2 in order to apply the boundary conditions. It sim plifies th e calculation to introduce the new variable Y Y = % dP = J S ' P - 11* Note th a t the m odified B essel function has the following properties = j f A + .M + V l M ) First dvgC/ d P will be expanded in terms of C \ and C 2 d v ,c _ dYdv^ = ^ dP p - l / t _ d [ c lh /! s { Y ) + C i h / d Y ) ] dP d Y = V s ;p - 1/4 dY C i , , , , ,, ,,, , C i, 1 (B.13) Substituting Equation (B .13) and ( B . l l ) in Equation (B.12) gives i(p,s) = g;„ [ i p - 1/a( c ,/2/3(y) + C2I-„3(Y)) + = V S ' P 1/4 [ ^ ( / s „ ( Y ) + /_ ,/3 (K )) + Y pl/2^ T T f ~ l,-'ls{Y) - + + y (/i/3 (n + /-5 /3 (V )) /s/3<y)1 - A /s d ')] y f S ' P XlA S a V s ! p > / 4 [ C l7_1/3( y ) _ C ,/„ „ !> ') - C 2I - 5I3( Y ) + C 2/ 1/3(V) 121 + C XI 5/3( Y ) + C XI . X/3(Y ) + C 2I 1/3(Y ) + C 2I - 5/3(Y)] = G'doay fS 'P xl A[CxI - X/z{Y ) + C 2I x/3(Y)} Now the ac voltage and current in the G CA region can be obtained in term s of C x and C2 vge( P ,S ) = C xI2/3( Y ) + C 2U f 3( Y ) (B.14) i ( P ,S ) = G’dotV S ' P 1' 4[C1I - x/3( Y ) + C 2I x, 3( Y )\ (B.15) Solving th e continuity and P oisson’s equations in th e saturation region gives th e ac voltage and th e ac current. Integrating th e continuity equation (2.9) in chapter II gives the ac current i(x') — i0 e ~ ^ x (B.16) Since the ac current is continuous at the G C A /saturation boundary, w e have i'o = = i(P , S )U = x s G ^ V S ’P ^ I C J ^ Y . ) + C 2I x/3(Y .)] (B .17) where Ps = Ya = (1 — k s ) 2 | ^ P S3/4 T h e channel potential can b e approxim ately obtained by solving th e Poisson equa tion along the 2DEG channel. (2.10) and integrating gives Substituting Equation (B .16) in P oisson equation 122 n ( a : ') = —fiio e~ j ~>x + ax' + b Since t h e ac electric fie ld is zero at th e G C A /saturation boundary, ^ j \ x= x s — 0? the unknov/n constant = a is jfiio u> (B .18) The ch a n n el voltage in th e saturation region is vK (x ') = P i o f c Y e - * * * + M o - x ' + b \W / ui (B .1 9 ) Now t h e a c potential a n d ac current for the entire region are know n if C\ and C 2 are known. A s mentioned in Chapter II, the boundary conditions are wflc(®) — v gs and vgc(Lg ) = vgs —vds. T h e channel p o ten tia l at th e drain end is show n in E quation (2.23) in C hapter II. uflc( 0 ) can be d irectly obtained from the E quation (B .14) by setting a: = 0 , P = 1 and Y = 4 \/5 '/3 . M O ) = C iI V3( ~ V f ' ) + C2/_ 2/3( | v ^ ) = U3S (B .2 0 ) Let us calcu late vgc(Lg) in term s of C i a n d C2. First th e voltage drop, Au(£), in th e saturation region is the difference of v ( x ' = i) and vgc(x' = 0). Augc(^) = #o0 ) e-*™*+ j / 3 i 0^ £ - p i 0 ( ^ = /3i0 0 ) 2 [e-j %e ~ l ] + j P i ^ e x a was given in Equation (2 .1 6 ) in C hapter II, X, — 1 , - .// . -- rV„c( j £ s ) V c c i X s f 3' (B .2 1 ) 123 Vq C is calculated in Appendix A .2: K ( l — 2^®) Vgs — Vr (1 - ksf X? Vg c ~ The calculation of v' ( X s ) in term s of Ci and C 2 will give us / . , V*c' S) dVgc = d P d V dVgQ = ~dx~dP~dY 2 ks — Xs + y { /i/3« ) + /- 5 /3 ( X ) } so that . = L---------- ^ *8(1 - j k . ) Vas_vT (1-fc.)4 X? . A Xs ----------- x + w y /S 'p -U * 1'-)} + y U i / a t n ) + / - 5 / 3( n ) } ] (1 - K f X . J S ' P ; ' ! * - k , { i - \ k , ) {va s - v Tf ' { , -''°(Y‘] + h '*(Y‘)) + c 2{ /1/3(y .) + /_ s/3(v;)}] (B .22) Substituting Equation (B .22), (B .21), and (B .14) into Equation (2.23) in Chapter II and m anipulating yields Ugc(-kg) = f i I d cE x s - ^ + C2{hi3,{Vs) + -f-5/3(K )}] + + j f - i + C i l 2 / 3(Y.) + C 2I-2/s(Ys) ” }■ X v g C( £ ) “ f* Vgc(Xs') ( - ( 1 7 k/ f X i f f ^ ~ 1/4) [ c . M \ ks (1 - j k sJ Vout W + W i'.) ) [ e J>'*< - l ] [<Zo. J S ' P } /4{ C i I - i /3{Y .) + C 2h /3(Ys)}} (B.23) 124 T w o equations (B .20) and (B .23) are used to obtain C i and C2. For sim plicity two equations m ay be rewritten as Vgc{Lg) v gc( 0) — A n C \ + A 1 2 C 2 = Vgs — Vds = A 21 C 1 + (B.24) A 22 C 2 — vgs where An = h M Y . ) + a ’^ V s ' p ; / ' - PhJ- ,(1 - k . f X s V S !P ; 0 u /3(y.) /4 ks ( l - Ik s) Vc r OUt A 12 = [ e - W - l] + ■ [ / - . / , « ) + / ./a « ) ] /_ 2 /3( n ) + ^ osv ^ P 1/4 ^ 0 ) 2 [ e - ^ - l ] + i ^ h , 3 ( Y 5) ,( 1 - k s ) 3X s V & P s - 1/4 - fild J 7--------- :-----X-----k s (1 - I k s ) o u t vc (fv^7) A 21 = ^2/3 A 22 = 1 -2 /3 From Equation (B.24) it is obvious that Cx A22(Vga Vds) A A 22 — A 12 A c2 = A n v as 122 35 - Aj2Vgs A 'Vds A 2\{VgS - v ds) A\x — A 21 A 21 'Vgs T A Vds A ' A where A = ^4iii422 — •'4i2-d.2i* For the calculation of Y u and F2i Vds have to be set to zero so that C \ and C2 will 125 be m odified as follow s n < (A 2 2 — A 1 2 ) ^1 — r > _ — ^2 V9* A ( A n - A 2i ) ^ v gs The gate and drain current w ill b e expressed in terms of C [ and C1'2 as Id — i{x' = £) = ioe G,1„ ' J S ' P t l l e - 1% ‘ [ C [ U i 3(Y,) + CS/1/ 3(y.)] ta — i(x = 0 ) — i(x' = £) = g^ v s 7 [ c ; / _ 1/3 ( | v ^ ) + c ; / , /3 ( ^ ) ] - G'to.T/S'PyU-WlCll-^Y.) + C M Y.)] T he Txi and v1 2 1 param eters for th e saturated M ODFET are — — Zd— Vgs Xu V 9 — ---- — ug s = G '^ y /S ' ~ P } /4e - j %e(ClgsI - l /3 (Ys) + C 2 gsh ,z { Y s))\ C i„ I- where C \ gs (A 22 _ A 12) A ( A n — A 21) C%g s A ,/3 ( | v ^ ) + C 2„ I 1/3 ( i s ? ) 126 vgs is set to zero to calculate Y12 and Y22 and we m ust introduce the new constant C'i and C" n" Cl a 22 — -^ V ds = si" -d-21 C 2 — ^ v ds The gate current and drain current are then id = i[x' = t) = i o e ~ ^ e = G'dosVS' P y 4e - ^ e[ C ';i-1/3(Ys) + c"/1/3(ys)] i(x = 0) — i(x' = t ) In = G L .V S ' c ; i . m +c ; i„ 3 - G '^ V S 'P ^ e -^ C U -^ Y ,) + c ; h ,3(y,)] + The Yi2 and Y22 param eters for the saturated M O D FET are 1d y22 Vds = Y t 2 G'doay /S 'P } f 4e - i %e[CldaI - 1/3(Y.) + C 2dsI 1/3(Ya)] = ^ds G'dn,VS' C u s l - n z ( 3 ^ ^dos ) + C 2dsI 1/3 ( - v ^ ) P.1/ 4e - J' * <[C'lli, / _ 1/ 3( y ,) + C 2ds l i /3(.YS)]} where r lds G2ds - — ^ 22 A i2l A p p en d ix C T h e F ourth Order F req uency P ow er-S eries S olu tion in th e G C A R eg io n The fourth-order power-frequency solution of the w ave equation can be obtained by expanding th e Equation (6a ),(6b ),(6c) in [10] up to v 4(x) terms d2 dxM V a c ( x - Vt ) vq( x ) = 0 (C .la ) - V r ) v i(s ) = v 0(x) (C .lb ) - VT ) v 2(x) = Vx(x) (C .lc ) - VT )v3(x) = V2( x ) (C .ld ) [h (V g c { x - VT )v 4(x ) = v 3(x) (C .le ) ^ H V a c(x dx2 The current in the channel is obtained from - ^ gWg- [ v ( x ) { V a c { x ) - VT)] (C .2) with u (s ) = v0(x) + j u jv i(x ) + (j w ) 2v 2(x ) + (ju>)3v 3( x ) + ( j u ) 4v 4(x) (C .3a) i (x ) = i0(x) + j(x>h{x) + (j u ) 2i 2( x ) + ( j u ) 3i 3(x) + ( ju ) * i4(x) (C .3b) i l l and i 2i are determ ined by calculating ig and i d for Vds = 0 w ith the boundary conditions no(0) = v0(L g) = u3s;v„(0) = vn(Lg) = 0 for n ± 0. 127 Ygd and Ydd are 128 calvn (fx, vdsn) := b lock ([i,m ,n,c,gdos,k,tm p,cl,c2], fx:in tegrate(fx*C *2*(l-y)/k /(2-k ),y), fx :in tegrate(fx*L *2*(l-y)/k /(2-k ),y) + C l*L *(2*y-y**2)/(2*k -k**2) + C2, fx :fx /g d o s /L /(l-y ), m :last(first(solve(ev(fx,y= 0),c 2))), fx:ev(fx,c2= m ) , n :last(first(so lv e(ev (fx ,y = k )-fv d sn ,cl))), fx :e v (fx ,c l= n ), return(factor(fx))) $ Figure 74: List of calvn program determ ined by calcu latin g ig and i d for vga = 0 w ith the boundary conditions vo(0) = 0,u o(L fl) = - v da, v n (0) = v n(Lg) = 0 for n ^ 0. T h e gate and drain currents (ig and id) are obtained from th e channel current i(x ) using id = ig = i{x = L g) (C.4) i{x = L g) — i(x = 0) (C.5) Due the tedious nature of this calculation the sym bolic m anipulator M ACSYM A was used. Two programs are used to calculated th e ac voltage and current, which are calvn and calix fu n ction , calvn function solves th e wave equation and calix function calculates the ac current from th e ac voltage for each com ponent. The programs are shown in Figure 74 and 75. In calvn function vdsn is a boundary condition for ac voltage. T he calculations gives: Y n = juFgg - ( j u ) 2 Sgg + ( j u f T g g ~ { j u J ^ D g g 129 calix (fx) : = block([tmp,id,k,m ,n], ix: -k * (2 -k )* d iff(fx * g d o s,L * (l-y ),y )/2 /L /(l-y ), return(factor(ix))) $ Figure 75: List of calix program w ith f - ZO09r fag — l'33 ^ 0 3(2 - b)2 6 6 fc + fc 2 _ _ 60 - 12* + 81b 2 - 21b3 + 2 b4 ~ 4 5 (2 - b)5 77560 ^ n - oo 226801b + 267301b2 - 156601b3 + 48071b4 - 7571b5 + 49b6 — r* — ^0 14175(2 - lb)8 = T*99 " 2 C q ------------------------------— --------------------------------- ------------- A 20196°0 - 80784001b + 135735601b2 - 124462801b3 + 6793245b4 ° \\ 233875(2 I -— lb) /vy11 33 22674901b5 - 4574111b6 + 51646b7 - 2527b8) \ 2338875(2 - lb)11 J Y12 = ju}Fgd - ( j u ) 2Sgd + (ju>)3Tgd - (j u ) 4D 3d with 3 - 41b + lb2 F g d = - 2 C 0 3(2 — b) 2 (1 - lb)(30 — 411b + 161b2 — 2*3) S g d = T g d = n _ 3d ~ 2 C 0 45(2 - lb)5 (1 - lb)(3780 - 8970b -f 79201b2 - 32471b3 + 637lb4 - 491b5 + 491b6) — C o 14175(2 - lb)8 ( ( * - * ) ( 1009800 - 3407580b + 4702830b2 -34307651b3 °V 2338875(2 - b ) U 1432920b4 - 346531b5 + 45486b6 - 2527b7) \ 2338875(2 - b )11 J 130 = ju F dg - (ju>fSdg + ( j u f T dg - ( j u f D dg with —2Co 30 - 45k + 20k2 - 3 k3 15(2 - k f 1 £ 105 II >21 II to DO - 1440A: + 1290A:2 - 540A;3 + 110A;4 - 9k5 II —46*0 225(2 - k)6 (415800 - 1405800A: + 1945350k 2 - 1424610A:3 779625(2 - k f V 599005A:4 - 146755A;5 + 19705A;6 - 1134fc7\ 779625(2 - k f + J (20196000 - 88387200k + 166148400A2 - 175707600A:3 ° \\ dg + + >22 iiu j 'ia i — 11694375(2 - ay A;)12 115222050A;4 - 48760380A:5 + 13410670A:6 - 2330920 A;7 11694375(2 - k)12 234250A:8 - 10449A:9\ 11694375(2 - k f 2 ) = ju F dd - ( j u f S d d + (ju j)3Tdd - (j u f D d d with (1 - fc)(20 - 15* + 3A;2) Fdd ~ 2Co 15(2 - k f ---------- ^ (1 - A:)(320 - 560A: + 340A:2 - 90k3 + 9A:4) dd ~ 0 0^/1 Tu = 225(2 - k f i \ ( 10560 - 290400A; + 315920A:2 - 175175A:3 779625(2 - k f ------------------- 53165A:4 — 8505A;5 + 567k6 \ + n D dd = 779625(2 - k f ) /10137600 - 38016000A: + 60064000A:2 - 52256000A3 C Q( l - k ) ^ -------------------------_ _ _ _ _ _ -----------------------27500560A:4 - 9035640A:5 + 1825520A:6 - 208980A:7 + 10449A8\ + 11694375(2 - A:)11 J A p p en d ix D D ev elo p m en t o f E quivalent C ircu it for th e v elo city -sa tu ra ted M O D F E T D .l E q u iv a len t circu it for t h e sa tu r a tio n reg io n T h e equivalent circuit for saturation region is based on the exact solution of the velocity-saturated M O D FE T wave equation so that th e procedure for solving the wave equation will be repeated. The exact solution for th e GCA wave equation is given in A ppendix B. vgs( Y ,S ) = C 1/ 2/ 3( F ) + C'2/ _ 2/ 3( F ) (D .la ) i(Y, S ) = G ' ^ P ^ V S ' i C . I ^ Y ) + C 2I 1/3( Y )) (D .lb ) where Y = \ V S iPa/A O P = l - ( 2 ka - k l ) ^ It is assum ed that vgc(Y3, S ) = vg3 — v'da and i = i'd at boundary. Now the developm ent of th e equivalent circuit for the saturation region requires to derive the relationship betw een v'ds and i d. Let us start with the voltage and current in the GCA 131 132 region. At th e boundary x = X s we have v (X .) = C J y s i Y , ) + C 2I-2I3(Y.) = vga - v'ds (D.2a) i(X .) = G'doaP t /4y / S ' { C 1I _ l/3(Ys) + C2I1/3(Ya)) = i'd (D.2b) The ac current and ac voltage in the saturation region is given by Equation (B.16) and (B.19) in Appendix B . In these equations z'o can be replaced i'd directly. For convenience th e se equations are rewritten i(x') = i'de~^~>x' (D.3a) v(x') = p i'd ( ^ ) \ - ^ + j p i ^ x ' + b \(jJ / U} (D.3b) The boundary condition at the drain side is vgc{ty v9• —Via- T h e gate to channel voltage vgc at drain side is given by Equation (2.23)(see Chapter II). Vac{Lg) = p i d j x s + A v gc(£) + Vgc( X a) (D.4) In order to derive the equivalent circuit Equation (D.4) has to be expressed in term s of vda and i d. x a is g iv en in Equation (2.16) in Chapter II X' = " V a c ( X s ) V^ Xs>> (D-5) where Vqc ( X s ) is given in Equation (2.17) in Chapter II. N ote that the derivative of th e Bessel functions verify = / „ + ,( * ) + ^ / „ M (D .6) 133 Using th ese properties one can o b tain v'gc( X s ) ' (V \ V“ {X S) d P d Y dv,gc dx d P d Y lx=Xv s <lVgc = v X=Xs 2 k a - k2 Xs S S ’P 3~1/4 [c7i { /-i/s(y .) - ^ r h / s ( Y s) + C 2 { l 1/3(YS) + -_ ^2/ I3 .U 2/ 3(F s ) 2 ks - k] Xs 3 Ys | V ^ P 5- 1/4 [ { C r l ^ / s i Y ) + C2I 1/3(Ys )} {C xIm {Ys) + (D.8) C 2 / _ 2 /3 ( F s ) } ] Substituting Equation (D .2a) and (D .2b) into Equation (D.8) yields vK ( X s ) = G ^ P ^ V S ' ( Z k . - k*) 2 ks - k2 / 7^ p _ i /4 2 uss — + Xs V iF ‘ G'dosX s P } /2 + i'd G '^ X s il-k .) i i V s ’P?'* 2&s - k2. IX sPs + (Vgs ~ Vjs) 2ka- k 2 2Xs ( l - k , ) * ' ( Vgs ~ (D.9) V'd s ) VqC( X s ) was given in Equation (2.17) in Chapter II so that x s can be obtain ed in terms o f i d and vds. 4 X |( 1 - ks)3 Xs " (2 ka - k 2) 2Vout , G'dosX s ( l - k s) 2k° ~ k s r.. + 7T^~T, 7T z{.vgs 2X 5(1 — ks )2 2X 5(1 - k.) 4 X 5 (1 - h f (Vgs — Vds) -i'd + G'd0, ( 2 k s - k2)2Vout d ’ (2k3 - k 2)Vout Vds) (D.10) where Vout = Vqs — Vt - For convenience the new constant A and B are introduced A = B = ^ X s(l ~ ks) (2k3 - k 2)V0Ut 4 X 5 (1 - h ) 2 G'dos(2ks - k ])W out 134 A sv 3C(f) is given in Equation (2.27) in Chapter II. Now Equation (D.4) can be rewrit ten in term s of v da and id. Vgc^^g) — Pldtfi^s “i"^SVgc(^) 4" ^(A s) = Vga Vda = p I dJ [ - B i ' d 4 A ( v ga - «$,)] + Pi'i ( 5 ) 2 [e_ j^ 4 jPi'd ~ !] i 4- Vga - v'da ( D .ll) From th e above equation one can easily derive th e relation betw een i d and v ds. v ds = /3ld<j*AVgS Vds 4 1 4 Pldc^A 1 4 fildcf-A Pi'd 1 4 P IdM (D.12) It is difficult to develop the equivalent circuit for th e saturation region from th e above equation so that th e exponential term is expanded in power-series up to fourth order term. Vds PldP'-A.Vgg Vds 4 1 4 pidc^A 1 4 Pldc^A Pi'd i dc£ B ~ ( ^ y { \ - j - i \(jJ J I v. 1 4 pj-dJLA /3Id<£Avga Vds 4 1 4 pidc^A 1 4 pidcf-A f3 P D I a J B + \ p e - j w r . S ^ . - (o,r.)2 ^ 24 6 1 4 pidJ-A (D.13) where rs = £ /v a T h e drain current is obtained from Equation (D .3a) by settin g x' = I. id = *'dt J“' T5 (D.14) One can easily ob tain ed the g a te current in th e saturation region from th e difference of th e channel current at drain side and the G C A /satu ration boundary. 135 S Figure 76: Equivalent circuit for the saturation region i , = i'd ( l - (D .15) T h e above three equations gives the equivalent circuit for the saturation region w hich is shown in Figure 76. Comparing Figure 76 w ith Equation (D .13) one can easily find 7 s and S3 7s = 1+ p ia A fiIdc£A (D ,16) . = T + Jh M (D '17) Let us calculate th e value of each elem ent in the im pedance part. T h e following equation can be found from equivalent circuit v da = 7sVds + S.vg, - ( R ai + i .1 •' ) i'd V + JwCs J (D .18) In order to com pare Equation (D.18) w ith Equation (D.13) we should expand th e denom inator in power-series up to the second order using the assum ption that 136 C s R s2 « OJ. v ds ~ 7 sVds + 6avga - [ if o + R a2 - ju>CaR 2a2 - u>2C 2R%2] i'd (D .19) Com paring Equation (D.13) and (D.19) and equating th e term of sam e order gives 'Idc£B + 1 + ftldc^A r,,/^ 2 = <D-21) 6(1 Solving the above equations yields p i dci B Rsl ~ R a2 = - l + 0 I dc£A (D>23) 2/y 3(1 + p I dc£A) (D .24) c■ = N ow ( ° - 25) is — ^av da -t- £>avga Z ai d (D .26) where Z a = i?si H— j/is2 + jio C a D .2 C a lc u la tio n o f th e Y -p a ra m eter s for th e tw o reg io n m odel T h e Y-param eters of the velocity saturated M O D FET , Yij(sat), w ill be obtained from the equivalent circuit in term s of the Y-param eters of the GCA region Yij(g) of 137 reduced gate length X s = Lg —£. The gate and drain currents are easily obtained by inspection from th e equivalent circuit (see Figure 39) ig = Ygg{g)vga -\-Ygd{g)v,d a-\-i,d{ \ - e ~ 3WT‘ ) (D.27) id = i'de~3wr‘ (D.28) i'd - Ydg(g)vga + Ydd(g )v da (D.29) where v'da is given by Equation (D .26). Substituting E quation (D.26) in to (D.29) yields id = Ydg(g')Vga ~ Ydd(g')('JaVda 8aVga Y>aid) Ydg^g)vga -)- Ydd{g)^av da + Ydd(g')8avga Ydd(g } Z ai d (D.30 ) From Equation (D .30) one can easily obtain i d in terms of vgs and vda as follows . Ydg(g) + Ydd(g)6a " i + , Ydd{g)^a v>‘ + i + Y M z . Vdl ( D3l) Replacing Equation (D .31) in (D .28) gives *= + (t W m ) ^ (a32) Substituting Equation (D .26) into (D.27) yields ig = Ygg(g)vgs + Ygd(g)(~jav ds + 6avga - Z si d) + i d ( l - e _ja,Ts) = ( Ygg(9 ) + Ygd{g)8a)vga + Ygd(g)l a v da + i'd { \ - e - i “T‘ - Z aYgd{g )) Substituting Equation (D .31) into (D.33) gives ig = (Ygg(g) + Ygd(g)8a)vga + Y g ^ g ^ v ^ + ( l - e~3“T’ - Z aYgd(g )) (D.33) 138 Yd3 (g) x + Ydd(g)&, 1 + Ydd( g ) Z s Y d d (g )is ^ 3‘ + 1 + Ydd(g)Za Vds) 99 M +rM (. + ^ “ e’,"T‘ “ Z,Y‘d(s)^ v.'gs v d, Ys i \ (D .34) By definition Equation (D .34) and (D .32) defined th e total Y-param eters of th e velocity-saturated M O D FET. M ^ Yi 3^ c- ,w . _ z sF5d(5 )) + Y _ Z. sy3(i(5 )) y , + T ¥ 0 ^ ( l - e -e-iu-r. ^ - Z Y ,M - YM Y ( o\ Yd9{S) — “ ( Ydgjg) + Ydd(g)Ss \ j WTt \ 1 + Ydd( g )Z s ) e Y“ {s) “ ( t W m ) ■ '* " t I1 r .M BIBLIOGRAPHY [1] H. Morkog and P. Solomon, “T he HEMT: A super fast transistor,” I E E E Spec trum, vol. 21, pp. 28-35, 1984. [2] C. P. Lee, S. J. Lee, D. L. Miller, and R. J. Anderson, “U ltra high speed digital integrated circuits using G aA s/A lG aA s high electron m obility transistors,” Proc. IEEE G a A s I C Symposium, pp. 162-165, 1983. [3] R. H. H endel, S. S. Pei, C. W . Tu, B . J. Rom an, N. Shah, and R. Dingle, “Realiza tion of sub 10 picosecond switching tim es in selectively doped (A l,G a)A s/G aA s heterostructure transistors,” IE E E IE D M Tech. Dig., pp. 857-858, 1984. [4] N. T. Linh, M. Laviron, P. D elescluse, P. N. Tung, D. D elagebeaudeuf, D . Diamand, and J. Chevrier, “Low noise performance of two dim ensional electron gas F E T s,” Proceedings o f the 10th Cornell Conference on Advanced concepts in High-Speed Sem iconductor Devices and Circuits, vol. 7, pp. 187-192, 1985. [5] U. K. M ishra, A. S. Brown, M. J. Delaney, P. T . Greiling, and C. F. K rum m , “The A lInA s-G alnA s H EM T for m icrowave and m illim eter-wave applications,” IEEE Transactions on Microwave Theory and Techniques, vol. 37, no. 9, pp. 1279-1285, 1989. [6] A. J. Tessm er, P. C. Chao, K. H. G. Duh, P. H o, M. Y. K ao, S. M. J. Liu, P. M. Sm ith, J. M. Ballingall, A. A. Jabra, and T. H. Yu, “Very high perform ance 0.15 p m gate-length In A lA s/In G aA s/In P lattice-m atched H E M T s,” Proceedings of the 12th Cornell Conference on Advanced concepts in High-Speed Semiconductor Devices and Circuits, 1989. [7] D. D elagebeaudeuf and N . Linh, “M etal-(n) A lG aA s-G aA s tw o-dim ensional elec tron gas F E T ,” IEEE Transactions on Electron Devices, vol. ED-29, pp. 955-960, 1982. [8] H. R. Yeager and R. W . D utton, “Circuit sim ulation m odels for high electron m obility transistor (H E M T s),” IE E E Transactions on Electron Devices, vol. ED33, p p . 6 8 2 -6 9 2 , 1986. [9] D. H. H uang and H. C. Lin, “Dc and transm ission line m odels for a high electron m obility transistor,” IE E E Transactions on M icrowave Theory and Techniques, vol. 37, no. 9, pp. 1361-1370, 1989. 139 140 10] P. Roblin, S. Kang, A . K etterson, and H. Morkoc, “Analysis of M O D FET m i crowave characteristics,” IEEE Transactions on Electron Devices, vol. ED -34, pp. 1919-1928, 1987. 11] D . B. Candler and A. G. Jordan, “A sm all-signal analysis of th e insulated-gate field-effect transistor,” International Journal o f Electronics, vol. 19, pp. 181-196, August 1965. 12] J. A. G eurst, “Calculation of high-frequency characteristics o f thin-film transis tors,” Solid-State Electronics, vol. 8 , pp. 88-90, January 1965. 13] J. R. Hauser, “Sm all-signal properties of field-effect devices,” I E E E Transactions on Electron Devices, pp. 605-618, 1965. 14] J. R. Burns, “High-frequency characteristics of the insulated gate field-effect transistor,” R C A Review, vol. 28, pp. 385-418, Septem ber 1967. 15] D . H. Treleaven and F . N . Trofimenkoff, “M O SFET equivalent circuit at pinchoff,” Proceedings o f the IEEE, vol. 54, pp. 1223-1224, Septem ber 1966. 16] J. V. N ielen, “A sim ple and accurate approxim ation to th e high-frequency charac teristics o f insulated-gate field-effect transistors,” Solid-State Electronics, vol. 12, pp. 826-829, 1969. 17] M . Bagheri and Y. T sivid is, “A sm all signal dc-to-high-frequency nonquasistatic m odel for th e four- term inal M O SFET valid in all regions of operation,” IE E E Transactions on Electron Devices, vol. ED-32, pp. 2383-2391, N ovem ber 1985. 18] M . Bagheri, “An im proved M O D FET microwave analysis,” I E E E Transactions on Electron Devices, vol. ED-35, no. 7, p. 1147, 1988. 19] A . V. D. Ziel and J. W . Ero, “Sm all-signal high-frequency theory of field-effect transistors,” IEEE Transactions on Electron Devices, vol. E D -11, pp. 128-135, April 1964. 20] V . Ziel and E. N. W u, “High-frequency adm ittance of high electron m obility transistors(H E M T s),” Solid-State Electronics, vol. 26, pp. 753-754, 1983. 21] H. Rohdin, “Reverse m odeling of E /D logic sub-micron M O D FE T s and predic tion of m axim um extrinsic M O D FET current cutoff frequency,” I E E E Transac tions on Electron Devices, vol. E D -37, pp. 920-934, 1990. 22] P. Roblin, H. Rohdin, C. J. Hung, and S. W . Chiu, “C apacitance-voltage anal ysis and current m odeling of pulse-doped M O D F E T ’s,” IE E E Transactions on Electron Devices, vol. E D -36, pp. 2394-2404, 1989. 23] D. J. W idiger, Two-Dimensional Simulation o f the High-Electron M obility Tran sistor. P hD thesis, U niversity of Illinois, 1984. 141 [24] H. Rohdin and P. R oblin, “A M O D F E T dc m o d el with im proved pinchoff and saturation characteristics,” IEEE Transactions on Electron Devices, pp. 664-672, 1986. [25] H. B. D w ight, Tables o f Integrals and Other M athem atical Data. Publishing Co. Inc., 1961. M acm illan [26] M. Bagheri, “An im proved M O D F E T microwave analysis,” I E E E Transactions on Electron Devices, vol. ED-35, p . 1147, 1988. [27] S. J. M ason, “Power gain in feedback amplifiers,” IRE Transactions on Circuit Theory, vol. CT-1, pp. 20-25, June 1954. [28] J. B. Gunn, “Transport of electrons in a strong built-in electric field,” Journal o f Applied Physics, vol. 39, no. 10, pp. 4602-4604, 1968. [29] M . J. Moloney, F. P onse, and H. Morkoc, “G ate capacitance-voltage character istics of M O D FE T ’s: Its effect o n transconductance,” I E E E Transactions on Electron Devices, vol. ED-32, no. 9 , pp. 1675-1684, 1985. [30] P. Roblin, L. Rice, and H. M orkoc, “Nonlinear parisitics in M O D FE T ’s and M O D FE T I-V characteristics,” I E E E Transactions on Electron Devices, vol. ED35, no. 8, pp. 1207-1214, 1988. [31] P. Wolf, “Microwave properties o f schottky-barrier field-effect transistor,” I B M Journal o f Research and Development, vol. 9, p p. 125-141, 1970. [32] J. B. Kuang, P. J. Tasker, G. W . Wang, Y. K . Chen, L. F . Eastman, O. A. A ina, H. Hier, and A. Fathim ulla, “Kink effect in subm icrom eter-gate M BEgrown InA lA s/InG aA s heterojunction M E SFE T s,” IEEE Electron Device Let ters, vol. EDL-9, pp. 630-632, 1988. [33] S. Y. Chou and D. A. Antoniadis, “Relationship between m easured and intrinsic transconductances o f F E T ’s,” I E E E Transactions on Electron Devices, vol. ED34, pp. 448-450, February 1987. [34] A. B. Grebene and S. K. Ghandhi, “General th eory for pinched operation o f the junction-gate fet,” Solid-State Electronics, vol. 12, p. 573, 1969. [35] W . Shockley, “A unipolar ’field-effect’ transistor,” Proceedings o f IRE, vol. 40, p. 1365, 1952. [36] K. Lee, M. S. Shur, T . J. Drum m ond, and H. Morkog, “Parasitic M E SF E T in (A l,G a)A s/G aA s m odulation d op ed FE T’s and M ODFET characterization,” IE E E Transactions on Electron D evices, vol. E D -31, pp. 2 9 -3 5 , January 1984. [37] M. B. Steer and R. J. Trew, “High-frequency lim its of m illim eter-wave tran sis tors,” IE E E Electron Device Letters, vol. EDL-7, November 1986. [38] P. Roblin, S. Kang, and W . Liou, “Improved sm all-signal equivalent circuit m odel and large-signal state equations for th e M O SF E T /M O D F E T wave eq uation ,” IE E E Transactions on Electron Devices, vol. E D -38, June 1991.

1/--страниц