Simulation and optimization of HEMTs

Transcription

1 1 Smulaon and opmzaon o HEMTs Hesameddn Ilakhameneh Reza Ashra Sna Khorasan * School o Elecrcal Engneerng, Shar Unversy o Technology, Tehran, Iran Emal: khorasan@sna.shar.edu Absrac We have developed a smulaon sysem or nanoscale hghelecron mobly ranssors, n whch he sel-conssen soluon o Posson and Schrödnger equaons s obaned wh he ne elemen mehod. We solve he exac se o nonlnear derenal equaons o oban elecron wave uncon, elecrc poenal dsrbuon, elecron densy, Ferm surace energy and curren densy dsrbuon n he whole body o he devce. For more precson, local dependence o carrer mobly on he elecrc eld dsrbuon s consdered. We urhermore compare he smulaon o a recen expermenal measuremen and observe perec agreemen. We also propose a graded channel desgn o mprove he ransconducance and hereby he hreshold requency o he devce. Keywords: Hgh Elecron Mobly Transsor, Smulaon, Opmzaon, Fne Elemen Mehod 1 Inrodcuon Nowadays, ulra-hgh speed crcus are mosly based on heerojuncon devces ncludng Heerojuncon Bpolar Transsors (HBTs) and Hgh Elecron Mobly Transsors (HEMTs). HEMT-based hgh speed Inegraed Crcus (ICs) and mllmeer-wave mcrowave ICs [1] need o be scaled down o small dmensons or hgher perormance. In 1994 or he rs me [] a monolhc HEMT IC desgn was presened, whch ncorporaed acve regulaed sel-bas. The operaon requency o monolhc ICs has recenly been exended well no he mllmeer-wave range [3,4]. An advanced desgn o a hghly negraed ransmer and recever Monolhc Mllmeer-wave ICs (MMICs) was repored n 005 [5], based on a commercal 0.15m, 88GHz (183 GHz MAX) GaAs phemt MMIC process and characerzed on boh chp and sysem levels. Whle mllmeer-wave applcaons call or acve devces wh hgher cu-o requences, HEMTs oer he hgh requency soluon due o her hgh requency operaon and large curren drvng capables. In hs regard, AlGaN GaN HEMTs are emergng as he promsng canddaes or he rado requency and mcrowave requency power amplers used n advanced wreless communcaon sysems [6]. Sngleheerojuncon (SH) HEMTs employng one dopng layer have shown excellen curren gan cu-o requences and exremely low nose gurers [7]. However, hey oen suer rom low curren denses due o he relavely small number o carrers n he channel. The shee carrer densy can be mproved o some exen by ncreasng he dopng n he donor layer a he expense o lower breakdown volages. A beer approach o acheve hgh curren drvng capably s by dsrbung he dopng no mulple donor regons and employng mulple heerojuncons. In hs way, mulple Two- Dmensonal Elecron Gases (DEGs) are ormed and a hgh curren densy can be expeced [8]. Hgh power compose channel GaInAs/InP HEMT [9] and dual-dela-doped power HEMTs [10] are ypcal nsances o mulple heerojuncon devces. For many deren crcus, desgn and applcaons o HEMTs, accurae models or varous characerscs o he devce are needed. In [11] he auhors repored a new emprcal and smple model ha can represen he curren volage (I V) characerscs o HEMT devces wh hgh accuracy. An nverse modelng echnque was nroduced n [1] o deermne he srucural and physcal parameers o HEMT rom he desred daa or maxmum ransconducance. Presened analycal model by [13] or I-V characerscs o sraned and lace mached HEMTs on InP subsrae usng a varaonal charge conrol model has resuled n an accurae descrpon o he devce. Ths also ncludes correc modelng o he subhreshold and sauraon regons, whch have a parcular mporance or dgal applcaons as well as he mcrowave power operaon o HEMTs. In hs approach, nsead o lnear capacance approxmaon [14,15] a polynomal channel charge densy versus gae-o-channel volage relaon has been adoped. Fg. 1. Typcal HEMT srucure. In general, or more relable modelng and smulaon o hgh speed and hgh perormance nanoscale heerojuncon devces, a sel-conssen and accurae couplng o quanum mechancal and elecrosac analyses should be addressed. Addonally, varous opmzaon approaches or mprovng he perormance o HEMTs have been repored up o now. In hs

2 regard, a novel and accurae mehod or smulang nonequlbrum gae curren and shee carrer concenraon n AlGaAs/GaAs HEMT srucures has been repored n [16]. A rs-prncples heorecal comparson o he perormance o dencal Al 0.3 Ga 0.68 As/GaAs and Al 0.15 Ga 0.85 As/GaAs pseudomorphc HEMTs based on an ensemble Mone Carlo smulaon coupled wh a D Posson solver has been publshed n [17]. A paper publshed n 1999 [18] presened a new and hgh-perormance InGaP/In x Ga 1-x As HEMT wh an nvered dela-doped V-shaped channel. Due o he presence o V-shaped nvered dela-doped InGaP/In x Ga 1-x As srucure, good carrer connemen and a la and wde ransconducance operaon regme could be expeced. Among noable works n he ranspor smulaon o semconducor devces [19], here has been a wde range o recen leraure dscussng quanum correced Mone Carlo [0] and quanum correced dr-duson models [1-4]. These days, advanced and expensve commercal sowares such as Slvaco [5] are able o ully ncorporae quanum eecs no he operaon o semconducor devces. In hs work, we solve he exac nonlnear coupled Schrödnger and elecrosac equaons (Posson and charge conservaon), n D geomery, usng he Fne Elemen Mehod (FEM). These equaons are solved sel-conssenly n an erave manner unl he soluon converges. As an example, we consder a GaN-based HEMT or deermne a ypcal cu-o requency o devce. As a new approach or mprovemen o HEMT characerscs, we also have consdered and smulaed he devce wh varous band edge energy proles n channel layer, resulng rom graded conrol over he Alumnum racon. We show ha s possble o acheve a 53% mprovemen n he ransconducance usng an opmzed prole o mpury n he channel layer. Ths sudy s based on he combnaon o Fne-Elemen solver FlexPDE 5 [6], whch s desgned or soluon o paral derenal equaons, and MATLAB codes. Theory A. Basc Equaons The quanum mechancal Schrödnger equaon governng he dsrbuon o elecrc charges s nonlnearly coupled wh he Posson and charge conservaon equaons, whch are n urn dependen on he probablsc wave uncons. Hence, hs sysem o equaons mus be solved sel-conssenly or a correc smulaon o HEMT devces. Soluon o he Schrödnger equaon resuls n energy egenvalues and elecron wave uncons, and hereby he elecron densy. Elecrc eld and poenal dsrbuon are hen exraced rom he soluon Posson equaon. Ferm surace dsrbuon and curren densy o devce are hen obaned when charge conservaon law s appled. Here, we smulae he HEMT wh wo deren approaches. In he rs approach, he rue dsrbuon o elecron curren s negleced and Ferm energy surace s supposed o be consan, whle n he second approach, he curren connuy equaon s appled along wh he selconssen sysem o Posson-Schrödnger equaons. Hence, he ormer s applcable o unbased srucures a equlbrum, he laer mehod may be used or analyss o ully based conguraons a unequlbrum. Alhough boh mehods are equally applcable o all ypcal HEMT srucures based on III- V compounds, we choose he (AlGa)N amly, whch has drawn parcular aracon n he recen years. The ypcal HEMT srucure under consderaon s shown n Fg. 1. B. Zero Bas The rs approach can be used or calculaon o elecron densy dsrbuon n he channel, and also dervaon o he relaed parameers such as gae-bulk capacance, when no dran-source volage s appled. A he begnnng o analyss we assume ha all he carrers are aached o her assocaed, 0 everywhere n he devce. Hence, he nal elecrc poenal can be esmaed rom he Laplace s equaon dopan aoms, whch mples zero charge densy x y 1 V, (1) where V s he D elecrc poenal. Ths poenal dsrbuon s obaned regardless o he derence beween he band-gap energes o semconducors. Thereore, usng he superposon law, he oal poenal energy U can be obaned n he orm o U qv Ec, () n whch q s he elecronc charge and E c s he conducon band energy. The nal uncon prole o E c could be approxmaed wh combnaon o smooh sep uncons ormed wh angen hyperbolc uncons as s shown n Fg.. Now, he Schrödnger equaon can be wren as ( x, y ) U ( x, y ) ( x, y ) E ( x, y ) m, (3) where ( x, y) s he elecron wave uncon, s he reduced Planck s consan, m s he elecron eecve mass, and E s he elecron energy. Here, he oal elecrosac poenal energy U ( x, y ) s smply subsued by qv. By solvng (3), he egenvalues E and egen-uncons o he -h sae could be obaned. Now consderng he ac ha he energy saes are dscree and do no orm a connuum [7], he oal elecron densy should calculaed by applyng he Ferm sascs o nd he summaon o elecron denses n he energy sub-bands [8]. Ths resuls n n x, y x, y n. (5) 0 Here ( x, y) s he correspondng wave uncon o he h energy sae, and n s he elecron densy a hs specc energy sae. In he case o D analyss, he elecron densy or he h energy sae s obaned rom he equaon [9,30] (Appendx) E E mkt n ln 1 exp kt. (6)

3 Fg.. Conducon band as a uncon o devce deph. where k s he Bolzmann s consan, T s he absolue emperaure, and E s he Ferm energy. The Ferm Energy E s here obaned rom he bulk dopng densy. Also, he conducon band o he channel layer E can be consdered as c he reerence energy or he whole o he sysem, and he derences beween he energy levels o E and E o he c channel layer semconducor s aaned usng E E n Nd NC exp kt. (7) The ypcal resulng energy band dagram s llusraed n Fg. 3. Now, he D elecron densy n x, y obaned rom (5) can be eravely plugged n he Posson equaon (1), resulng n q N d n x, y V. (8) Aer solvng (8), he oal poenal dsrbuon s obaned usng he new poenal, and qv replaces U x, y n he Schrödnger equaon (3). Thereore, he energy saes and egen-uncons are approxmaed usng he new poenal energy dsrbuon. In hs way, he eraon loop s closed and repeaed unl he elecron densy n x, y converges o a seady dsrbuon. A lowchar oulnng he menoned procedure s shown Fg. 4. I should be noed ha he Schrodnger equaon s solved n D subjec o he boundary condons gven n Table 1. Ths wll ypcally resul n a dscree specrum, wh each sae beng hghly occuped due o n-plane momenum o carrers. Ths s because he connemen occurs manly n y-drecon, and hese are reaed by negraon o energy saes over normal drecons by approxmang he dense specrum as a parabolc subband, whch auomacally resul n (6). Ths has been elaboraed n Appendx. Furhermore, oher nondeal phenomena such as polarzaon and sran-nduced eecs, whch are que ypcal n III-V heerosrucures are here gnored. Such eecs only nd mporance n opoelecronc devces where accurae ranson energes mus be known and lgh absorpon or emsson specrum s sough. For nsance, n he opmal desgn o GaN Lgh-Emng Dodes [31] and sronglycoupled AlGaAs quanum dos [3], boh polarzaon and sran eecs mus be calculaed usng 4 4 marx echnques, perurbaon mehod and envelope approxmaon. Ths wll enable one o nd accurae soluons or elecrons, heavy- and lgh-holes as well as holes n he spl-o band. For oher applcaons where ransons do no sgncanly ake place and only one ype o carrer s nvolved, such as he conducng nerace modulaor [33] and hs work, hese may be hereore saely gnored. The reason s ha, here, we look or charge densy o he D elecron gas (DEG), whch s by (6) only logarhmcally dependen on he values o egen energes. Ths wll also help us o jusy he accuracy o numercal smulaons, whch s here observed and descrbed n he nex secon. C. Based Conguraon The second approach or dervaon o he basc parameers n HEMTs akes care o he poenal derence beween he dran and source elecrodes o he devce, and hence can be used or analyss o based conguraons. As a resul, he curren densy dsrbuon n he bulk o he ranssor s aaned. The only derence beween he new suaon a unequlbrum (n he presence o V ) and he prevous orm DS a equlbrum (n absence o V ) s ha he Ferm energy DS level s no longer la and consan hroughou. Indeed, he Ferm energy needs o be replaced by he quas-ferm level. The quas-ferm energy level a he source and dran conacs are equal o her respecve appled volages, whle n oher regons wll be derved rom where 3 J 0, (9) J ne, (10) Fg. 3. Energy saes n quanum well o channel layer. Fg. 4. Flowchar o rs smulaon approach algorhm.

4 D. Boundary Condons The appled boundary condons or he Schrödnger, Posson and curren connuy equaons are shown n Fg. 7 and also reeraed n Table 1 or more clary. The reader may noce ha here he Schrödnger equaon s solved n D a once, nsead o only solvng or he waveuncons a he band edges n 1D across he quanum well. 4 Fg. 5. Elecrc eld dependency o GaN elecron mobly, based on daa aken rom [3]. One can also consder he elecrc eld dependence o elecron mobly E e n GaN, where E s he eecve e elecrc eld, hrough curve ng o he avalable expermenal daa [34]. Ths s done accordng o he polynomal as shown n Fg. 5. Thereore, he mobly becomes a local uncon o coordnaes, and he dvergence condon (9) changes no ( x, y) n( x, y) E ( x, y) 0, (11) In he above equaon, n x, y s replaced rom (5), whch s dependen on he Ferm energy. Fnally, (11) recass no he non-lnear derenal equaon readng, E E x y ln 1 E 0 0 kt. (1) By solvng he above derenal equaon, he Ferm level energy dsrbuon n he whole body o he ranssor s deermned. Thereore, n x, y and he resulng curren densy are aaned a each node separaely. The lowchar used by he program s shown n Fg. 6. Snce he second approach s more comprehensve and gves he complee I-V characerscs o he devce, needs sgncanly more compuaon. Fg. 6. Flowchar o second smulaon approach algorhm. Number Table 1. Boundary condons n he smulaon. Deals 1 (Dran) V V, 0, E qv d d ms d (Gae) V V, 0, E qv, E 0 g g ms dn 3 (Source) V V, 0, E qv s s ms d d 4 V 0, 0, E 0 dn dn Table. Dmensons o he smulaed HEMT. W W W W l l l l D G SU SD 0nm 8nm 15nm 10nm 30nm 0nm 5nm 5nm Fg. 7. Boundary condons deermnaon. 3 Resuls A. Elecron Densy The dmensons o he double heerojuncon devce shown n Fg. 1 are enlsed n Table. All equaons are solved usng he sandard FEM, and he soluon converges quckly only aer our cycles o repeang he ouer loop n he lowchar, as shown n Fg. 6. Only he rs our elecron wave uncons as ploed n Fg. 8 are aken no accoun. Ths s because o he ac ha he rs ew egen-uncons are domnan n deermnaon o overall elecron densy dsrbuon, and he res have neglgble occupaon and hereore are only o mnor mporance. Aer evaluang he elecron wave uncon, we can calculae he elecron densy va (5) whch s shown n Fg. 9. In addon, he elecron densy o a sngle heerojuncon smulaed HEMT wh same dmenson s ploed n he Fg. 10 or comparson purposes. The elecron densy n he sngle heerojuncon srucure s more ouspread n comparson wh he double heerojuncon srucure. Obvously he correspondng peak elecron densy s lower, oo. As can be seen rom Fgs. 9 and 10, n he double heerojuncon srucure he maxmum carrer densy s approxmaely one order o magnude larger han ha o he sngle heerojuncon srucure; addonally, n he double heerojuncon HEMT, elecrons are more conned. Ths arses rom he ac ha elecrons spread more unormly over he channel layer.

5 5 Fg. 9. Toal elecron densy dsrbuon n he double heerojuncon devce. Fg. 10. Toal elecron densy dsrbuon n he sngle heerojuncon devce. In conras o Fg. 9 or he double heerojuncon srucure, he densy s much more unorm. Fg. 8. The rs our normalzed elecron wave uncons o he double heerojuncon devce. The ground sae s calculaed or every bas pon n D, whch s naurally subjec o asymmerc boundary condons across he channel. Furhermore, he curren does no low compleely horzonal as Fg. 11 clearly shows. Whle smple non-degenerae 1D quanum sysems are known o have no nodes n her ground saes (and he order o saes s characerzed by he number o zeros o waveuncons), here s no reason ha he same concep sll could be expeced under D and srongly asymmerc condons. As s normally expeced, he elecron curren pah s observed o be hrough he channel layer. Ths eec s seen clearly n Fg. 11. Here, neglecon o hgher-order egenuncons resuls n a prole wh locally rapd varaons. B. I-V Characerscs The smulaed DC I-V characerscs s shown n Fg. 1 and compared wh he expermenal resul by Kwon e al [8]. A remarkable agreemen beween he numercal smulaon and expermenal resuls s observed. We can also esmae he cuo requency usng he expresson [35] Fg. 11. Absolue value o curren densy n devce. g m c, (13) C gs where g m s he ransconducance o he devce. Also, he Gae-Source capacance C gs can be obaned rom Taylor seres expanson [36] as gs gs1 gs gs gs3 gs. (14) C C C V C V For a barrer hckness o 30nm, he coecens o he above seres or GaN HEMTs are [36] known o be Cgs pf mm, Cgs pf mmv, and Cgs pf mmv. The ransconducance s hen calculaed a Vgs 0.5V and Vds 1V usng he obaned Id Vds characerscs (Fg. 13). Calculaed cu-o requences or double and sngle heerosrucure are 189.5GHz and 171.1GHz, respecvely.

6 Fg. 1. I-V characersc o he double heerosrucure GaN HEMT: numercal smulaons (blue lnes) versus expermenal daa (black sold lnes) rom [8]. The agreemen s very remarkable beween he numercal smulaon and expermenal daa and he s perec. n Fg. 1. Conducon band edge energy proles o hs sandard srucure due o a conrolled Alumnum racon and our proposed srucure, along y-drecon are respecvely shown n Fgs. 14 and 15. We employ a smple lnear nerpolaon or he bandgap dependency o he ernary compound Al x Ga 1-x As [37], and esmae he Al racon needed a each deph or consrucng hs srucure has also been calculaed and shown n Fgs. 14, 15. I should be possble o realze such a graded prole o dopng usng layer-by-layer growh echnques, such as Molecular Beam Epaxy (MBE) or Meal-Organc Chemcal Vapor Deposon (MOCVD). Reerrng o Fg. 1, he dmensons o hese wo smulaed srucures are gven n Table 3. Through numercal smulaon we nd ha he carrer connemen n he proposed srucure s sgncanly hgher han he convenonal double heerojuncon HEMT (see Fg. 16 and Fg. 17). Hence, he eecve channel hckness s reduced roughly by a acor o.1; he eecve hckness o channel s here dened as he lengh scale o decay o carrer densy across he y-drecon (see Fg. 18). Thereore, a channel wh unorm carrer densy and hckness L e would suppor he same oal number o carrers. 6 Fg. 13. Smulaed he ransconducance o HEMT a bas volages Vgs 0.5V and V ds 1V : (a) Double Heerosrucure; (b) Sngle Heerosrucure. 4 A Novel Srucure In hs secon, we show ha would be possble o ncrease he ransconducance o he HEMT ranssor by more han 53%, hrough engneerng he prole o Alumnum n he channel layer. We rs noce ha based on he resuls or sngle and double heero juncon HEMTs wh he same dmensons, was observed ha double heerojuncon HEMT has a larger ransconducance. As a resul, enjoyed hgher cu-o requency, oo. One o he causes or such an mprovemen s beer carrer connemen n he channel, whch s ypcal or double heerojuncon devces. Ths eaure s clearly demonsraed n Fgs. 9 and 10, n whch carrers have gher connemen and hgher concenraon n channel layer or he double heerojuncon srucure. Hence, we can conclude ha a proper desgn o poenal well, and speccally, he prole o band edge energy, would conrbue o a superor carrer densy dsrbuon and connemen. Here, we have smulaed and compared wo srucures wh he same dmensons, bu deren conducon band edge energy proles across he channel layer. The rs smulaed srucure s he same double heerojuncon HEMT llusraed Fg. 14. Prole o conducon band energy and Al racon o he smulaed srucure (he same double heerojuncon HEMT ha was nvesgaed above jus wh smaller dmensons). Fg. 15. Prole o conducon band energy and Al racon o our suggesed novel srucure. Table 3. Dmensons o he smulaed and compared wo HEMTs. W W W W l l l l D G SU SD 3nm nm 3nm 4nm nm nm 0.5nm 0.5nm

7 The eecve channel hcknesses or he wo compared srucures are shown n Fgs. 16 and 17, or bas volages o Vgs 0V and Vds 0.5V. Calculaed I ds -V gs curve or 0 Vgs 0.35V and Vds 0.5V are also calculaed and ploed n Fg. 19. As can be clearly seen here, our suggesed srucure has a hgher ransconducance due o he much beer carrer connemen. Based on Fg. 19, he maxmum enhancemen n ransconducance can be easly esmaed o be abou 53%. (b) (a) 7 Fg. 19. Calculaed ransconducance o wo compared HEMTs a bas volages V ds 0.5V and 0 Vgs 0.35V : (a) Double Heerosrucure; (b) Our novel Heerosrucure. 5 Concluson Fg. 16. Toal elecron densy dsrbuon n he double heerojuncon wh conducon band edge energy prole shown n Fg. 14. The eecve channel hckness s calculaed o be around 1.875nm. Fg. 17. Toal elecron densy dsrbuon n our suggesed novel srucure wh conducon band edge energy prole shown n Fg.15. The eecve channel hckness n hs case s only 0.88nm. Fg. 18. Denon o creron or calculaon o eecve channel lengh. The FEM has been appled o smulaon o GaN HEMTs. Two deren approaches or smulaon o HEMT devces, under equlbrum wh zero bas, and a unequlbrum when based, were repored. Posson-Schrödnger equaons were solved sel-conssenly unl he soluon converged. Through addon o he sem-classcal curren connuy equaon, he eec o dran-source volage could be consdered. Obaned I ds -V ds characerscs o HEMT devces rom hs smulaon, was shown o be n complee agreemen wh he repored expermenal resuls. Sngle and double heerojuncon HEMT srucures were compared and superor perormance o double heerojuncons was conrmed by numercal smulaons. Based on he smulaon resuls, double heerojuncon HEMTs had hgher ransconducance and hereore hgher cu-o requences, whch was a resul o beer connemen o elecrons n he channel layer. Through engneered desgn o dopng prole n he channel layer, he possbly o a sgncan enhancemen n ransconducance has been esablshed. Appendx The densy o -h energy sae as gven n (6) can be obaned by rs nong ha he D densy o saes o conned elecrons s ndependen o her energy, gven n energy or momenum spaces as m g D E de de, (A.1) g D k d k de, (A.) where he acor s nsered o ake accoun or he spndegeneracry. Ths corresponds o assumng an soropc connemen normal o he y-drecon n xz-plane, wh an elecron energy or a parabolc band as E 1 k E. m k (A.3) Now, he elecron densy n he -h subband can be ound va negraon as

8 8 n E k ; E d k de g E D E E E 1 exp kt m E E ln 1 exp, kt kt n whch E; E s he Ferm-Drac dsrbuon. Reerences (A4) 1. Suemsu, T., Yokoyama, H., Umeda, Y., Enok, T., Ish, Y.: IEEE Trans. Elecron Dev. 46, 1074 (1999).. Kobayash, K. W., Esandar, R., Nelson, B. L., Mno, K., Jones, W. L., Bendenbender, M., La, R., Tan, K. L., Berenz, J.: IEEE Trans. Mcrowave Th. Tech. 4, 610 (1994). 3. Wang, H., La, R., Chen, T. H., Chow, P. D., Velebr, J., Tan, K. L., Sre, D. C., Lu, P. H., Ponchak, G.: IEEE MTT-S In. Mcrowave Symp. Dg., 519 (1993). 4. Kwon, Y., Pavlds, D., Brock, T., Sre, D. C.: IEEE Trans. Mcrowave Th. Tech. 41, 36 (1993). 5. Saccon, F., D Carlo, A., Lugl, P., Morkoc, H., IEEE J. Sold-Sae Crc. 40, 174 (005). 6. Ja, S., Dkme, Y., Wang, D., Chen, K. J., Lau, K. M., Heuken, M.: IEEE Trans. Elecron Dev. 6, 130 (005). 7. Duh, K. H. G., Chao, P. C., Lu, S. M. J., Ho, P., Kao, M. Y., Ballngall, J. M.: IEEE Mcrowave Gud. Wave Le. 1, 114 (1991). 8. Kwon, Y., Pavlds, D., Brock, T. L., Sre, D. C.: IEEE Trans. Elecron Dev. 4, 1017 (1995). 9. Boudrssa, M., Delos, E., Wallaer, X., Théron, D., De Jaeger, J. C.: IEEE Trans. Elecron Dev., 57 (001). 10. La, Y. L., Chang, E. Y., Chang, C. Y., Chen, T. K., Lu, T. H., Wang, S. P., Chen, T. H., Lee, C. T.: IEEE Trans. Elecron Dev. 17, 9 (1996). 11. Chen, Y. C., Ingram, D. L., Yen, H. C., La, R., Sre, D. C.: IEEE Mcrowave Gud. Wave Le. 8, 34 (1998). 1. Ahn, H., El Nokal, M. A.: IEEE Trans. Elecron Devces 4, 598 (1995). 13. Guan, L., Chrsou, A., Halkas, G., Barbe, D. F.: IEEE Trans. Elecron Dev. 4, 61 (1995). 14. Moon, B. J., Byum, Y. H., Lee, K., Shur, M.: IEEE Trans. Elecron Dev. 37, 908 (1990). 15. Chang, C. S., Feerman, H. R.: IEEE Trans. Elecron Dev. 34, 1456 (1987). 16. Takano, C., Yu, Z., Duon, R. W.: IEEE Trans. Comp.-Aded Desgn 9, 117 (1990). 17. Park, D. H., Brennan, K. F.: IEEE Trans. Elecron Dev. 37, 618 (1990). 18. Lu, W. C., Chang, W. L., Lour, W. S., Pan, H. J., Wang, W. C., Chen, J. Y., Yu, K. H., Feng, S. C.: IEEE Trans. Elecron Dev. 0, 548 (1999). 19. Hess, K., Leburon, J. P., Ravaol, U. (eds): Compuaonal Elecroncs: Semconducor Transpor and Devce Smulaon, Sprnger, Wnsead, B., Tsuchya, H., Ravaol, U.: J. Comp. Elec. 1, 01 (00). 1. Hossen, S. E., Faez, R., Sadogh Yazd, H.: Jpn. J. Appl. Phys. 46, 747 (007).. Acharyya, A., Goswam, J., Banerjee, S., Banerjee, J. P.: J. Comp. Elec. 14, 309 (015). 3. Vasleska, D., Goodnck, S. M., Klmeck, G.: Compuaonal Elecroncs: Semclasscal and Quanum Devce Modelng and Smulaon, CRC Press, Sheng, Y., Xa, C.-S., L, Z.-M.-S., Ru, G.-P.: Op. Quan. Elec. 47, 659 (015). 5. Slvaco hp:// 6. FlexPDE hp:// 7. Sreeman, B. G., Banerjee, S.: Sold Sae Elecronc Devces, 6h ed., Prence Hall, Trellaks, A., Zbold, T., Andlauer, T., Brner, S., Ken, R., Smh, Morschl, R., Vogl, P.: J. Compu. Elecron. 5, 85 (006). 9. Sern, F., Sarma, S. D.: Phys. Rev. B 30, 840 (1984). 30. Saccon, F., D Carlo, A., Lugl, P., Morkoc, H.: IEEE Trans. Elecron Dev. 48, 450 (001). 31. Khoshnegar, M., Sodagar, M., Eekharan, A., Khorasan, S.: IEEE J. Quan. Elec. 46, 8 (010). 3. Sodagar, M., Khoshnegar, M., Eekharan, A., Khorasan, S.: J. Phys. B: A. Mol. Op. Phys. 4, (009). 33. Khorasan, S., Nojeh, A., Rashdan, B.: Fb. In. Op. 1, 173 (00). 34. Ye, P. D., Yang, B., Ng, K. K., Bude, J., Wlk, G. D., Halder, S., Hwang, J. C. M.: In. J. Hgh Speed Elecron. Sys. 14, 791 (004). 35. Gupa, R., Aggarwal, S. K., Gupa, M., Gupa, R. S.: Mcroelec. J. 37, 919 (006). 36. Faraclas, E. W., Islam, S. S., Anwar, A. F. M., Sold-Sae Elecron. 48, 1849 (004). 37. Agrawal, G. P., Dua, N. K.: Semconducor Lasers, Van Nosrand Renhold, nd ed., 1993.