A TRANSIENT MODEL FOR INSULATED GATE BIPOLAR. TRANSISTORS (IGBTs) Mohsen A Hajji. B.S. in E.E., University of Pittsburgh, 1988

Transcription

1 A ANSEN MODEL FO NSULAED GAE OLA ANSSOS Gs y Mohse A Hajji.S. i E.E., Uiversiy of isburgh, 988 M.S. i E.E., Uiversiy of isburgh, 996 Submied o he Graduae Faculy of he school of Egieerig arial fulfillme of he requireme for he degree of Docor of hilosohy Uiversiy of isburgh he auhor gras ermissio o reroduce sigle coies. Siged

2 COMMEE SGNAUE AGE his disseraio was reseed by Mohse A. Hajji was defeded o July 3, ad aroved by Sigaure Commiee Chairerso Mahmoud El Nokali, Associae rofessor Sigaure Commiee Member Dierich Lager, rofessor Sigaure Commiee Member Amro El-Jaroudi, Associae rofessor Sigaure Commiee Member George Kusic, Associae rofessor Sigaure Commiee Member arick Smoliski, Associae rofessor ii

3 ASAC A ANSEN MODEL FO NSULAED GAE OLA ANSSOS Gs Mohse A. Hajji, h.d. Uiversiy of isburgh, he sulaed Gae iolar rasisor is widely acceed as he referred swichig device i a variey of ower coverers ad moor drive alicaios. he device combies he advaages of high curre desiy biolar oeraio ha resuls i low coducio losses wih he advaages of he fas swichig ad low drive ower of MOSFE gaed devices. he basic idea behid G is o icrease he coduciviy of a hick lighly doed eiaial layer, hus reducig he o-resisace ad losses associaed wih ower MOSFE. his reducio i resisiviy resulig from high level of carrier ijecio is referred o as coduciviy modulaio. he he G is ured OFF, however, ijeced carriers mus be eraced firs before he device ca susai he reverse blockig iii

4 volage. Several models have bee roosed i he lieraure o describe boh DC ad rasie behaviors of Gs. hese models ca be broadly classified io wo mai caegories: hysics based models ad behavioral or comac models. he disseraio comares he various aroaches made i he lieraure o model he rasie behavior of Gs. A ew hysics-based model for a No uch hrough N G durig rasie ur OFF eriod is reseed. he seady sae ar of he model is derived from he soluio of he ambiolar diffusio equaio i he drif regio of he G. he rasie comoe of he model is based o he availabiliy of a ewly develoed eressio for he ecess carrier disribuio i he base. he rasie volage is obaied umerically from his model. Aleraively, a aalyical soluio for he rasie volage is reseed. he heoreical redicios of boh aroaches are foud o be i good agreeme wih he eerimeal daa. he model is used o calculae he isaaeous ower dissiaio ad he swichig losses E off i a G for differe device carrier lifeimes. DESCOS High-level ijecio Ambiolar Diffusio Equaio hysics-based model No-uch hrough G Aalyical Modelig rasie ur-off Losses iv

5 ACKNOLEDGMENS ake his ooruiy o deely hak my advisor, rof. Mahmoud El Nokali, for helig me esablish a srog foudaio i he hysics of G ower device. orkig uder his suervisio, leared how o be resosible, orgaized ad self-deede. His iovaive hikig ad ousadig eachig olicy were highly simulaig. May God always bless him, his family ad relaives. would like o hak Dr. Dierich Lager, Dr. Amro El-Jaroudi, Dr. George Kusic ad Dr. arick Smoliski for beig my h.d. commiee members. Also, my haks go o Dr. H.K. Ah for his ime ad valuable guidace. am hakful o my family ad my coury for suorig me i various ways. am also hakful o he chairma of he Elecrical Egieerig Dearme rofessor Joel Falk for his hel ad suor. v

6 ALE OF CONENS age ASAC.iii ACKNOLEDGMENS.v LS OF ALES...i LS OF FGUES.. NOMENCLAUE...iii. NODUCON.... ASC ACKGOUND.6. G Srucure.6. G Oeraio..3 asic ools For he Aalysis vi

7 .3. he Seady Sae Codiio rasie Oeraio of G 8 3. LEAUE EVE ANALYSS OF SOME ANSEN MODELNG OF G 4. A NE HYSCS-ASED ANSEN MODEL he Seady Sae 3 4. ur-off rasie Eressio For dv CE d he saaeous ower ad Swichig Losses E off A Aleraive Aalyical Soluio For he rasie dv CE d ESULS AND DSCUSSON 65 vii

8 6. SUMMAY AND FUUE OK...69 AENDX A AENDX AENDX C LNEA CHAGE CONOL 7 EXESSON N CASE..75 EXESSON N CASE..77 AENDX D DEVAON OF AN ANALYCAL SOLUON FO HE dv CE...79 d LOGAHY 86 viii

9 LS OF ALES able No. age G Model arameers 49 hysical arameers...5 i

10 LS OF FGUES Figure No. age. A cross secio schemaic of he G half-cell.7. Equivale circui model of he G D coordiae sysem used i he modelig of he N G 5 4. yical G ur-off rasie showig ur-off hases ad 5. he collecor-base caaciace formaio. 6. Adjused coordiae sysem used for amamurhy G aalysis [5].3 7. he seady sae charge Q as a fucio of carrier lifeimeτ Liear ecess carrier coceraio rofile as a fucio of Comariso of aode volages wih eerimeal daa for differe lifeimes..46. Large iducive loads wih o freewheelig diode i a G circui.47. ducive loads wih a free wheelig diode i a G circui 48. he isaaeous ower dissiaed by he G for differe lifeimes for 3. off he umerical model..5 E as a fucio of ime for τ. 3 S...53 µ 4. off E as a fucio of ime for τ. 45 S.54 µ 5. off E as a fucio of ime for τ 7. S 55 µ 6. Comariso of he eergy losses for hree differe lifeimes i a G

11 for he umerical model ur-off swichig losses E off as a fucio of aode curre for he umerical model Comariso of aode volages wih differe lifeimes bewee he aalyical model ad eerimeal daa Aode curre rofiles..6. Comariso of aode volages wih he referece model for differe lifeimes 63. a ma values for differe lifeimes for N G srucure.64. ouu as a fucio of he bus volage ad lifeime hrough he raio of..83 L 3. Saber sofware simulaio for as a fucio of V bus ad τ...84 i

12 NOMENCLAUE b A Device acive area cm. µ Ambiolar mobiliy raio. µ cm µ, µ Elecro, hole mobiliy. VS cm D, D Elecro, hole diffusiviy. S D DD, Ambiolar diffusiviy D D cm. S, Elecro, hole curre A. se Emier elecro sauraio curre A. Device aode curre A. 3 N ase doig coceraio cm. 9 q Elecroic charge.6 C. Q Q oal ecess carrier base chargec. Seady-sae base charge C. 3 Ecess carrier coceraio a cm. 3 Ecess carrier coceraio cm. τ ase high-level lifeime S. ii

13 L ε si Dτ, Ambiolar diffusio legh cm Dielecric cosa of silico F/cm Disace i base from emier cm. Meallurgical base widh cm. Collecor-base deleio widh cm. C bcj Aε si, Collecor-base deleio caaciace F. bcj V CE Device aode volage V. iii

14 . NODUCON he sulaed Gae iolar rasisor G is a ower-swichig device ha combies ower MOSFE iu characerisics wih biolar ouu characerisics. his device is corolled a he iu by MOSFE volage; however, he ouu curre is characerisic of ha of a biolar jucio rasisor J. Cosequely, he device is called a isulaed gae biolar rasisor G. he combiaio of he low o sae resisace of ower biolar rasisor ad he high iu imedace of ower MOSFE is oe of he advaages of his device. he high iu imedace of ower MOSFE allows corolled ur-o ad gae corolled ur-off []. Several models have bee roosed i he lieraure o describe boh he DC ad he rasie behaviors of he G. hese models ca be classified io wo caegories: hysics-based ad behavioral comac models. he model oulied i his hesis disseraio belogs o he caegory of hysicsbased models i ha he basic semicoducor hysics of he G device is used o develo relaios bewee he ecess mioriy carrier s disribuio, he aode curre, he base curre ad he ouu volage of he device V V CE V. here are C o eac soluios whe cosiderig hysics-based modelig aroach. Aroriae mahemaical rereseaios should be foud o roughly aroimae he soluio. is ossible o reach a eac soluio if cerai assumios are me whe cosiderig he imosed boudary codiios. he aalyical modelig aroach uses arameers, which racke refereces laced i he fro of he lie of e refer o bibliograhy.

15 have hysical meaigs ad are relaed o each oher based o he device hysics. Neverheless, develoig a hysical model is ime cosumig. Sice here are some simlifyig aroimaios made whe develoig a hysics-based model, accuracy may o be %. Furhermore, differe srucures of he G device or ew devices require ew modificaios or ew models. he aalyical model develoed by Hefer e al. [-3] is he mos comrehesive i he caegory of hysics-based G model. rivedi e al. [4] rovided a umerical soluio o obai dv d i erms of he oal curre durig he ur-off for hard ad sof swichig G alicaio. he sweeig ou acio of he ecess carriers i he drif regio due o he wideig of he collecor-base deleio was used o obai dv d eressio. Sice decreases or shriks due o he wideig of he deleio regio i he collecor-base, he charges are forced o be ake away ad d h q υ q. d amamurhy e al. [5] roosed a aalyical eressio for he volage variaio wih ime dv d for he ur-off of he G. he simle eressio for he variaio of he ieral charges ad he boudary codiios wih volages is a o-liear oe. For he ur-off aalysis, amamurhy liearized he seady sae carrier coceraio eressio. amamurhy used a osiive value for d d isead of a egaive sig, which coradics he equaliy. bcj

16 Faemizadeh e al. [6] develoed a semi-emirical model for G i which he DC ad he rasie curre rasor mechaisms of he device were aroimaed by simle aalyical equaios. he model equaios of he G were isalled o he SCE simulaor usig aalog behavioral modelig ools. However, he accuracy of he model is limied by he fac ha he aalyical semi-emirical model is over simlified. Yue e al. [7] reseed a aalyical G model for he seady sae ad rasie alicaios icludig all levels of free carrier ijecio i he base regio of he G. he auhors have show ha he low ad high ijecio models sigificaly overesimae he G curre a large ad small bias codiios resecively. Kraus e al. [8] cosidered a N G wih a higher charge carrier lifeime ad lower emier efficiecy. he ijecio of elecros io he emier of he G isead of carrier recombiaio i he base deermied he -V characerisics. he aode hole curres ad he emier hole curres are assumed cosa. Due o his eglecio of he recombiaio i he base, a liear charge disribuio isead of he hyerbolic fucio was derived. Kuo e al. [9] derived a aalyical eressio for he forward coducio volage V F for N ad Gs akig io accou he coduciviy modulaio i he base of he G. he MOSFE secio has o bee icluded i he aalysis. Sice MOS corols he ur-off rocess of he G, he MOSFE secio should have bee icluded. Sheg e al. [] solved he wo dimesioal -D carrier disribuio equaios o model he forward coducio volages, which ca be used i circui simulaio ad 3

17 device aalysis for DMOS G. he drawback of his modelig aroach is ha i coais comle erms ha would imose some requiremes o he simulaor. Sheg e al. [] reviewed G models ublished i he lieraure; he he aalyzed, comared ad classified models io differe caegories based o mahemaical ye, objecives, comleiy ad accuracy. Alhough [] claimed ha may mahemaical models are accurae ad able o redic elecrical behavior i differe circui codiios comared o he behavioral models, i failed o rovide a comrehesive hysical model for udersadig he device mechaisms. Leurcq [] roosed a simlified aroach o he aalysis of he swichig codiio i G. he aroach was based o a ew mehod for solvig he ambiolar diffusio equaio akig he movig deleio boudary i he base io cosideraio. his aer heled i udersadig he swichig rocess i G. he above lieraure survey deal maily wih G havig laar srucures, amely he No-ach hrough N ad he uched hrough G, alhough oher srucures were roosed i he lieraure o imrove he -V characerisics of Gs, emeraure oeraio or develo beer hysics-based models [3-3]. his h.d. disseraio deals wih modelig he rasisor -V characerisics of he G device durig he swichig or urig-off eriod. Chaer describes he basic srucure of a G ad rovides a descriio of he G oeraio. addiio, he chaer iroduces he ambiolar rasor aalysis, which is he key feaure i modelig he G device. he coclusio of he chaer highlighs he basic ools for he aalysis: he seady sae ad he rasie oeraio of a G. Chaer 3 covers he lieraure review aalysis of he mos imora rasie modelig of G. Chaer 4 4

18 iroduces ew hysics-based models for N G wherei he seady sae ar of he model is derived by solvig he ambiolar diffusio equaio i he base. he ur-off rasie ar of he model is based o he availabiliy of a ew eressio for ecess carrier charge disribuio i he base. he rasie volage is obaied umerically from his model. Aleraively we have roosed ad imlemeed a aalyical soluio for he rasie volage. addiio, he ower dissiaio ad he swichig losses E off i G are calculaed ad simulaed for differe device carrier lifeimes. eergy losses. Chaer 5 discusses he ouu resuls for volage, curre, ower dissiaio ad Chaer 6 rovides a summary of he work accomlished ad ossible fuure research area alicaios. 5

19 . ASC ACKGOUD. G Srucure Figure shows a cross secio schemaic of a yical G. Figure shows he discree equivale circui model of he device, which cosiss of a wide base -N- biolar rasisor J i cascade wih a MOSFE. he srucure of he device is similar o ha of a verical double diffused MOSFE wih he eceio ha a highly doed - ye subsrae is used i lieu of a highly doed -ye drai coac i he verical double diffused MOSFE. A lighly doed hick -ye eiaial layer N 4 cm 3 is grow o o of he -ye subsrae o suor he high blockig volage i he reverse bias mode sae. A highly doed -ye regio N A 9 cm 3 is added o he srucure o reve he acivaio of he NN hyrisor durig he device oeraio. he ower MOSFE is a volage-corolled device ha ca be maiulaed wih a small iu gae curre flow durig he swichig rasie. his makes is gae corol circui simle ad easy o use. A highly doed buffer layer could also be added o o of he highly doed subsrae. his layer hels i reducig he ur-off ime of he G durig he rasie oeraio. he G wih a buffer layer is called a uch hrough G while he G wihou a buffer layer is amed a o-uch hrough N G. 6

20 G Gae Cahode K Emier MOSE Chael ody bcj C J Drif egio cm 4 3 Subsrae Collecor Aode A E J Figure A cross secio schemaic of he G half-cell. 7

21 Cahode Emier Gae -MOSFE S MOS V C - - D -N- rasisor - V V DS V C V V V V V CE E Sice VE C C is very small ase curre V E Aode Collecor Figure Equivale circui model of he G. 8

22 because a he case of G, he eilayer is o as hick less hick as N G layer is laced over he layer. is very likely o have uched hrough he J jucio o he J jucio. his ad ac as a shield o he J jucio. his layer ca hadle some of he uch hrough buffer layer occuies some saces i he base of G, which leaves less sace for he oal charges i he base regio durig he G o sae ur-o oeraio. As a resul, hese charges are removed by he layer more quickly whe swichig occurs. his layer is a recombiaio ceer which hels holes ge recombied wih elecros before reachig he base regio ad fewer holes, as a resul, will be ijeced io he base regio lower efficiecy. As a resul, he carrier lifeime is reduced ad he swichig frequecy is icreased. However, sice fewer carriers are ijeced io he base regio comared o he N G, he coduciviy modulaio elaied laer is reduced higher base resisace ad he osae volage is icreased. he rade OFF bewee he reduced ur-off ime ad he icreased o-sae volage should be accoued for. For N G cosidered i his hesis disseraio, he eilayer hickess is hick eough sice o layer is iroduced o o of he highly doed layer subsrae. ecause he hickess of he layer is greaer ha G, he resisace is high i his regio ad, as a resul, a higher reversed volage ca be susaied whe J is reversed biased. he uch hrough of he J o he J is less likely o occur, i.e. a ouch hrough siuaio. f he uch hrough occurs, he he G breakdow volage will be degraded causig he device o break dow. he he emier base jucio J is 9

23 forward biased ad he gae volage is greaer ha he hreshold volagev ; ijecio of holes from he subsrae is icreased. he resisace of he - layer is reduced whe he ijeced hole desiy becomes much greaer ha he backgroud doig N. he ur-off ime of he N G is loger ha he G because he removal of so may sored charges i he base regio is slow due o he absece of recombiaio ceers. Carriers do o recombie as fas, which meas hey have loger lifeimes. he N G is more efficie ha he oes sice more charges are ijeced from he layer o he. he ijeced holes do o ge recombied as fas as i G, which has a buffer layer o o of he layer.

24 . G Oeraio he a osiive gae volage greaer ha he hreshold volage V is alied, elecros are araced from he body owards he surface uder he gae. hese araced elecros will iver he body regio direcly uder he gae o form a chael. A ah is formed for charges o flow bewee he source ad he drif regio. he a osiive volage is alied o he aode ermial of he G, he emier of he G secio is a higher volage ha collecor. Mioriy carriers holes are ijeced from he emier regio io he base drif regio. As he osiive bias o he emier of he J ar of he G icreases, he coceraio of he ijeced hole icreases as well. he coceraio of he ijeced holes will eveually eceed he backgroud doig level of he drif regio; hece he coduciviy modulaio heomeo. he ijeced carriers reduce he resisace of he drif regio ad, as a resul, he ijeced holes are recombied wih he elecros flowig from he source of he MOSFE o roduce he aode curre o sae. he a egaive volage is alied o he aode ermial, he emier-base jucio is reversed biased ad he curre is reduced o zero. A large volage dro aears i he drif regio sice he deleio layer eeds maily io ha regio. he MOSFE gae volage corols he G swichig acio. he ur-off akes lace whe he gae volage is less ha he hreshold volage V. he iversio layer a he surface of he body uder he gae cao be ke ad herefore o

25 elecro curre is available i he MOSFE chael while he remaiig mioriy carriers of holes, which were sored durig he o sae of he G, require some ime i order o be removed or eraced. he swichig seed of he G device deeds uo he ime i akes for he removal of he sored charges i he drif regio which were buil u durig he o sae curre coducio G ur-o case..3 asic ools For he aalysis he hysics-based modelig aroach is used i his disseraio o beer udersad he effec of he carriers o he G characerisics. he followig ois have o be ake io accou i erformig he aalysis:. he carrier disribuio i he -drif regio of he G is described by he ambiolar diffusio equaio because of he high level ijecio of holes i his regio N : L D where N is he base backgroud doig coceraio, is he hole coceraio, L Dτ is he ambiolar diffusio legh, D is he ambiolar diffusio cosa ad τ is he hole carrier lifeime.

26 . he rasor of he biolar charge is assumed o be oe-dimesioal -D for he ease of aalysis. 3. he emier regio of he J ar of he G has a very high doig coceraio level 8 cm 3. his regio also acs like recombiaio ceers for mioriy carriers comig from he lighly doed base regio elecro i his case. 4. he sace charge regio, which is deleed of mioriy carriers, suors he eire volage dro across he collecor-base ermials based o oisso s equaio. he effec of mobile carriers i he deleio regio is o accoued for i his disseraio..3. he Seady Sae Codiio he equivale circui model ad -D coordiae sysem of Figure 3 is used i he modelig aroach of he N G aalysis. From his Figure, is he G oal curre, is he hole curre of he J ad is he base or MOS elecro curre. ca be eressed i several ways:. he G oeraes uder low gai ad high-level ijecio codiios. he curre equaios are give as J qµ E qd - 3

27 J qµ E qd - where J ad J are he elecro ad hole curre desiies resecively. he firs erms i equaios - ad - are due o he drif comoe while he secod erms are due o he diffusio comoe. he he ecess carrier coceraio is larger ha he backgroud coceraio, he rasor of elecros ad holes are couled by he elecric field i he drif erms of he rasor equaios. he mioriy carrier curre desiy J cao be egleced ad eds u affecig he majoriy carrier curre desiy J. Hece, equaios - ad - cao be decouled. he oal curre desiy is give by J J J qµ qµ E q D D. Subsiuig for he elecric field from he above equaio io equaio - yields J qµ J qµ qµ qµ D qµ D q qµ qµ. 4

28 Emier E ase e Deleio regio Collecor C bcj Figure 3 -D coordiae sysem used i he modelig of he N G. 5

29 A ambiolar diffusio coefficie D is defied as qµ D qµ D D. qµ qµ he eressio for J ca be rearraged ad rewrie as J b J qd -3 b where b µ µ. eeaig he same rocedure sarig wih equaio -, J is obaied J J b qd -4 where J is he oal curre desiy J J. he ecess hole carrier disribuio ca be obaied by solvig he seady sae hole coiuiy equaio L -5 Cosiderig he coordiae sysem give i Figure 3, he boudary codiios for he ecess hole carrier disribuios are []

30 Equaio -6 resuls from he collecor-base jucio beig reversed biased for forward codiio ad equaio -6 reflecs he fac ha he emier-base jucio is forward biased. is he ecess carrier coceraio a, is he quasi-eural base widh give by bcj -8 where is he meallurgical base widh ad bcj is he collecor-base deleio widh. Solvig oisso s equaio -9 yields a eressio for he collecor-base jucio deleio widh as i equaio - d V d qn ε si -9 bcj ε V qn si bc - where N is he doig coceraio of he lighly doed regio of he G ad ε si is he dielecric cosa of silico. he jucio volage V V V V bcj bc bi, bc V is he collecor-base jucio volage dro of he J ar of he G ad V bi is he buil i oeial. Hece from Figure ε V si - qn 7

31 where V V V V is he collecor-base volage ha aears across he drif bc CE A regio..3. rasie Oeraio of G hile he ur-o ime of he G is quie fas, he ur-off ime ca be slow because of he oe base of he N rasisor durig he ur-off eriod. Figure 4 shows he yical G ur-off rasie where is he o-sae curre before he iiial raid fall ad is he curre afer he iiial raid fall. he iiial raid dro i he aode curre is due o he sudde removal of he MOS chael. his is followed by a slower decay due o he removal of he carriers sored i he lighly doed layer. he ur-off rocess is iiiaed whe we lower he gae volage o a value lower ha he hreshold volage V firs hase. his removes he formed elecro chael from uder he gae ad blocks he MOS comoe of he curre MOS. MOS i his case ad he collecor-base volage V bc icreases resulig i a wideig of he deleio regio a he base-collecor side or source of MOSFE. he relaio bewee ad is hrough β r []. is he raio of he curre immediaely afer he iiial raid fall o he magiude of he fall ad is show alog wih he raio of o L L i he aedices. he swichig losses of he G are domiaed by he losses, which occur durig he much slower secod hase of he ur-off eriod rasie. his is because of he ime required removig or eracig he ijeced carriers i his hase. his is a 8

32 major disadvaage of he G device as i suffers from high-swichig losses. his ca be overcome by reducig he lifeime of he carriers i he base hrough recombiaio or eracio rocesses as quickly as ossible before he device reaches is blockig volage sae. he collecor-base jucio is reversed biased ad is deleio regio wides durig he ur-off of he G. he he G is o, he saus of he base-collecor jucio is reversed biased as ca be see from Figure. he he G is OFF, he saus of he base-collecor jucio is reversed biased ad V bc is icreased leadig o he icreme of he deleio regio sice he curre decreases. he wideed regio suors he eire volage dro across he device as meioed reviously based o oisso s equaio. Sice he quasi-eural base widh of he G chages wih ime ad decreases wih he icrease of V bc, we ca fid a eressio for he rae of rise of he volage across he device dv d varyig of he ouu volage durig he swichig-off of he G from he collecor-base jucio deleio widh bcj eressio as show. From bcj ε V si bc ad he fac ha bcj qn if we ake he ime derivaive of bcj, we ge d ε si d qn V dv d bcj 9

33 ad d d d d bcj ε si dv - qn V d his equaio shows he ime rae of chage of he quasi-eural base widh ha covers almos all he legh across he drif regio durig he ur-off sice he collecor-base jucio is reversed biased. Aode CurreA Q αfq ime Figure 4 yical G ur-off rasie showig ur-off hases ad.

34 3. LEAUE EVE ANALYSS OF SOME ANSEN MODELNG OF G Hefer e al. [3] rasie modelig aroach, he geeral ambiolar rasor elecro curre eressio dv eressio for he volage rise d qad was used o fid a b. MOS as show i Figure 3 ad i is imora sice i corols he oeraio of G. Sice he reverse bias o jucio J i Figure does o icrease raidly ad he deleio caaciace of jucio J is arially charged i a shor eriod of ime, MOS curre is isaaeous. A eressio for MOS ca be obaied if we cosider he collecor-base jucio deleio caaciace as i Figure 5. For he volage V bewee he laes, he charge er ui area ε si V q, where d is he disace bewee he laes ad he rae of q chage is d dq d curre Q q C V bcj dq d Cbcj V d d dcbcj dv V Cbcj d d

35 ad i erms of he jucio caaciace of he reverse biased jucio, he dislaceme curre is dcbcj dv V Cbcj. 3- d d d C V Figure 5 he collecor-base deleio caaciace formaio.

36 he firs erm o he righ had side of equaio 3- was igored by Hefer. he rae of chage of C bcj should be icluded i calculaig he dislaceme curre sice he caaciace varies wih ime as he deleio widh chages wih volage [5]. From equaio -3 ad he fac ha J A, where A is he device acive area ad from Hefer s aroach b b AqD C bcj dv d qad b b dv C qad 3- bcj b d Hefer used equaio 3- o obai V for he rasie oeraio of G. He imlemeed he coce of movig he redisribuio curre. his rasie aroach, he eiher used he seady sae eressio for show laer or did he liearize he seady sae eressio for. His eressio cosiss of wo ars 3 d 3-3 D 6 3 d From equaios -3 ad 3-3 3

37 4 qad b b b qad b 3-4 ad isead of equaio 3-, Hefer alied d dv C bcj i his aroach. egraig equaio 3-3 i he base ad mulilyig by qa, he oal charge Q D qa qa d d D qa d d D qa d qa Q qa Q 3-5 as ca be see d d has o effec o Q calculaio. e ca fid d from equaio 3-3 as d d D d d D 6

38 d 6D d he hole curre is qa [3] ad from he above equaio qad qad qad 6D d d qad qa 6 qa Q Q d d QD Q d qad d he firs erm o he righ had side of equaio 3-6 is caegorized by Hefer as he charge corol comoe ad he secod erm is caegorizes as he movig boudary redisribuio comoe of he hole curre. From equaios 3-4 ad 3-6 dv QD 4 Q d C. d 3 d b b his equaio ca be eressed i a differe way if qad i equaio 3-6 is modified as qad QD Q d qad 3 d b QD Q d qad b 3 b d 4D DQ Q d qad D D b 3 b d 5

39 6 d d b Q Q D qad From equaio 3-4, we have b qad b d dv C his ca be rearraged as qad d dv C b. Now usig equaio 3-7, ad he fac ha V N A A C C si bcj si bcj ε ε ad d dv qan C d d, he above equaio yields d d b Q Q D d dv C b bcj 3 4 b Q D N qa Q d dv C bcj A qn Q b C Q D d dv bcj 3-8 where for large iducive loads ad 4 Q D [Aedi A] for he cosa aode volage i which d dv idicaig ha he volage ad

40 are cosas. Equaio 3-8 is Hefer s rasie dv d model for G ad Q is eressed by solvig he followig o-liear Hefer differeial equaio dq d Q 4Q 3-9 τ A se q i where se is he emier elecro sauraio curre A ad i is he irisic carrier 3 coceraio cm. Hefer s aroach, he egaive of he colleced hole curre cosiss of a charge corol curre CC ad redisribuio curre, which make his model dq Q 4Q se more comle. he Eressio is o simle ad Q d τ A q cao be easily deermied sice here is o eressio for which ca be subsiued for i Q equaio o evaluae Q magiude. Also, his model did o cosider he rae of chage of C i he calculaio of he dislaceme curre. rivedi e al. [4] alied he seady sae carrier disribuio equaio sih[ L] for aalyzig he ur-off rocess i he G o derive sih L i he dv d eressio. he e curre followig io he collecor of he G is C h β h h α 3- where h J base curre similar o i equaio 3-, β h curre similar o i Hefer s aroach ad α sech []. L J collecor 7

41 From equaio 3-, d h q υ q ad d d d eressio, dv d was obaied as dv d Dcosh sech L L Lsih L τ qn V e ε si 3- e τ was added o ake care of he decay of charges due o he recombiaio i he drif regio. rivedi e al. [4] mulilied τ i he eoe erm by a cosa κ o accou for he effec of boh carrier recombiaio ad elecro curre ijecio se. his aroach of mulilyig κ wih τ alers he real value of τ. he modelig aroach for he dv d i [4] is differe from Hefer s aroach due o differe key assumios ad equaios used for he ambiolar elecro ad hole curres o aalyze he ur-off rocess i he G. his model igored he diffusio effecs of he carriers i he quasi-eural base i he G aalysis besides usig a osiive eressio for d d which should be egaive sice. bcj amamurhy e al. [5] used differe coordiae sysems for modelig he rasie ur-off of G as show i Figure 6 i.e. ad d. he seady sae carrier disribuio equaio eressio. sih[ L] was alied o fid sih L 8

42 was calculaed from he oal diffusio curre desiy J wih he use of boudary codiios for elecro curres J, he hole curre J ad he oal curre J as J 3- J J J J 3-3 J J J. 3-4 From equaios 3- o 3-3 ad he eressio for eressed as Lah J L J µ J µ qd J qd b µ µ., is Subsiuig for as i [5], oe ges µ µ Lah J J L µ µ. 3-5 qd dv he sweeig-ou acio of charges was alied o fid d he ur-off aalysis d qa d 3-6 Q bcj ad bcj ε V si qn 9

43 d ε si dv. d qn V d From equaios 3-5, 3-6 ad qa ε si qn V dv d dv d qa qn V ε si dv d N J N d 3-7 ε A ε si si amamurhy e al. [5] used a osiive value for d d isead of a egaive value which coradics he equaliy. bcj 3

44 Emier E ase Deleio regio Collecor C bcj d Figure 6 Adjused coordiae sysem used for amamurhy aalysis [5]. 3

45 4. A NE HYSCS-ASED ANSEN MODEL 4. he Seady Sae As meioed i secio.3., he seady sae eressio for ca be obaied by solvig he ambiolar diffusio equaio i he base L uder he boudary codiios [] his equaio has he soluio of he form L L Ae e Solvig for he cosas A ad, is eressed as [ L] L sih. 4- sih egraig equaio 4- ad mulilyig by qa, he oal charge sored Q i he base is obaied as Q cosh L qa d qa sih L Usig he ideiies cosh sih ad sih cosh sih leads o 3

46 his Q resuls from o-sae rasie. L Q qal ah 4- which resuls from he ur-o sae, ca be calculaed by usig he geeral ambiolar rasor hole curre, he seady sae ecess carrier disribuio equaio 4-, ad he followig boudary codiios for elecro, hole ad he oal curres Q. he assumio of is valid for elecro carriers i he ur-o sae because he quasi-eural base widh sice V. 5 vols equals he bcj CE meallurgical base widh. As a resul, elecros do o have sufficie ime o reach he emier side of J ar of G o recombie wih he ijeced hole carriers ad cosiue elecro curre. Usig equaios -4 ad 4-, a eressio for is L ah L 4-5 qad where is he seady sae ur-o curre, which is used as he iiial curre a he sar of ur-off eriod. Solvig equaios 4- ad 4-5, Q ca aleraively be eressed as 33

47 L Q [ sech L ]. 4-6 D his is he seady sae ur-o charge ad is cosa i associaio wihq. For a give, Q τ curves ca be roduced as show i Figure 7 for differe values. As ca be see, Q is roorioal o ad deede τ hrough L Dτ o ha sems from he erm sech L sech Dτ. e ca also oice ha he volage V i ε V si erm corresods o he seady qn sae volage across he J, which amous o a value of aroimaely.5 vols. 4. ur-off rasie Durig he ur-off of he G he ecess carrier base charge, which was buil u durig he ur-o, will decay by recombiaio i he base ad by elecro curre ijecio io he emier: dq d Q τ. Assume he assumio of is valid for elecro carriers i ur-off rasie gae volage V because he sudde removal of he MOS chael will reve he aracio of elecros from he body direcly uder he gae. As a resul, o elecro carriers will be available o reach ear he emier ar side of J ar of G o 34

48 recombie wih he ijeced hole carriers ad cosiue elecro curre. dq d Q τ his has he soluio of he form τ Q e Q 4-7 where Q α FQ ad Q was obaied from he seady sae aalysis where [ L] L sih which is used as a iiial codiio for he rasie aalysis. sih is almos cosa i associaio wih Q. As ca be see from Figure 4 regio, a sudde dro i curre from o will chage he magiude of Q associaed wih o a lower value Q α FQ associaed wih ad F should be less ha.. regio where here is a slow decay of curre, equaio 4-7 alies because ha is where recombiaio akes lace. 4.. Eressio For dv CE d he ur-off eriod for N G sice here are o recombiaio ceers, he ambiolar diffusio legh is larger ha he drif layer which imlies L ad. his is rue as carrier lifeimes icrease. his is show if we calculae L Dτ for differe lifeimes ad assume V V 4 vols. US 35

49 3 3 Forτ. 3 S, L.3 cm, L 6.6 cm for τ. 45 S ad µ µ 3 L.3 cm for τ 7. S. Sice 4. cm, L as carrier lifeimes icrease. µ he hole disribuio is o he same as give by equaio 4-. ca however be esimaed by eadig equaio 4- usig aylor series o yield a firs order aroimaio as sih, sih [ L] L sih [ L] L L sih L 3 3 L L..., L L,. 4-8 Equaio 4-8 is loed for as i Figure 8 which shows a liear carrier rofile a a d give ime. From his Figure he sloe is egaive sice he d behavior of eds owards he mius -direcio as he device volage V chages wih ime movig boudary [Aedi A]. 36

50 y iegraig equaio 4-8 hrough he base ad mulilyig by qa, he oal charge Q is foud as qa Q qa, d. Usig he above equaio, he carrier coceraio a he emier edge of he base is relaed o Q as Q 4-9 qa he accuracy of he volage waveform i rasie is highly correlaed o,. Hece addiioal imroveme o he hole carrier rofile is required by subsiuig equaio 4-8 io ime deede diffusio equaio,,,, L D by iegraig wice wih resec o, by usig he boudary codiio,,, as well as usig he followig relaio [Aedi A] d d,, we ge C L D., comes from equaio 4-8 as, d. d Subsiuig he above equaio i he revious iegral yields 37

51 38, C d d d d D L 4- Now iegraig equaio 4- wih is cosa C oce wih resec o, we ge , C C d d D L Usig he followig boudary codiio,,,, we ge C d d D L C , 3 3 d d D L d d D L d d D d d D L L , , 3 3 d d D L L L , 3 3 d d D L L L

52 , 3 3 D d d L L L he refied versio for he hole disribuio durig rasie is herefore eressed as , 3 3 D d d L L L 4- egraig equaio 4- ad mulilyig by qa yields he charge Q 3 4 L qa Q Sice L, he secod erm ca be egleced leadig o he revious equaio 4-9 qa Q. 4-9 Subsiuig for i equaio 4- leads o, as fucio of ],, [ Q. Equaio 4- is similar o Hefer s eressio e al. [] ece for hree addiioal erms which modify he liear deedece of o. Sice L, we subsiue for i equaio 4- by is value from equaio 4-9 o calculae d, eressio i he equaio of he dislaceme curre. he base curre is a dislaceme curre i aure ad resuls from he discharge of he reverse-biased collecor-base deleio jucio caaciace C bcj d dv C d dc V V C d d CE bcj bcj CE CE bcj

53 4 where V N q A A C CE b si bcj si bcj ε ε. Sice bcj he d dv qan C d d CE bcj ad he dislaceme curre is d dv C CE bcj. 4- his ye of curre flow is usually called he dislaceme curre. he rae of chage of C bcj mus be icluded whe calculaig his curre, sice he caaciace varies wih ime as he deleio widh chage wih volage. Usig equaio 4-, 4- ad he dislaceme curre equaio, qad b,, qad is eressed as d d D qa L qad qad 6 6, d d qa L qad b d dv C bcj bcj Q ad si bcj qn V ε ad V N q A A C b si bcj si bcj ε ε Q

54 d Cbcj dvce. 4-4 d qan d dv Subsiuig equaios 4-9 ad 4-4 io 4-3 yields he CE d eressio dv CE d 4D Q 6L Cbcj Q b 3qAN 4-5 where τ αfqe Q 4-7 L. 4-6 D ad Q [ sech L ] he fac ha he curre dros suddely o requires adjusig he iiial sored charge o α FQ where α is a adjusig arameer ad F is a facor ha is less ha uiy derived from referece [] by assumig [Aedi C] where F sech / ad is almos cosa i associaio wih Q. he ouu L L collecor-emier volage is obaied by simulaeous iegraio of equaios 4-5 ad 4-7. i ch L se erm is almos cosa sice Q is cosa. he aode curre is give by / τ / τ e e / τ e F 4-6 4

55 ad corresods o he ime whe he ouu volage V CE reaches he suly bus volage. he above equaio alies o he slow decay of curre because i is here where recombiaio akes lace while he oal curre i he ambiolar rasor curre equaio, qad relaes he hole curre o he dislaceme b curre whe calculaig he aode volage eressio. is isrucive o oe ha by mulilyig Q by he correcio facor α F, he eraced charges i he base are reduced by his facor. he dislaceme curre C bcj dvce should herefore be also reduced by he same facor α F for scalig urose i d order o accou for he reduced charge i he base durig he ur-off rasie. his ew hysics-based model has several advaages over Hefer s model:. he o-liear caaciace is icluded i he calculaio of he dislaceme curre. his is ulike Hefer s model where he caaciace was assumed cosa.. he ambiolar diffusio equaio ime-deede is oally imlemeed wihou eglecig he firs ar o he righ had side of ha equaio. However, Hefer L e al. [] did o ake his oi io accou.. he iiial seady sae base charge Q for each τ ca be comued easily i ay commercial sofware rograms sice he eressio for dq d is simler ha Hefer s dq d eressio. he ew model, is o used as a adjusig arameer. 4

56 However, i Hefer s model, he iiial seady sae base charge Q magiude a differe dq τ mus be rovided searaely i he sofware rogram sice d eressio is o-liear ad has more ha oe deede variable τ,.. he eressio for is simle ad comuable due o he boudary codiios. However, was used as a arameer i Hefer s model ad does o have a simle eressio.. he model is o a comle fucio of he redisribuio curre, which is due o he redisribuio comoe of he carrier disribuio. Figure 9 comares he ouu volage waveforms for differe carrier lifeimes for his model wih measured eerimeal daa ouus ad model i [] for large iducive loads A. heoreical redicios of he ouu volages have bee obaied wih differe carrier lifeimes ad comared wih eerimeal daa available i he lieraure ad good agreeme has bee obaied. 43

57 Figure 7 he seady sae charge Q as a fucio of he load curre ad carrier lifeimeτ. 44

58 Figure 8 Liear ecess carrier coceraio rofile as a fucio of disace. 45

59 Figure 9 Comariso of aode volages wih he eerimeal daa for differe lifeimes. 46

60 LLarge V US 4 V V G ulse V - V N G Figure Large iducive loads wih o freewheelig diode i a G ur-off circui. 47

61 LLarge Diode V US 4 V V G ulse V - V N G Figure ducive loads wih a freewheelig diode i a G ur-off circui. 48

62 able G Model arameers MODEL AAMEES VALUES UNS 9.3m cm A. cm N 4 3 cm 3 τ τ. 3 S -> / cm µ µ S τ 45 > / cm. µs τ > -- / cm 7. µs D D D D D D b 8 cm L Varies τ deede cm 49

63 able hysical arameers µ 5 cm /Vs µ 45 cm /Vs µ b µ cm i 3 ε.5 F / cm si 5

64 4.. he saaeous ower ad Swichig Losses E off : rasie losses occur whe swichig acio akes lace i a G. Sice he G susais high aode curre desiy ad volage simulaeously, high ower dissiaio occurs. he isaaeous ower dissiaed by he G is he roduc of he volage ad he aode curre V *. From volage equaio 4-5 ad CE curre equaio 4-6, differe ower rofiles for differe device lifeimes τ are roduced as show i Figure. A higher lifeimes, because of a larger umber of carriers, he curre is high ad as a resul he corresodig ower dissiaio is larger. he rasie ur-off eergy E off resuls from iegraig he isaaeous ower over a ime ierval. For a give value of he device aode curre, oe daa eergy oi is calculaed. he eergy resuls for differe device lifeimes are show i Figures 3, 4 ad 5 searaely. he comariso of heses eergy level losses is show i Figure 6. o roduce E off as a fucio of he aode curre, hese eergy ois for aricular aode curre values a differe device lifeimes are loed i Figure 7. 5

65 Figure he isaaeous ower dissiaed by he G for differe lifeimes for he umerical model. 5

66 Figure 3 E as fucio of ime for τ. 3 S off µ 53

67 Figure 4 E as fucio of ime forτ. 45 S. off µ 54

68 Figure 5 E as fucio of ime forτ 7. S. off µ 55

69 Figure 6 Comariso of he eergy losses for hree differe lifeimes i a G for he umerical model. 56

70 Figure 7 ur-off swichig losses E for he umerical model. off as a fucio of aode curre 57

71 4..3 A Aleraive Aalyical Soluio For he rasie dv CE : d A aleraive aalyical soluio for he roosed rasie ur-off of he N G is reseed. he soluio is derived by cosiderig equaio 4-5, by assumig a iiial soluio ad usig oe ieraio o obai a closed form eressio for he rasie volage. A cosa β is used o adjus he seady sae charge Q. ased o his aroach, he deleio widh ca be eressed as bcj a 4-7 ma where ε si qn ad a ma is he sloe of he iiial volage wih ime ad differs from srucures ad carrier lifeime. From equaio 4-5, he gradie of aode volage wih ime is give as dv CE d 4D Q 6L Cbcj Q b 3qAN 4-5 For he ur-off rasie, L for which << he equaio 4-5 L becomes dvce d 4D Q Cbcj Q b 3qAN

72 59 he above equaio ca be aleraively be simlified uder he assumio ha 3 >> qan Q o read 3 4 / / qan Q e b C Q e D d dv bcj CE τ τ β β. 4-9 y relacig ma a d dv CE ad usig he followig relaios: 3 V qan b r A r A C CE bcj r si bcj si bcj ε ε equaio 4-9 becomes 4 / / / ma V e Q Q e D V Q e a CE r CE r τ τ τ β β β. Mulilyig boh sides of he above equaio by V Q e CE r / τ β, collecig erms ad doig some algebraic maiulaios [Aedi D]; he aode volage V CE ca be obaied as a fucio of ime ad charge Q as. 4. A C A C A CE V 4-

73 A βq e r / τ. where βq e r / τ. C 4D βq e / τ Equaio 4- ca be successfully alied o G by addig era erms i Q so ha i covers all he Gs regardless of srucures modified by ay reaso. equaio 4-, he value for Q is subsiued from equaio 4-5. he arameers used for develoig he above models are summarized i ables ad he ouus of he aalyical model of equaio 4- for differe lifeimes are show i Figure8. he resuls are comared wih he umerical model ad he eerimeal daa []. Figure9 shows he aode curre rofile for τ.3us,. 45 ad µ 7.µ S. he aalyical model aode volage is comared wih he umerical oe ad wih he model i [] i Figure 8 for τ us, 4uS ad 6 us resecively. rasie losses occur whe swichig acio akes lace i a G. Sice he G susais high aode curre desiy ad volage simulaeously, high ower dissiaio occurs. he isaaeous ower dissiaed by he G is he roduc of he collecor-emier volage ad he aode curre VCE. he aode curre S is give by / τ / τ e e / τ e F 4-6 6

74 Figure 8 Comariso of aode volages wih differe lifeimes for he aalyical model ad eerimeal daa. 6

75 Figure 9 Aode curre rofiles. 6

76 Figure Comariso of aode volages wih he referece model for differe lifeimes. 63

77 Figure ama values for differe lifeimes for N G srucure for he aalyical model. 64

78 5. ESULS AND DSCUSSON A ew hysics-based model for N G for he rasie ur-off was develoed i chaer 4. he isaaeous ower dissiaio ad he swichig eergy loss rofiles have bee roduced. A aleraive aalyical soluio for his model has bee geeraed beside a umerical soluio. he umerical aode ouu volages are comared wih he eerimeal daa ad referece model [] i Figure 9 for 3 differe lifeimes τ.3µ S,. 45 S ad µ τ as measured by Hefer for a device whose arameers are lised i ables 7. µs ad. he ouu resuls show good agreeme over he wide rage of lifeimes. referece [], a large iducive load was used i he simulaio wih o freewheelig diodes. he circui used is show i Figure. Sice he curre hrough a iducor ca o chage isaaeously, i follows ha i his case aearig i equaios 4-5 ad 4- is cosa ad equal o. is isrucive o oe ha he aure of i he model deeds o he load codiio i he circui. f he load is urely resisive, he curre ca o be cosa. he device lifeime deermies he ouu volage behavior. he ijeced holes from he emier of he J ar of he G ge recombied wih he elecros comig from he chael of he MOSFE orio of he G. he ecess carrier lifeimes i he base of he G deermie he amou of curre available i he G. For lower carrier lifeimes, he curre is lower as well ad, as a resul, he ouu aode volage 65

79 builds u faser ad reaches he suly bus volage. O he oher had, as he carrier lifeime icreases, he amou of he roduced curre icreases as well ad he ouu volage is decreased as show i Figure 9. he average error bewee he heoreical redicios of umerical simulaio comared wih he eerimeal daa for lower carrier lifeimes is wihi 6% wih he accuracy icreasig for higher carrier lifeime. he corresodig average error for he aalyical model is wihi 4%. yical carrier lifeime rage is from o µ S for racical alicaios. he ew models reveal good agreeme for his rage. Commercial MALA sofware was used i develoig differe ouu resuls [6]. he isaaeous ower dissiaio ad he swichig eergy losses i he case of free wheelig diode for differe aode curres ad lifeimes are show i Figures ad 6. he circui used is show i Figure. he isaaeous ouu ower is he roduc of he aode volage ad aode curre. he aode curre eressio 4-6 is used i cojucio wih he ouu volage o roduce he ower ad eergy losses. Figure 9 shows he aode curre rofiles for hree differe device lifeimes τ whe usig a circui ha has a free wheelig diode i arallel wih a iducor. he ur-on of G, a seady sae curre flows i he G. A he ur-off swichig, his seady sae curre dros iiially o a value of F he decays eoeially due o he recombiaio i he base ad should reach a zero value a a ime whe he collecor-emier volage reaches he suly bus volage. Carrier lifeime deermies he eoeial decay rae wih faser decay whe carrier lifeime is smaller. Higher carrier lifeime requires more ime for eracio or removal whe he device urs-off. his ower ouu deeds uo he ijeced carrier lifeime wih higher ower 66

80 dissiaio for higher carrier lifeime. From Figure 6, he swichig eergy losses are he iegral of he ower ouu over he ime ierval. he swichig eergy losses ca also be dislayed as fucio of he aode curre as show i Figure 7. For may commercially available G ackages, daa are rovided for he swichig losses as a fucio of he aode curre. he aode curre for heses ackages rages from 5 A o 5 A. However, for lower rages of aode curres, daa are o readily available. he curre of Figure 9 has a differe decay rae durig he ur-off for differe carrier lifeimes. ih lower carrier lifeime, he sloe of curre decay is very shar hece he sored base charge eracio is abru. As a resul, he eergy loss is low ad visa versa. his cofirms he behavior of he swichig of eergy verses he device aode curre. From hese resuls, he ower dissiaio ad eergy losses icrease wih he icrease of he carrier lifeime ad aode curre. he average error of differe comarisos is wihi 5%. For he aleraive aalyical soluio of he umerical model, he rasie behavior of maimum available gradie of volage a ma is show i Figure. he rasie charge Q value is reduced from Q o β Q. he deleio widh i Q eressio comes from equaio 4-5 ad a ma is lifeime ad srucure deede. A value of β. 9 was used o develo he -V characerisics of he N G. Figure 8 shows a ecelle agreeme bewee he aleraive aalyical model ad he eerimeal daa available i lieraure for he hree lifeimes. his cofirms he validiy of he assumios made o develo he simle aalyical model. Figure rovides addiioal comariso bewee he aalyical, umerical ad daa obaied from subsiuig i he equaios of referece [] sice eerimeal daa 67

81 are o readily obaiable. he aalyical model of [] wih he give arameers ca be used as a referece for he comariso sice i is he mos develoed comrehesive model i he lieraure. From his Figure, he average error for lower carrier lifeime τ. 3 S is wihi 5% wih he accuracy icreasig for higher carrier lifeime. he µ ouus are, i geeral, i good agreeme. 68

82 6. SUMMAY AND FUUE OK his hesis disseraio deal wih he modelig ad simulaio of he sulaed Gae iolar rasisors, which is a referred ower-swichig device. A hysics-based model for he N G durig he ur-off eriod was develoed based o he soluio of a oe-dimesioal ambiolar diffusio equaio which resuls from high ijecio levels of mioriy hole carriers. he ambiolar rasor equaios are aalyzed i deail. his model was comared wih Hefer s model el al. [] ad foud o be much simler ad ca be imlemeed easily i commercial sofware rograms. he ouu redicios are i good agreeme wih he available lieraure eerimeal daa. A aalyical soluio was iroduced i addiio o he umerical aroach. his aroach has a very simle liear form ha is easily solvable. he advaage of his model is ha i is suiable for ad ca be alied o differe srucures of a G G by alyig aroriae charge models. Sice he G susais high curre ad volage simulaeously durig he swichig-off rasie, here is high ower dissiaio i he device. he ur-off ower dissiaio ad eergy losses were iroduced ad simulaed. he ouu characerisics for boh he collecor-emier volage ad he aode curre ca be comared i ime oi by oi ad loed. he ierceio of he DC lie from KCL i a circui wih he roduced -V characerisics will deermie he oeraig oi. he validiy of he ewly develoed models was cofirmed by comariso o he available eerimeal daa or he referece model []. 69

83 A fuure oic of research would be o ead ad aly he ideas used i hese models o ew device srucures ad he imleme hem i SAE sofware simulaio o cofirm he validiy of he develoed model comared o he eerimeal daa available i he lieraure. 7

84 AENDCES

85 AENDX A LNEA CHAGE CONOL From he seady sae aalysis we have [ L] L sih A- sih Sice he diffusio legh L is larger ha he drif layer hickess because of he high lifeime of carriers, oe ca aroimae he carrier rofile for he holes i he drif layer by a liear eressio as show reviously,. A- ih he aid of he geeral ambiolar rasor hole curre eressio qad b A-3 we ca fid a eressio for from equaios A- ad A-3 as A-4 Q ad qad b b 7

86 qad b b qad. A-5 Sice oal aode curre is cosa, from he above equaio we ca fid as qad d d. Subsiuig for from equaio A-5, he above equaio becomes d. A-6 d From Figure 8 his sloe is egaive sice he behavior of eds owards he mius -direcio as he device volage V CE varies wih ime movig boudary. he oal charge i he base of he G is Q qa d qa Q qa Q qa qa Q. A-7 73

87 Combiig equaios A-5 ad A-7 yields Q qa Q qa qad 4D Q. A-8 Equaio A-8 is he liear charge corol curre ad is cosa i his case sice dv CE d. he charge corol curre A-8 was derived uder much simler assumios ha Hefer e al. []. 74

88 75 AENDX EXESSON N CASE is he curre afer he iiial raid fall whe he G is swiched OFF. From he curre desiy of elecros J, he seady sae eressio ad he fac ha MOS i he ur-off case, we ca fid a eressio for qd J b b J or b qad b - ad [ ] L sih L sih - Se MOS [ ] L sih L sih b qad b [ ] L sih L cosh b L qad b L Lsih qad

89 qad. -3 Lsih L Usig equaio 4-5 derived reviously, ad are relaed as L Q ah L -4 qad ah sih L L cosh L. -5 Sice sh L bias effec, co, as ca be see from Figure 4. efore he reverse ad sh L co is cosa ad much greaer ha. Equaio -5 is similar o he eressio foud i aliga [] bu wih a differe ad simler aroach for he firs ime. Sice he curre i he large base decays eoeially a a rae deermied by he lifeime of hole carriers, ca be relaed o as τ e -6 where τ is he lifeime of he hole carriers. 76

90 AENDX C EXESSON N CASE he seady sae base curre ca be obaied from equaios -3 ad 4- of he develoed model as q AD L coh L [ L ] L cosh sih he seady sae aode curre ca be obaied as well from equaios - 4 ad 4- as L coh[ L ] b sih L q AD coh L. For he develoed model, i which, ad are eressed as se. q AD coh L L sih L q AD coh L b sih L L C-. C- y addig C- ad C- we ca ge he oal seady sae aode curre as qa D coh L b L 77

91 qad L coh L. C-3 he charge corol curre for equaio A-8 for is 4D Q C-4 ad he ecess sored seady sae charge i he base Q was obaied from 4- L Q qal ah. C-5 From equaio C-4 ad C-5, a eressio for he curre immediaely afer he iiial raid fall i he ur-off is 4D 4D Q qal ah Dividig he above equaio by equaio C-3 yields L [ sech L ] L. C-6 his raio was foud o be he correcig facor F. F [ sech / L ] / L C-7 is loed as a fucio of L for differe lifeimes hrough L ad differe V US. Figure shows his relaio. As τ is icreased wih he icrease of V US, is icreased as well. is a fucio of he bus volage. Figure 3 shows he simulaio resuls comared o he umerical ouus, which show similar reds. 78