Revisiting the Integral Charge Control Relation (ICCR)

Transcription

1 Rvstg t Itgral Carg Cotrol Rlato. Atcts Dffrt bas rsstac valus av b obta for t M BIC-MXRAM a Hcum bas mols by rformg t xtracto from g-frqucy two-ort aramtrs wt a ovl mto scrb [4]. Partcularly, comutg t bas rsstac from t xrsso ~ ~ I coj cf rb cf rc r ~ I o fs rb Hcum > rb M ~ aramtrs ot masurmt ata mb from t xtral arastc rsstacs rbx a rcx a mof by rmovg t xtral collctor caactac. Paramtr cf s a corrcto factor trm by t tral collctor caactac cjc, t trsc trascouctac gm a trast frqucy ω of t vc. rst of t otatos av tr usual mags. Clarly t tral collctor rsstac rc wat xsts M but s zro t rst Hcum quvalt must lay a ky rol t araox. Sc aartly vry otr aramtr must av t sam valus t root of t scracy as b susct to b rlat to t fuamts of t mol cocts. Wtout gog to t tals of comact mol formulas t basc startg ot of all rst bolar mols, t tgral carg cotrol rcl of H. K. Gumml as b rvst wt rsct to ts alcato at t varous mol famls.. Som Basc Smcouctor quatos Posso s quato lctrostatc ottal satsfs t sco orr.. q x ε s [ x x N x N x ] a Dfto of t Quas-Frm Pottals It follows a l l x 3 x 3 zolta.uszka@austramcrosystms.com

2 Currts x 4 q x 5 q x 5 3. rasstor Structur B C Zo trasto B- QBF QBR Zo trasto B-C BL QB0 QjC Qj Qm QmC xj x 0 xc 0 x jc x x0 x x b X x j x Bb x Bbc x b x BM xc bc x Cbc xjc xc xc x C x c - x - - x - x b - x BM 0 - x Cbc - x C c - - x c Fg. Dog, carg strbuto a rgy agram of a trasstor at rvrs collctor bas. Collctor carg Q mc s rst at quassaturato. cor of t ur sm-fgur: courtsy of B. Arou, XMOD o-msoal trasstor taks lac wt t mttr cotact x a t collctor cotact x c as sow o Fg.. formr s t bouary btw t olycrystall a t moocrystall mttr, t lattr s t g of t gly o bur layr. lctro a ol quas-frm lvls a rsctvly ar sktc ur t og rofl a carg strbuto lots o a vrt scal. ormal actv mo of orato s sow wr t B jucto s forwar-, t tral CB jucto s rvrs bas. Morty carg Q m s always rst t mttr ur ormal oratg cotos wl Q mc vlos.g. at quas or saturato. jcto or utralty - ots t mttr x a t collctor x C ar t locatos wr t lctro a ol quas-frm ottals collas. zolta.uszka@austramcrosystms.com

3 Prcl I a trasstor t trasfr currt s carr by lctros trougout t wol structur. bas currt s saratly scuss t artcular mollg tals but as t rlatv loss of s small t s glct t formulato. From 5 a 4 x q x q 4. Mor tratmt x q Substtutg 4 for t. rouct x q x 6 Rarragg x q x x 7 s quato s val aywr btw t mttr a collctor cotacts. Itgratg bot ss btw two abscssa a suc tat [ ] x c x,, 8 w gt q x x x x 9 ottal rfrc ca b arbtrarly slct. A covt coc s t ol quas- Frm ottal t ml ot x BM of t mtallurgcal B a BC juctos bcaus s costat trougout t bas rgo: 0 x BM 0 tgral o t LHS s t Gumml umbr of t volum btw t tgrato bouars x, x G arttog t Gumml umbr wll ot b tal r. Substtuto 9 yls, x x G q

4 4 s s t rcl. currt s varat agast t slcto of t tgrato bouars utl coto 8 s satsf. varous mol cocts bascally ffr t scfc fto of ts tgrato lmts. It ca b obsrv o Fg. tat t rgo bou by t xtral ss of t sac carg layrs [x b,x Cbc ] t ol quas-frm lvl s ractcally costat. Wt t rfrc slcto 0 t xotal trm t tgra of ca b omtt. Gumml umbr r ca b comut as t ol carg wgt by t osto t omator. ffct of trojuctos ca b tak to accout troug t trsc coctrato. I t jcto rgos s ot costat ay mor a t Gumml umbr ts trvals as to b comut by scal cosratos. s tals o ot affct t gralty of ts scusso a wll ot b alt wt r. 4. Classcal tratmt Gumml70 [] I t orgal ublcato [] o t a trasstor was vstgat. aalyss blow wll b ma o trasstors wt a fw mor cags ot affctg t rsults of t rfrr ar. Gumml assum a lctrc fl t moblty t form of x v D v s s 0 0 factor o t LHS of 7 covrts wt x x to s s x v D x v D x x x x x x ffuso costat D ts xrssos s t - og t - low fl valu. lctrostatc ottal taks ts gatv xtrmum t bas at x BM wt a valu of m. major cotrbuto to t tgral btw t mtallurgcal juctos [x j,x jc ] coms omatly from t art clos to m tus for t frst trm t Gumml umbr t followg uqualty ols < m B jc j D w x x G x, tgral of t sco, fl-t trm s rform saratly t trvals [x j,x BM ] a [x BM,x jc ] to rsolv t absolut valu fucto. xotals at t juctos ar glgbl comar to t valu at t ottal mmum tus m s jc j D v D x x G x,

5 5 7 D Wt rou valus of v s 0 cm / s a D 0cm / s w av 0m. I 970 vs w Gumml wrot s ar ts was glgbl comar to t cotmorary bas wts. oay owvr t mtallurgcal bas wt v for a morat frqucy trasstor of f 50GHz s w B 8m []. At mor g frqucy trasstors t cotrbuto of t moblty fl t bas carg trm s comarabl to t bas-gumml umbr a must b tak to accout. 5. Bolar rasstor Mol Catgors 5.. Rgoal Aroac Gtru, 978 [3] I s book Mollg t Bolar rasstor 978 [3], Gtru locat t tgrato bouars to t mttr s of t B a t collctor s of t BC sac carg layrs. As oos to t orgal mol formulato of Gumml a Poo [4] wt t bouars at t mttr a collctor cotacts, ot tat ts way "... o assumto s ma about t morty carrr quas-frm lvls t utral mttr a collctor rgos...". It s trstg to rcall tat Gumml al also ts bouars w formulatg t []. It s a mystry wy cag s m alf a yars latr [4] trasfr currt wt xb a xcbc bcoms Bb, b Bbc, Cbc x x q 3 G xb, xcbc argumts t umrator wt t ottal rfrc slcto 0 ar t bass across t two sac carg layrs as s o t quas-frm lvl agram. jucto voltags rctly trm t morty carrr coctratos at t lto layr gs troug t Boltzma rlato. Cosqutly t Gumml umbr t omator s also trm as a fucto of ts ar of bas voltags. Bb,b a Bbc,Cbc ar t two t cotrol voltags of t trasstor gratg t trasfr currt xlctly troug 3. By ts mov Gtru ot oly crat a fully masurmt-cosstt coct but o t way for t vlomt of a succssful mol srs SGP-BIC-MXRAM wt crasg rfcto t rgoal collctor scrto. c - rlatos t layr - wt t trval [x Cbc, x c ] - s aalyz DC mo o t bass of t xt Kull-Nagl tory wt t ror bouary cotos at x Cbc. I otr wors a bas-ot t varabl tral collctor rsstac s lk to t mol. AC bavour s aroratly ajust by t stor carg t layr. All ossbl cass.. t total lto, ltoomc, jctoomc a jctoot carrr ar cosr bot DC a AC. Mxtram as a mor rf collctor mol ta BIC. SGP rgars t wol rgo from x Cbc to t xtral collctor cotact ot sow o Fg.. - a sgl costat collctor rsstac rc. I BIC a MXRAM a atoal xtral collctor rsstac rcx rrsts t fx omc rgo btw x c a t xtral collctor cotact. zolta.uszka@austramcrosystms.com

6 6 5.. Global Aroac Gumml-Poo70 [4], Scrotr-R86 [5], Stübg-R87 [6] I ts aroac t wol trasstor s clu t carg cotrol tgral by takg t tgrato bouars at t osto of t mttr x a collctor x c cotacts. fx omc rgos from ts ots to t yscal cotacts ca b rrst by atoal xtral rsstacs t mol scmatc lk t rgoal aroac. trasfr currt bcoms Bb, Bbc, c x x q 4 G x, xc s coct was suggst mlctly toug - t orgal GP mol formulato by Gumml a Poo [4]. Howvr t actual GP mols aot t rgoal aroac of Gtru [3] a t a was ot ralz utl t troucto of Hcum [5], [6]. Isrtg t voltag ros across t crystall mttr a t taxal collctor layrs from Fg. to 4 Bb, b Bbc, Cbc c x x q 5 G x, xc Bb, a Bbc,c ar ot t cotrol voltags ay mor bcaus a c o t trasfr currt. o rsrv a xlct xrsso for ts voltag ros must b glct. I t actv mo of orato t collctor s rvrs bas a t sco xotal ca b omtt. Du to t g couctvty of t rcrystallz mttr rgo t voltag ro across [x, x b ] s glgbl. Hc ormal actv mo wt rvrs tral collctor bas 5 s r-formulat Hcum as Bb, Bb, b x x q q 6 G x, xc G x, xc Gumml umbr s comut rgoally across [x, x b ], [x b, x Bb ], [x Bb, x Bcb ], [x Bcb, x Ccb ] a [x Ccb, x c ] rsctvly: G x, xc G x, xb G xb, xbb G xbb, xbbc G xbbc, xccbc G xcbc, xc 7 Prformg t comutatos accorg to t ffct of baga varatos ca b cosr troug t baga c of t trsc coctrato. s way HBs ca also b moll. ar of rlatos 6 a 7 ar trm G.. t gralz Scrotr- Frrc-R93 [7]. outstag ovato of Hcum s to xtract t Gumml umbr total carg of t carrrs tractg wt t bas trmal from t masur trast tms of t trasstor. fuamtal coct of t mol s tat t trasfr currt automatcally rsults by substtutg t Gumml umbr - xtract from t trast tms to 6. s s oost to t rgoal aroac wr t DC bavour as t rorty. At g s crcuts t accurat mollg of t AC bavour may b artcularly mortat c Hcum s clam to b mor sut for suc alcatos. xtracto coct rovs a suror trast frqucy scrto comar to otr xstg mols. zolta.uszka@austramcrosystms.com

7 7 Omttg c.. rgarg t tral collctor rsstac zro owvr s ot fully stragtforwar. assumto of rc0 ca trouc rrors t scrto of bot of t DC a AC bavor. 6. Pottal Cosqucs of Nglctg rc 6. ffct o t DC bavor ublcato of Stübg-R87 [6] wat s a comrsv troucto to t omsoal Hcum, lmts t alcablty of t mol ts quato to c r was o xlaato gv for t rstrcto but t raso ca b urstoo from t scusso blow. valty of t quato 3 follows ot oly from t strct tortcal rvato of Gumml [] but was cofrm umrcally as wll [8] a was fou to b a farly goo aroxmato far to t g currt rgo. It follows from 3 tat at t coto Bb, b Bbc, Cbc.. w t tral collctor-mttr voltag s zro c 0 t trasfr currt gos also zro 0. Nglctg t trasfr currt of t Hcum aroac 5 ca b r-wrtt as Bb, b x c c _ Hcum q x 9 G x, xc Hc at 0 Hcum rovs a ozro ostv trasfr currt tus c _ Hcum > 0 Hcum ovrstmats t trasfr currt t quas/ saturato rgm. rror ca b lmt by t coto c >> c Not tat 8, or a smlar xlct rstrcto s mssg from t rst Hcum ocumtato [9] st of t fact tat t fuamtal mol axoms 6 a 7 av ot cag t matm. 6. ffcts o t AC bavour 6.. Collctor tm costat quato Stübg-R87 [6] s also quato 5 Scrotr-R85 [0] - troucs a mof trast tm τ * f τ f rc cjc for takg to accout t ffct of t omtt tral collctor rsstac o t stor carg. xlct obsrvato of t collctor tm costat owvr ca ot b fou t rct mol quatos [9], []. Sc bot rc a cjc ar bult o t sam cross scto, t collctor tm costat s t of t latral vc szs t frst orr: εε W 0 τ c rc cjc 3 q ND W zolta.uszka@austramcrosystms.com

8 8 wr for smlcty t lto wt W s assum to tak lac trly t collctor W 0 Cbc, Bbc B εε. qnd 0.8 Itrsc Collctor m Costat W 0.5um 0.7 tau_c rc*cjc; [s] cb 0 cb 0.75 cb.5 cb.5 cb og [cm -3 Fg. Collctor tm costat Fg. sows τ c for a wt of 0.5um t rag of tycal ogs. At a f 50GHz trasstor scrb [] t low bas trast tm was fou to b 0.68s. At mor vcs t collctor tm costat s comarabl to t total trast tm a must b tak to accout. low-currt trast tm comot τ f0 s scrb Hcum [9] by τ f 0 vb' C' τ 0 τ 0 c τ Bvl 4 c wt t ormalz tral BC lto caactac c C jc, t B' C' C jc0. Now 3 ca b r-wrtt as τ c τ τ 5 c c oug t yscal backgrou of τ Bvl s ffrt carrr jam t BC sac carg layr- t scrto of τ c formally ca b urstoo t rst trast tm mol of Hcum. fact of clug τ c t trast tm formulato soul b xlctly stat t mol ocumtato. c0 zolta.uszka@austramcrosystms.com

9 9 6.. wo-ort aramtrs C cotact B rbx rb cbcx cjc C vvbl BL vx v rcx 0 rc vx vbl vrc.icvbl HICUM: vvxvbl rcx Fg. 3 Hcum aroxmato A tutv ctur of t Hcum aroxmato s sow o Fg. 3. vrtcal rsstac rcx to t collctor s of t BC jucto s lss ta rc ur t mttr. Actually t s rgar zro all mols. Assum ormal actv orato w t collctor s of t BC sac carg layr s far from t BL. Obvously, voltag v at t o of cjc s ffrt from vx o cbcx. Hcum assums vvx a bot qual to t ottal vbl at t BL bouary. Ntwork lmts cbcx, cjc a rcx mt o sgl o rmovg rby a yscally xstg twork lmt from t quvalt. Atoally t DC bas of t two caactacs s forc to b qual wat - scally at crasg collctor currts - ot t cas s. Hcum usrs mgt tk at t frst glac tat aramtr rc0 rsolvs t roblm. Howvr rc0 s a trast tm aramtr wat as a fluc o t carg t omator of 5 oly. voltag cotos - umrator of 5 ar ot affct by ts aramtr. ffct of omttg rc o t two-ort ybr aramtrs wll b vstgat o t cor mols of Fg. 4. B rb r c cjc ro rc C B rb r c cjc ro C vb gm * vb vb gm * vb r r Pyscal quvalt Fg. 4. Smlf wo-port quvalts Hcum aroac Y g s c gm s cjc cjc s cjc go s cjc Z z rb r z r z z r rc r followg tts wll b aly to twork#: zolta.uszka@austramcrosystms.com

10 0 z z z z Dotg t crcut blocks t surscrts rb r z z r rb r z z r Substtutg y w av rb r z r y It wll b assum tat t currt ga of twork# ca b qually st to t masurmt valu H bot t lft a t rgt Hcum mol. Quatts rfrrg to t Hcum aroac wll b subscrt accorgly. ffrc of t ut ybr aramtrs bcoms _ Hcum _ Hcum 6 y currt ga of twork# ca b uc from z r z rc r troug rc z rc r z as rc r rc H z z ffrc of t two aroacs bcoms rc _ Hcum H z xrssg t outut mac wt t amttac aramtrs a glctg go from y w gt from 6 y y Hcum H rc H s cjc rc y y y y _ Abov t bta brakot frqucy s s ω 0 ω Hc t rlatv vato of t ut ybr aramtrs rsults as _ Hcum π f τ c 7 Rag τ c 0. 5s from Fg., t rlatv vato of at f 50GHz a 00GHz rsctvly s 6% a 3%. Cosqutly, t s ot ossbl to st bot a Hcum qual to t M aroac. zolta.uszka@austramcrosystms.com

11 6..3 Maxmum oscllato frqucy, f max owr ssat t layr s ot coutr for Hcum. us t mol s xct to ovrstmat t ultmat frqucy of orato. It as b sow [3] tat t assvty fucto * γ γ P γ 4 R γ R γ gos uty at t sam frqucy for ay sts of t two-ort aramtrs z, y,, s a g ta t Maso s ga U scfyg t uty ga frqucy of t trasstor. O gts wt t z aramtrs of t Hcum aroac rgt a of Fg. 4 for t assvty fucto of t yscal quvalt o t lft z P 4 R z z * R z * z z rc 4 R z R z rc R z zolta.uszka@austramcrosystms.com I t vcty of f max a -0B/D cl f ca b assum. Sc t LHS rrsts fmax t yscal P, t Hcum f max ca b obta as rc fmax_ cum fmax 8 R z z s t sum of r a t mac lookg t outut of twork# o t rgt a of Fg. 4 wt t bas o floatg. Nglctg ro for smlcty a otg t amttac of r cjc by y t voltag vb for a al outut forc voltag of s gv as s cjc vb ; s jω y s cjc outut currt rsults as s cjc s cjc gm y y s cjc y s cjc a t outut mac bcoms z y s cjc r s cjc gm y Not tat t trsc currt ga s gm y us z r s cjc gm Substtutg t trsc currt ga rolloff s s ω 0 ω w gt z r ω s cjc gm ω Nglctg r

12 ω rc ω ω cjc cjc ω τ c; cf R z cf gm c cjc gm τ f Substtuto 8 rsults cf π f τ c fmax_ cum fmax 9 f max cf π f τ c f Assum cf0. a as a cosrvatv stmato, f max f. Rag τ c 0. 5s from Fg., t ovrstmato of f max by Hcum at f 50GHz a 00GHz rsctvly yls 4% a 3%. 7. Summary I. I rct tcologs t cotrbuto of t moblty fl t bas carg trm s comarabl to t bas-gumml umbr. s soul b arss uto-at comact mols lk M a Hcum. II. Hcum ts to ovrstmat t trasfr currt t quas/ saturato rag. Bas lmts a vato magtu ar mssg from t mol ocumtato. III. fact of clug τ c toug formally - t trast tm scrto soul b xlctly stat t mol ocumtato. I. Wl Hcum scrbs t bst amog t rstly avalabl mols t s ot ossbl to st cocurrtly to b qual to tat M. rocss a bas t vato ca asly go u to valus xcg 5%. a av ot b vstgat but ts ar also subjct to vatos.. Hcum ovrstmats f max u to omttg t owr ssat t layr. rror crass wt t f max /f rato a ca asly xc 5%. I. cosstt two-ort matrcs may la to urctabl scracs at som AC alcatos lk rb xtracto [4] rfrr to t Atcts. Rlocatg rc to t quvalt at t cost of a atoal tral collctor o - woul lmat t roblms abov. Sc t bas t tral collctor rsstac s scrb for t trast tm formulato t ugra mgt b o v w/o atoal mol aramtrs. zolta.uszka@austramcrosystms.com

13 3 Rfrcs [] H. K. Gumml, A Carg Cotrol Rlato for Bolar rasstors, Bll Systm ccal oural, 970,. 5-0 [] M. Scrotr, H ra, W. Kraus, Grmaum rofl sg otos for SG LC HBs, Sol-Stat lctrocs [3] I.. Gtru, Molg t Bolar rasstor, lsvr, 978 [4] H.K. Gumml a H.C. Poo, "A Itgral Carg-Cotrol Mol for Bolar rasstors", Bll Systms ccal oural, ol. 49, 970, [5] M. Scrotr a H. M. R, A Comact Pyscal Larg Sgal Mol for Hg S Bolar rasstors Iclug t Hg Currt Rgo, NG Mtg, Würtzburg, May 986. [6] H. Stübg H.-M. R, A comact yscal larg-sgal mol for g-s bolar trasstors at g currt sts - Part I:, I ras. lctro Dv., ol. 34, , 987. [7] M. Scrötr, M. Frrc, a H.-M. R, A gralz Itgral Carg-Cotrol Rlato a ts alcato to comact mols for slco bas HB's, I ras. lctro Dv., ol. 40, , 993. [8] H.-M. R, H. Stübg a M. Scrötr, rfcato of t Itgral Carg-Cotrol Rlato for Hg-S Bolar rasstors at Hg Currt Dsts, I ras. lctro Dv., ol. 3, No. 6, , u 985. [9] tt:// [0] M. Scrotr a H. M. R, wo-dmsoal Molg of Hg-s Bolar rasstors at Hg Currt Dsts Usg t Itgral Carg-cotrol Rlato, Pysca 9B [] M. Scrötr a.-y. L, Pyscs-Bas Morty Carg a rast m Molg for Bolar rasstors, I ras. lctro Dv., ol. 46, , 999. [] M. Malory, M. Scrötr, D. Cl, D. Brgr, A mrov mto for trmg t trast tm of S/SG bolar trasstors, Proc. BCM,. 9-3, 003 [3] Z. Huszka,. Sbacr a K. Molar, stmato of fmax by t Commo Itrct Mto, Proc. BCM, 4.3 [4] Z. Huszka,. Sbacr a W. Pflazl, A xt wo-port Mto for t Dtrmato of t Bas a mttr Rsstac, I BCM005,.3 zolta.uszka@austramcrosystms.com