IX.2. A Semiconductor Device Primer

Transcription

1 1 IX.2. Smcoductor vc Prmr blograhy: 1. Grov,.S., Physcs ad chology of Smcoductor vcs (Joh Wly & Sos, w Yor, 1967) 2. Sz, S.M., Physcs of Smcoductor vcs (Joh Wly & Sos, w Yor, 1981) K S988, IS Kttl, C., Itroducto to Sold Stat Physcs (Joh Wly & Sos, w Yor, 1996) QC176.K5, IS Carrr Coctratos h robablty of a lctro stat th coducto bad bg flld s gv by th rm-rac dstrbuto f ( 1 ) ( ) / + 1 h trasto broads as th thrmal rgy crass, as llustratd for 0.005, ad 0.1 V. rm-rac strbuto ucto 1.0 Probablty of Ocuacy (300K) rgy Uts of th rm Lvl Itroducto to Radato tctors ad lctrocs, 30-Mar-99 IX.2.a. Smcoductor vc Prmr - og ad ods Hlmuth Slr LL

2 2 h dsty of atoms a S or G crystal s about atoms/cm 3. Sc th mmum carrr dsty of trst ractcal dvcs s of ordr to cm -3, vry small ocuacy robablts ar qut mortat. rm-rac strbuto ucto Probablty of Occuacy V (300K) V (400K) rgy Uts of rm Lvl I slco th bad ga s 1.12 V. If th rm lvl s at mdga, th bad-dgs wll b 0.56 V abov ad blow. s s aart from th lot, rlatvly larg dvatos from th rm lvl,.. xtrmly small occuacs, wll stll yld sgfcat carrr dsts Itroducto to Radato tctors ad lctrocs, 30-Mar-99 IX.2.a. Smcoductor vc Prmr - og ad ods Hlmuth Slr LL

3 3 h umbr of occud lctro stats s dtrmd by summg ovr all avalabl stats multld by th occuato robablty for ach dvdual stat m f ( ) Sc th dsty of stats ar th bad dg tds to b qut hgh, ths ca b wrtt as a tgral f ( ) g( ) d whr g() s th dsty of stats. c Soluto of ths tgral rqurs owldg of th dsty of stats. ortutously, to a good aroxmato th dsty of stats ar th bad dg has a arabolc dstrbuto g( ) d ( c ) 1/2 s th rgy crass byod th bad dg, th dstrbuto wll dvat from th sml arabolc form, but sc th robablty fucto dcrass vry radly, th tgral wll hardly b affctd. h scod obstacl to a sml aalytcal soluto of th tgral s th tractablty of tgratg ovr th rm dstrbuto. Itroducto to Radato tctors ad lctrocs, 30-Mar-99 IX.2.a. Smcoductor vc Prmr - og ad ods Hlmuth Slr LL

4 4 If - s at last svral tms, th rm dstrbuto ca b aroxmatd by a oltzma dstrbuto 1+ ( ) / ( ) / f ( ) ( )/ rm-rac strbuto vs. oltzma roxmato 1 Probablty of Occuacy 0.1 rm-rac strbuto oltzma roxmato rgy Sacg from rm Lvl Uts of t rgs byod 2.3 of th rm lvl th dffrc btw th oltzma aroxmato ad th rm strbuto s <10%, for rgs >4.5 t s lss tha 1%. lyg th aroxmato to th occuacy of hol stats, th robablty of a hol stat bg occud,.. a valc stat bg mty s f h ( ) 1 f ( ) ( )/ )/ Sc th bad ga s of ordr 1 V ad at room tmratur s V, th codtos for th oltzma aroxmato ar fulflld for xctato across th bad ga ( Itroducto to Radato tctors ad lctrocs, 30-Mar-99 IX.2.a. Smcoductor vc Prmr - og ad ods Hlmuth Slr LL

5 5 Wth ths smlfcatos th umbr of lctros th coducto bad thrmal qulbrum s ad th lctro coctrato ( ) 3/ 2 ( c ) / c ( c )/ whr c s th ffctv dsty of stats at th bad dg. Corrsodgly, th hol coctrato v ( )/ v h ffctv dsts of stats S ad G ar Slco: c 2.8 x cm -3 v 1.04 x cm -3 Grmaum: c 1.04 x cm -3 v 6.0 x cm -3 I a ur smcoductor ach lctro th coducto bad corrsods to a hol th valc bad, whr s th carrr coctrato trsc to a dal crystal whr th oly oulato mchasm s xctato across th bad ga. Usg th abov rsults c ( c )/ ( v )/ v whch o ca solv to obta ( ) c v log c / If th bad structur s symmtrcal ( c v ), th trsc rm lvl ls th mddl of th bad ga. v Itroducto to Radato tctors ad lctrocs, 30-Mar-99 IX.2.a. Smcoductor vc Prmr - og ad ods Hlmuth Slr LL

6 6 v rathr substatal dvatos from a symmtrcal bad structur wll ot affct ths rsult sgfcatly, as c / v trs logarthmcally ad s much smallr tha th bad ga. Slco ( g 1.12 V): 1.45 x cm -3 at 300K Grmaum ( g 0.66 V): 2.4 x cm -3 at 300K or comarso: h urst smcoductor matral that has b grow s G wth actv murty lvls of about cm -3. ot that th trsc carrr coctrato oly als to dal crystals, whr th oly sourc of mobl carrrs s thrmal xctato across th bad ga wthout ay addtoal murty atoms or crystal mrfctos that would allow othr oulato mchasms. rmarabl rsult s that th roduct of th lctro ad hol coctratos 2 ( c v )/ g / c v c v dds oly o th bad ga g ad ot o th rm lvl. hs rsult, th law of mass acto, s vry usful smcoductor dvc aalyss. It rqurs oly that th oltzma aroxmato holds, so t s ot lmtd to udod or dal crystals. Qualtatvly, t says that f o carrr ty xcds th qulbrum coctrato, rcombato wll dcras th coctratos of both lctros ad hols to mata 2. Itroducto to Radato tctors ad lctrocs, 30-Mar-99 IX.2.a. Smcoductor vc Prmr - og ad ods Hlmuth Slr LL

7 7 2. Carrr Coctratos od Crystals h qualty h oly holds for ur crystals, whr all of th lctros th coducto bad hav b thrmally xctd from th valc bad. I ractcal smcoductors th rsc of murts ts th balac towards thr th lctros or hols. Imurts ar a uavodabl byroduct of th crystal growth rocss, although scal tchqus ca achv astoudg rsults otably ultraur G whr th t murty coctrato s about cm -3,.. a rlatv coctrato of I smcoductor dvc tchology murts ar troducd ttoally to cotrol th coductvty of th smcoductor. Lt d + b th coctrato of ozd doors ad a - th coctrato of ozd acctors. Ovrall charg utralty s rsrvd, as ach ozd doat troducs a chargd carrr ad a oostly chargd atom, but th t carrr coctrato s ow or + d a ssum that th actvato rgy of th doors ad acctors s suffctly small so that thy ar fully ozd h + ad + +, Itroducto to Radato tctors ad lctrocs, 30-Mar-99 IX.2.a. Smcoductor vc Prmr - og ad ods Hlmuth Slr LL

8 8 2 whch, usg, bcoms If th acctor coctrato >> ad >> +.. th coductvty s domatd by hols. + 1, 2 << Covrsly, f th door coctrato >> ad >> th coductvty s domatd by lctros. If th coductvty s domatd by oly o ty of carrr, th rm lvl s asy to dtrm. or xaml, f >>, th xrsso ca b wrtt + + yldg c ( )/ c c log c If >>, th c - must b small,.. th rm lvl ls clos to th coducto bad dg. I ralty th murty lvls of commo doats ar ot clos ough to th bad dg for th oltzma aroxmato to hold, so th calculato must us th rm dstrbuto ad solv umrcally for. vrthlss, th qualtatv coclusos drvd hr stll aly. Itroducto to Radato tctors ad lctrocs, 30-Mar-99 IX.2.a. Smcoductor vc Prmr - og ad ods Hlmuth Slr LL

9 Itroducto to Radato tctors ad lctrocs, 30-Mar-99 Hlmuth Slr IX.2.a. Smcoductor vc Prmr - og ad ods LL 9 It s oft covt to rfr all of ths quatts to th trsc lvl, as t accouts for both c ad v. h ad th rm lvl th -rgo Varato of rm lvl wth dog ad tmratur, cludg arrowg of th bad ga wth tmratur: (from Sz) c c / ) ( / ) ( v v / ) ( / ) ( log

10 Juctos - jucto s formd at th trfac of a - ad a -ty rgo. (from Kttl) Sc th lctro coctrato th -rgo s gratr tha th -rgo, lctros wll dffus to th -rgo. Corrsodgly, hols wll dffus to th -rgo. s lctros ad hols dffus across th jucto, a sac charg du to th ozd door ad acctor atoms bulds u. h fld du to ths sac charg moss a vlocty comot th oost drcto. h stuato s dyamc: h coctrato gradt causs a cotuous dffuso currt to flow. h fld du to th sac charg drvs a drft currt th oost drcto. Itroducto to Radato tctors ad lctrocs, 30-Mar-99 IX.2.a. Smcoductor vc Prmr - og ad ods Hlmuth Slr LL

11 11 qulbrum s attad wh th two currts ar qual,.. th sum of th dffuso ad drft currts s zro. h t hol currt dsty s whr s th dffuso costat for hols ad s th lctrc fld th -rgo. o solv ths quato w ma us of th followg rlatoshs: h hol coctrato s d J q + q µ dx so ts drvatv d dx ( ) / d dx, d dx Sc th forc o a charg q du to a lctrc fld s qual to th gatv gradt of th ottal rgy, dc dv d q dx dx dx s oly th gradt s of trst ad c, v ad dffr oly by a costat offst, ay of ths thr masurs ca b usd. W ll us th trsc rm lvl sc t als throughout th saml. h rmag grdt s th st rlatosh, whch rlats th moblty to th dffuso costat µ q Itroducto to Radato tctors ad lctrocs, 30-Mar-99 IX.2.a. Smcoductor vc Prmr - og ad ods Hlmuth Slr LL

12 12 Usg ths rlatoshs th t hol currt bcoms d J q µ dx ccordgly, th t lctro currt J d q dx µ d dx Sc, dvdually, th t hol ad lctro currts qulbrum must b zro, th drvatv of th rm lvl d dx d dx 0 thrmal qulbrum th rm lvl must b costat throughout th jucto rgo. or th rm lvl to b flat, th bad structur must adat, sc o th -sd th rm lvl s ar th valc bad, whras o th -sd t s ar th coducto bad. If w assum that th doats ar xclusvly doors o th -sd ad acctors o th -sd, th dffrc th rsctv rm lvls s log hs corrsods to a lctrc ottal 2 V 1 q V b oft rfrrd to as th bult- voltag of th jucto. Itroducto to Radato tctors ad lctrocs, 30-Mar-99 IX.2.a. Smcoductor vc Prmr - og ad ods Hlmuth Slr LL

13 13 s thr or crass rlatv to, th rsctv rm lvl movs closr to th bad dg, crasg th bult- voltag. Wth crasg dog lvls th bult- voltag aroachs th quvalt ottal of th bad-ga g /q. (from Sz) Itroducto to Radato tctors ad lctrocs, 30-Mar-99 IX.2.a. Smcoductor vc Prmr - og ad ods Hlmuth Slr LL

14 14 h hrt ottal dstrbuto th jucto lads to a dlto rgo, whos wdth ca b crasd by alcato of a xtral ottal,.. rvrs basg th jucto. hs was dscussd a rvous lctur. (from Kttl) ow th forward bas mod wll b tratd mor dtal. Comlcato: lyg a xtral bas lads to a codto that dvats from thrmal qulbrum,.. th rm lvl s o logr costat throughout th jucto. Itroducto to Radato tctors ad lctrocs, 30-Mar-99 IX.2.a. Smcoductor vc Prmr - og ad ods Hlmuth Slr LL