JUNTIONS I. HARNESSING ELECTRICAL CONDUCTIVITY IN SEMICONDUCTOR MATERIALS I.1
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1 OS A. La Rosa Physcs Dartmt P JUIOS I. HARSSIG LCRICAL CODUCIVIY I SMICODUCOR MARIALS I. Itrsc coductty ur slco I.2 trsc coductty slco dod wth slctd dffrt atoms II. RGY LVLS DIAGRAM II. II.2 Itrsc matrals trsc matrals matral, matral III. QUAUM SAISICS of LCROS III. h rmdrac Dstrbuto ad th Chmcal Pottal III.2 Alcato of th rmdrac dstrbuto to smcoductors III.2. Itrsc cas. Locato of th chmcal ottal III.2.2 trsc cas. Locato of th chmcal ottal IV. JUCIOS IV. Dod: ad matrals lacd togthr IV.2 Sac charg at a jucto IV.3 Chmcal ottal at juctos V. ORWARD IAS, RVRS IAS VI. COMPARISO of CHARGCARRIR POPULAIO at ACH SID of th JUCIO I. HARSSIG LCRICAL CODUCIVIY I SMICODUCOR MARIALS Lt s ta Slco as a tycal aml of smcoductor matral. A Slco atom has 4 alc lctros. Jod by coalt bods, S atoms form a lattc structur costtutd by two trtratg rodc fcc lattcs, hag Satomscm 3 8 atoms a cub of sd 5.43 Agstros. At zro tmratur Slco s a sulator bcaus all ts alc lctros artcat th bodg. It lacs lctros that ca wadr fr aroud th crystal structur. At g.. S atoms arragd a crystal twor.
2 =0, coductty = 0. I. Itrsc coductty Hr w cosdr a S matral wthout forg atoms just ur slco atoms. hrmal grato of coducto lctros At room tmratur ~ V, thrmal agtato of th crystall host rods rgy to lctros ad thr s a oashg robablty ~ g2 that som of thm wll b abl to bra away from th g =.2 V bodg ad, cosqutly, st almost fr ad wadr aroud th crystal: thy bcom coducto lctros. hs lctros ar g.2 Itrsc coductty. abl to artcat th coductty of slco udr lctros ad hols cotrbut th rsc of a tral lctrc fld. to lctrcal coductty. A tycal alu of th coctrato of coducto lctros gratd ths mar, at room tmratur, s o ~.40 0 cm 3. Grato of hols otc that th crato of a w coducto lctro las thr S atoms wth o bodg acacy. hs bodg acacy s calld a hol. Hols cotrbut to th coductty. Idd, udr th rsc of a tral lctrc fld, a bodd lctro from a S atom arby wll ha a tdcy to fll ths bodg acacy th fact that th S atom hostg th hol s a ost o hls th rocss ad ds ry lttl rgy to do that. Wh that has a w hol wth ts assocatd ost o s lft bhd, whch trggrs th crato of aothr hol, ad so o. h t rsult: a t ost charg s trasortd alog th fllg of hols. Sc a hol s cratd smultaously wth a coducto lctro, th coctrato of hols o quals th coctrato of lctros: o = o., both cotrbutg to what s calld trsc coductty of slco that s, wh fr of forg atoms. h trsc coductty stuato could b chagd f slctd forgr atoms ar troducd th slco crystall structur, as dscrbd th t scto I.2 trsc coductty ty: S crystal dod wth door atoms 5 thr alc shll. our artcat th bodg wth S atoms, whl th ffth bcoms almost fr of ay bodg. Wth a lttl bt of rgy ths ffth bcoms fr to wadr aroud th tr slco matral. aml of dog ll: 0 6 arsc atomscm 3. otc th ablty to dod S wth forgr atoms wth coctratos css of th trsc rgm. ty: S dod wth acctors atoms 3 alc. oro atoms, for aml.
3 h thr artcat th bodg wth S atoms, cratg a bod acacy to b flld. hat s, acctor atoms crat hols, whch also artcat th coductty as lad abo. Accordgly, th rsc of doors ad acctors crass th lctrcal coductty of th matral. Acctors Doors g.3 trsc coductty. Harssg lctrcal coductty by srtg to th S crystal matr forg murts II. RGY LVLS DIAGRAM As solatd atoms ar brought togthr to form a sold, arous tractos occurs btw ghborg atoms. I th rocss, mortat chags occur th lctro rgy cofguratos. Wh th sold s formd th forcs of attracto ad rulso btw atoms fs a balac at th ror tratomc sacg th slt rgy lls dslay sstally cotuum bads of rgy. rgy 4 stats carbo atoms brought togthr Coducto bad 6 stats Carbo atom 6 atom 2 g 4 2s stats 2 stats Valc bad 2 2 stats stats s Damod lattc sacg brought togthr Atomc sarato g.4 ormato of rgy bads by brgg togthr tally solatd carbo atoms. At = 0 0, smcoductor matrals ha bascally th sam structur as sulators:
4 a flld alc bad saratd from a mty coducto bad by a a badga of oallowd rgy stats mty g lld Isulator matral mty g lld Smcoductor matral g.5 ad dagram at = 0 K for two dffrt matrals. h dffrc btw a sulator ad a smcoductor ls th sz of th badga g. h lattr s much smallr smcoductors. Comar, for aml,. V for S wth 5V for damod. Wth th bad structur modl, udr th ffct of a tral lctrc fld, flld bads do ot cotrbut to th lctrcal coductty of th matral. At 0 K, th rlatly small badgas of smcoductors allow for ctato of lctros from th alc bad to th ur coducto bad a thrmal ctato. hs gs rs to th crato of lctrohol ars, whch ar th carrrs for lctrcal coductty. h grato of h ars ca b qualtatly sualzd as th brag of coalt bods th crystal lattc g.5. h rgy rqurd to bra th bod s th badga rgy g. [A shortcomg of th bro bod ctur s that th fr lctro ad th hol aar dctly localzd th crystal lattc. Istad, thr osto srad or sral lattc uts udrgog robablstc quatum mchac jums.] h robablty for such a jum S s roortoal to. V. h robablty for such a jum damod s roortoal to 5 V. hus, du to thrmal ctato, smcoductors th umbr of charg carrrs aalabl for lctrcal coducto s much gratr tha sulators. W wll s latr that thr ar othr ways to cras th lctrcal coductty.
5 At = 300 K r lctro Hol g g.6 hrmal grato of lctrohol ar carrrs. II. Itrsc slco A rfct smcoductor crystal wth o murts or lattc dfcts s calld a trsc smcoductor. I such matrals thr ar o charg carrrs at =0 K. h alc bad s flld ad th coducto bad s mty. At 0 K, lctrohol ars ar gratd s also g. 6 abo. Slco lctro rgy Coducto bad g badga Valc bad orbdd lls rgy lls aalabl to b occud by lctros g.7 rgy ll dagram for ur slco. umbr of lctros th coducto bad s qual to th umbr of hols th alc bad. or trsc smcoductor matral Coducto bad Valc bad lctro coctrato = coctrato of hols lctroscm 3 holscm 3
6 II.2 trsc slco matral matral lctro rgs Acctor s rgy lls acctors atoms door atoms r lctros door s rgy lls lctrcally utral lctrcally utral matral matral g.8 I th matral matral th rgy ll of murts ar locatd ar th alc coducto bad. III. QUAUM SAISICS of LCROS III. h rmdrac Dstrbuto ad th Chmcal Pottal gur 8 shows a systm tractg wth a hug thrmal rsror. h coct of thrmal rsror s a dalzato coyg that o matr what amout of rgy s trasfrrd to or ta from th rsror ts tmratur rmas costat. Rsror at tmratur rgy chag P oltzma dstrbuto Small systm A rgy g.9 A systm A trchagg rgy wth a thrmal rsror. h robablty of fdg th systm A a stat of rgy s roortoal to.
7 W scalz to th cas whch w ha a systm comosd of otractg lctros. hat s, lctros ar dstrbutd amog th dffrt ossbl olctro stats. h fgur blow shows two ossbl artcular cofguratos corrsodg to th cas =3. Sc may cofguratos ar ossbl g.9 shows two of thm, lt s call th rgy assocatd to a g cofgurato or statoary stat. rgy d 2 rgy g.0 wo ossbl statoary stats of th lctro systm. or smlcty w ha chos =3. h robablty of fdg a lctro systm a artcular stat o of rgy o s g by, α α α α P o o 2 I th domator th sum s all or th ossbl statoary stats. h domator s ow as th artto fucto ad t s rlatd to th Hlmholtz fr rgy = US whr U s th tral rgy ad S th troy by, α α Dfto of 3 hus, P o o o α α α W ca omt th sub d o th. α α α P 4
8 W ha to cororat th Paul s cluso rcl, whch rmts at most o lctro to occuy ay sgl lctro ll. Accordgly, to costruct a lctro stat o must fll dffrt olctro lls. hus ach lctro statoary stat ca b scfd by lstg whch of th olctro ll ar flld that stat. or aml [3,5,7] ad [2,3,6] rsctly th amls g g. 9 abo, for =3. A quatty ry usful to calculat s f, th robablty to fd a lctro th artcular olctro ll, wh th lctro systm s thrmal qulbrum. f P 5 Summato tds all or th lctro stats whch thr s a lctro th olctro ll. otc, f P 6 Summato tds all or th lctro stats whch thr s o lctro th olctro ll. otc, y tag ay lctro stat whch thr s a lctro th olctro ll, w ca costruct a lctro stat whch thr s o lctro th ll, by smly rmog th lctro th th ll, lag th occuato of all th othr lls ualtrd. urthrmor, Ay lctro stat wth o lctro th olctro ll ca b costructd from just o lctro stat wth o lctro th ll. dtly, th rgs of ay lctro stat ad th corrsodg lctro stat dffr by just. hus, th st of rgs of all lctro stats wth th ll uoccud th stats dfd 6 abo, s th sam as th st of rgs of all lctro stats wth th ll occud lt s call thm th stats rodd that ach rgy th lattr st s rducd by.
9 hat s to say, γ P = P Summato tds all or Summato tds all or th lctro stats th lctro stats whch thr s o lctro whch thr s a lctro th olctro ll th olctro ll. Hc, P f 7 Summato tds all or th lctro stats whch thr s a lctro th olctro ll. Usg 4 χ χ ε P f f f f f f Sc th summato tds all or th lctro stats whch thr s a lctro th olctro ll, o obtas,
10 f f h chmcal ottal at tmratur s dfd by = 9 rsso 8 tas th form, f f 8 0 Sc s of th ordr of 0 22, addg o lctro wll hardly altr th robablty hus, f f f. Hc, f Probablty to fd a lctro th artcular olctro ll, of rgy, wh th lctro systm s thrmal qulbrum at tmratur. otc f s also th ma umbr of lctros th olctro ll, f = 0 f f f 2 Ma umbr of lctros th olctro ll, I subsqut rssos, th lct rfrc to s drod t wll b mlctly assumd. f rmdrac dstrbuto 3 Imlctly, th chmcal ottal carrs formato of.
11 f 2 =0 2 > > 0 2 g. h rmdrac dstrbuto III.2 Alcato of th rmdrac dstrbuto to smcoductors or a gral frmos systm, t s dffcult to aluat lctly th alu of th chmcal ottal. or th cas of smcoductors, w wll follow a tut grahc trrtato aroach. III.2.. Grahc mthod to locat th chmcal ottal : Itrscmatral cas I th trsc cas, th coctrato of lctro cm 3 th coducto bad s qual to th coctrato of hols holscm 3 th alc bad, lctroscm 3 = holscm 3 Rcall that w cosdr a hol as th absc of a lctro. hus, If f s th robablty to fd a lctro a ll of rgy, th f s th robablty that th ll of rgy s mty. I othr words, f s th robablty to fd a hol at th rgy ll. 4 otc also th symmtry of th rmdrac dstrbuto f aroud : th robablty f that a stat of rgy abo s flld s th sam as th robablty f that a stat of rgy blow s mty 5 or th trsc cas, rssos 4 ad 5 mly that th chmcal ottal must b locatd ar th mddl of th badga.
12 2 f 0 Probablty to fd a hol Probablty to fd a lctro Carrr coctrato g g. 2 Locato of th chmcal ottal for th trsc cas. III.2.2. Grahc mthod to locat th chmcal ottal : trscmatral cas 2 f 0 Probablty to fd a hol Probablty to fd a lctro Carrr coctrato g g. 3 Locato of th chmcal ottal for th ty trsc cas.
13 f Probablty to fd a hol 2 0 Probablty to fd a lctro Carrr coctrato g g. 4 Locato of th chmcal ottal for th ty trsc cas. III.3 umbr of carrrs lctros ad hols thrmal qulbrum: Gral trsc cas III.3. h law of mass acto h most mortat rorty of ay smcoductor at tmratur s c, th umbr of lctros r ut olum th coducto bad, ad, th umbr of hols r ut olum th alc bad. c, ad dd crtcally o th rsc of murts doors ad acctors. Howr, thr ar crta rlatos that hold rgardlss of th urty of th saml, l, for aml, th law of mass acto whch s cosdrd t. Lt g c b th dsty of lls th coducto bad g b th dsty of lls th alc bad c d c g 6 d g d g
14 Imurts doors ad acctors affct th alus of c, ad oly through th alu of th chmcal ottal. Cas: odgrat aromato W scalz th cas satsfyg th followg codto, c 7 I ths aromato, 8 rsso 6 bcoms, c d g c d g c c d g c d g 9 hat s, whr c c c c c d g c d g 20 O caot stll fr th alu of c or from 20 utl th alu of th chmcal ottal s dtrmd. Howr, otc th followg, c c c g c h law of mass acto 2 Vald for a gral trsc cas udr th odgrat aromato
15 IHOMOGOUS SMICODUCORS Sgl crystals whch th local coctrato of of doors ad acctors murts ars wth osto. O way to ma such crystals s to ary th coctrato of murts th mlt as th growg crystal s slowly tractd, thus roducg a arato murty coctrato alog o satal drcto. Dlcat mthods of fabrcato ar dd bcaus t s grally mortat, for ffct orato, that thr b o grat cras lctroc scattrg assocatd wth th arato murty coctrato. IV. JUCIOS A jucto s a smcoductg crystal whch th murty coctrato ars alog a g drcto tas as th as hr. or < 0 t has door murts, hc calld ty or > 0 t has acctors murts, hc calld ty Dog rofl cas of a abrut jucto a d cost cost A D for for 0 0 matral matral Imurty dsty o A D 0 g. Dog rofl. o: ad matrals lacd togthr. ottom: Dog rofl for th cas of a abrut jucto. h ouformty murty coctrato ducs ouformty, th dsts of coducto bad lctros 2 c th dsts of hols th alc bad
16 matral matral lctro rgs A acctors cm 3 D doorscm 3 rgy dagram lctrcally utral matral Chargd rgo lctrcally utral matral g. rgy dagram. Wh th ad samls ar lacd togthr, lctros ad hols dffus ad rcomb ad, th rocss, la bhd ubalacd gat ad ost rgos that d u costtutg what s calld th dlto rgo. Cas: Cas: A > D A > D = ult lctrc fld Cross scto ara A W Dlto rgo Practcally thr ar ot fr carrrs ths rgo g. 9a ormato of a dlto rgo. Wh th ad samls ar lacd togthr, lctros ad hols dffus ad rcomb ad, th rocss, la bhd ubalacd gat ad ost rgos that d u costtutg what s calld th dlto rgo. otc, sc th t charg o ach sd s th sam, th lowr th coctrato of murts o sd, th wdr th dlto rgo that sd.
17 IV.2 Sac charg at a jucto Wth th trasto rgo, lctros ad hols ar trast from o sd of th jucto to th othr. lctros dffus from to, but thy ar also subjctd to th bult lctrc fld that ushs thm from from th rgo to th ; at th d thr s a qulbrum btw ths two trds. Corsly for hols. hr ar ry fw carrrs wth th trasto rgo at ay g tm, sc th lctrc fld srs to sw out carrrs whch ha wadrd to W. o a good aromato, w ca cosdr th sac charg wth th trasto rgo W as du oly to th ucomsatd door ad acctor os. hs aromato s rflctd g. 9b. Charg dsty Q= A o A o D A o Q=A o D g. Charg dsty s osto across a abrut jucto. A s th cross scto ara of th jucto. Imurty dsty o A D 0 Carrr dsty rom th Mawll quato, = q A th ty rgo, ad = q D th ty rgo
18 d q ε o q ε A A d o d q D d ε o q D ε o o q ε A o q ε D o lctrc fld o W o d q A d ε d q d ε D o g.9c h lctrc fld crats a ottal dffrc V 0 across th jucto. dv d o o or o dv d o o V d o = W o 2 o Sc ad W A o D o o o
19 V o 2 q ε A D A D W 2 hs rsso udrscor th rlatosh btw th oltag across th jucto ad th wdth of th dlto layr. IV.3 Chmcal ottal at juctos At tmratur, smcoductor matrals ar charactrzd by thr chmcal ottal, a rgy ll locatd somwhr th badga. Its act locato dds o th dog acctors ad doors coctratos. lctro rgs Chmcal ottal Chmcal ottal rgy lls bfor of th ddual ty ad ty slco bfor jog thm. ach matral has ts ow chmcal ottal ll. Wh th matrals ar jod, lctros dffus from to lag bhd ubalacd ost door os ad rcomb wth hol th rgo thus cratg gat acctor os. hols dffus from to. Whl dffuso tas lac, a lctrc fld that ooss th dffuso rocss buldsu utl a stady stat s rachd. I ths stuato a sgl rm ll charactrzs th whol jucto systm.
20 rgy rm ll ult tral lctrc fld = W lctrostatc ottal rgy of V o V o rm ll
21 V. ORWARD IAS, RVRS IAS orward bas V f rgy of f rgy of V f V f V o V o V f Rrs bas V f V R rgy of R rgy of V R V R V o V R V o V R X X
22 Org of th drft currt I o V f I o f rgy of Drft of thrmally gratd lctros hrmally gratd h ars V f hrmally gratd h ars Drft of thrmally gratd hols I o X lctros thrmally gratd ar th jucto ar swt by th tral lctrc fld that sts across th jucto. Smlar stuato has to th thrmally gratd hols. h drft mchasm llustratd th fgur abo costtut th org of th uaodabl drft currt I 0. otc, th lctros ad hols wll b swt by whatr lctrc fld s rst across th jucto, rgardlss of th actual alu of. hat s, th rat of chargs crossg th jucto s somhow ddt of th forward or rrs bas oltags. Oc thy ar gratd arby th jucto, ay lctrc fld wll sw thm across. hus, I 0 dds oly o how fast ar th h ars gratd, whch s flucd by th bad ga g ad tmratur.
23 VI. Comarso of chargcarrr oulato at ach sd of th jucto : coctrato of th sd : coctrato of th sd : coctrato of hols th sd : coctrato of hols th sd h subd 0 stads for alus wh th systm s qulbrum rgy of o V o o sd = o sd = o carrr dsty oulato rgo rgo X rom thrmodyamcs cosdratos: o = o qv o
24 Smlarly: lctrostatc ottal o sd = o V o P o sd = o rgo rgo X hol carrr dsty oulato rom thrmodyamcs cosdratos: rom ad 2 : o o = o o 3 I othr words: o o sd = o o sd Morty ad majorty carrrs : coctrato of hols th sd : coctrato of hols th sd h subd 0 stads for alus wh th systm s qulbrum
25 majorty carrrs o o morty carrrs lctrostatc ottal o rom thrmodyamcs cosdratos: V o o = o qv o o rgo rgo htt:
26 Rfrcs:. J. R. Cogdll, oudatos of lctrocs, Prtc Hall, 999.\ G. Strtma, Sold Stat lctroc Dcs, Prtc Hall, Ashcroft ad Mrm, Sold Stat Physcs, Saudrs Collg G. Strtma, Sold Stat lctroc Dcs, Prtc Hall990.
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