Izt erges S, Ge, GAs
xtrsc Semcuctrs A extrsc semcuctrs s efe s semcuctr whch ctrlle muts f secfc t r murty tms hve bee e s tht the thermlequlbrum electr hle ccetrt re fferet frm the trsc crrer ccetrt. Oe tye f crrer wll be remt. Ag murtes wll chge the strbuts f electrs hles the mterl sce the erm level s relte t the strbut fuct wll ls chge. I the fgure: >g/ > -tye V c ex ex v
rrer strbut f -tye semcuctr <g/ > -tye semcuctr
xtrsc Semcuctrs The exresss revusly erve fr the therml-equlbrum ccetrts f electrs hles re geerl: V v ex ex fr -tye semcuctr electrs re mjrty crrers hles mrty fr -tye semcuctr hles re mjrty crrers electrs mrty ex ex ex G V V V The ruct s lwys cstt fr gve semcuctr t gve temerture ly whe the Bltzm rxmt s vl.
vlty f the Bltzm rxmt If c - >> the - >> the erm rbblty fuct reuces t the Bltzm rxmt f 1 1 ex [ ex If the Bltzm rxmt es t hl - very clse t the cuct b : * 3 / 4π m c h 3 c 1 ex If we g mke chge f vrble η ls efe η
vlty f the Bltzm rxmt 4π m h η * 3 The tegrl s efe s 1/ 0 3 / 0 1/ η 1 ex 1/ η 1 ex [ η η ] [ η η ] η η Ths fuct s clle erm-drc tegrl s tbulte fuct f the vrble η. The sme geerl meth c be use t clculte the therml equlbrum hle ccetrt ' 1/ η ' V V η η η ' ' [ η η ] * 3/ 4π m h ' 3 0 1 ex
xmle x: S t 300 K η - 5meV 5meV - KT 0 0 1/ η.8 10 π 19.8 10.3 7.7 10 π 19 19 cm cm 3 3 1/.3 te f we wul use the Bltzm rxmt t vl ths cse 0 0.08 10 cm 3
Degeerte egeerte semcuctrs I the revus scuss f g t tms t semcuctr we hve mlctly ssume tht the ccetrt f t tms e s smll whe cmre t the semcuctr tms ~10 4 cm -3. The murty tms re sre fr eugh rt s tht there s t terts betwee r electrs. We hve ssume tht the murtes truce screte, terctg r ccetr eergy sttes -tye -tye semcuctr egeerte semcuctr. If the murty ccetrt creses the stce betwee the murty tms ecreses t e t the r ccetr electrs wll beg t terct wth ech ther. Whe ths ccurs the sgle escrete r ccetr eergy stte wll slt t b f eerges my verl the bttm f the cuct b. Whe the electr hle ccetrts excees c the les wth the cuct b egeerte semcuctr.
egeerte Degeerte ergy-bs egeerte semcuctr Degeerte semcuctr Smll r the g Hgh r
Sttstcs f rs ccetrs The ervt f the erm-drc rbblty fuct ws erve cserg the Pul exclus rcle. Ths rcle ls les t the r ccetr sttes. The sttstcs f fllg b g levels ffers slghtly frm the sttstcs f fllg eergy b levels. A r ste c ccmte ly e electr hwever the r-ste electr c hve ether s-u s-w g s egeercy. Ths fct creses the wys f rrgg electrs the r-level stes The rbblty fuct f electrs ccuyg the r stte s: 1 1 ex g g where s the electr esty ccuyg the r level s the r eergy level esty f the r tms esty f ze rs
Sttstcs f ccetrs The sme fr ccetr tms: g 1 1 g 4 ex where s the hle esty ccuyg the ccetr level s the ccetr eergy level gx4 fr S GAs ue t tw-fl hevy lght-hle egeercy f the vlece b esty f the ccetr tms - esty f ze ccetr tms
umber f electrs the r stte ex ex 1 If we ssume >> I ths ct the Bltzm rxmt s ls vl the fr the electr the cuct b we hve: ex We wul lke t eterme the reltve umber f electrs the r stte cmre t the ttl umber f electrs: ex 1 1 ex ex ex
mlete Izt x: we wt t eterme the frct f electrs re stll the r stte t T300 K the cse f P r tms S 10 16 cm -3. 1 1 0.0041 19.8 10 0.045 ex 16 10 0.059 0.4% Ths exmle shws tht t RT ly few electrs re the r stte cmre wth the cuct b. Oly 0.4% f the r sttes ct electrs: the r sttes re s t be cmletely ze. The sme s vl fr the ccetr sttes. Therefre t RT ll r ccetr tms hve te electr t the cuct b.
reeze-ut The ste ccurs t T0 K. All electrs re ther lwest ssble eergy stte r 0 > 0 ex 0 ; ex ex 1 1 g. 4-13
meste semcuctrs I therml equlbrum the semcuctr crystl s electrclly eutrl. Ths chrge-eutrlty ct s use t eterme the therml-equlbrum electr hle ccetrts s fuct f the murty g ccetrt. A cmeste semcuctr s e tht cts bth r ccetr murty tms the sme reg. -tye cmeste semcuctr ccurs whe > -tye cmeste semcuctr ccurs whe < If we hve cmletely cmeste semcuctr chrcterstcs smlr t trsc.
ergy-b grm meste semcuctr
fr cmeste semcuctrs I therml equlbrum the semcuctr crystl s electrclly eutrl the et chrge esty s zer. If we ssume cmlete zt t RT 0 0 0 r trsc semcuctr 0 fr >
Determt f cmeste semcuctr x: -tye S t T300 K wth 10 16 cm -3 10 1 cm -3 Assume 1.5x10 10 cm -3 10 16 10 16 10 16 3 1.5 10 10 cm mjrty crrer ccetrt OT: 10 1.5 10 4 3.5 10 cm 16 1 10 mrty crrer ccetrt > < electr restrbut lg the vlble eergy sttes >
0 cmeste semcuctr The electr ccetrt the cuct b creses bve s we r tms the mrty crrer hle ccetrt wll ecrese.
versus T Betwee T0 K freeze-ut T300 K cmlete zt, we hve rtl zt f r ccetr tms. r 0 Hgh T the semcuctr shws trsc behvur
vs T strg ees frm T. By cresg T electr hle rs re thermlly geerte the the term strt t mte. The semcuctr wll lse the extrsc chrcterstcs.
Pst f the erm eergy level fr -tye Determt f the st f the erm level s fuct f g ccetrt s fuct f temerture. We ssume tht the Bltzm xmt s vl. c ex l If we cser - tye semcuctr l c D c ll r tms re ze mves t c f the r ccetrt creses. r cmeste semcuctr term s relce wth -.
fr cmeste semcuctrs 0 0 0 If we ssume cmlete zt > Therml-equlbrum mjrty crrer hle ccetrt -tye semcuctr r whe >. Mrty crrer ccetrt :
Pst f the erm eergy level fr -tye X: S T300 K 10 16 cm -3 0 1.5x10 10 cm -3 -tye semcuctr 19 c.8 10 l 0.059l 0.06eV >> 16 10 vlthe Bltzm r. We evel fferet exress fr fr -tye semcuctr ex l wth r semcuctr wth - 0 cmletely cmeste the I > sme chrcterstcs f trsc semcuctr.
Pst f the erm eergy level f -tye r -tye semcuctr V V A v V V v ex l tye semcuctr cser - we If l s the creses mves clser t the vlece b r l ex
vs r -tye > > r -tye > >
vs T l l V ex g g. 4-19 strgly ees frm T s fuct f T ls. As T creses creses t > At hgh T the semcuctr lses the extrsc chrcterstcs behves s trsc. At lw T freeze-ut ccurs, Bltzm rxmt s t lger vl. The equts we erve the erm level st lger ly. -> fr -tye -> fr - tye.
vs g -tye semcuctr -tye semcuctr l c V l V
Relevce f hs relts re sgfct - juts r ther semcuctr evces. I therml-equlbrum s cstt thrughut system. g. 4-0