LECTURE 13 Filling the bands. Occupancy of Available Energy Levels

Transcription

1 LUR 3 illig th bads Occupacy o Availabl rgy Lvls W hav dtrmid ad a dsity o stats. W also d a way o dtrmiig i a stat is illd or ot at a giv tmpratur. h distributio o th rgis o a larg umbr o particls ad its chag with tmpratur ca b calculatd by mas o statistical cosidratios. h probability o lctros occupyig a lvl at rgy is giv by: rmi - Dirac distributio uctio + / Whr is a paramtr calld th rmi rgy, which rprsts th highst rgy that th lctros assum at K. At ay tmpratur abov absolut zro th probability o occupatio o th rgy lvl at is ½ ot thr may ot b a actual stat at th rmi rgy! h uctio is plottd blow. h highr th th mor smarig. W ca s rom this igur that at all lvls that hav a rgy smallr tha ar compltly illd with lctros, LUR 3 illig th bads Sptmbr

2 whras highr rgy stats ar mpty. hus i a rgy lvl is compltly occupid by lctros, th rmi distributio uctio quals crtaity; or a mpty rgy lvl o obtais. h igur also shows that th probability o occupatio o stats abov is iit or > K ad that thr is a corrspodig probability that stats blow ar mpty. alculatig o calculat th carrir coctratios i rgy bads w d to ow th ollowig paramtrs: i h distributio o rgy stats or lvls as a uctio o rgy withi th rgy bad g. ii h probability o ach o ths stats big occupid by a lctro. So w ca writ dsity o lctros g d - probability o occupacy at g - dsity o stats i rgy at th hol dsity is giv by: p dsity o hol [ ] g d I th occupatio probability o a lvl is giv by, th probability that this lvl will ot b occupid by a lctro, i.., th probability o occupatio by a hol is: [ - ] - probability o a missig lctro at o occupacy LUR 3 illig th bads Sptmbr

3 oductio ad Dsity o stats Prviously w calculatd total umbr o stats, ot distributio. osidr a smicoductor or which th bottom o th coductio bad has sphrical costat rgy suracs or r lctros with a mass m*. Sphrical is 3D vrsio o parabolic.! m* or m*! z x + y + z y x L A sphr o radius i -spac cotais π 3 3 π allowd stats which ar uiormly distributd i 3D -spac. umbr o rgy lvls pr uit volum divid by L 3 is: 3/ 4π m* π! 3/ h dsity o stats i rgy is thror 3/ d m* g 3 d π! / LUR 3 illig th bads Sptmbr 3

4 LUR 3 illig th bads Sptmbr 4 ot that th dsity o stats g may b did as th umbr o rgy stats pr uit rgy pr uit volum. Rturig to th calculatio o carrirs coctratios: W hav that th dsity o lctros i coductio bad is d g or >> w us th ollowig approximatio or oltzma distributio: c c + which is valid or th tail o th distributio wh is wll away rom. w gt d g Di:, d g h solutio o this itgral ca b show to b 3/ * 4 π! m with - ctiv dsity o stats i coductio bad.

5 ad thus givig: W ca also gt or p: p With 4 m * π! 3/ Itrisic Smicoductors A prct smicoductor crystal cotaiig o impuritis or lattic dcts is calld a itrisic smicoductor. As th carrirs ar gratd i pairs, th coctratio o lctros i th coductio bad quals th coctratio p o hols i th valc bad, p i, i - itrisic carrir coctratio w ca us p to gt: ad, l + givig, LUR 3 illig th bads Sptmbr 5

6 + l th scod trm o th right had sid is usually vry small ad i ½ g. lis just blow midgap w also hav: p i calld th mass actio law. ad i or i G / It is importat i dvics to cotrol ad p coctratios ad supprss th iluc o itrisic coctratio. hs quatios ar importat i stablishig uppr limits i smicoductor opratig tmpratur. W grally rquir i << miimum dopig dsity i dvic ad, practically, this mas w d dopig coctratios abov 4 cm -3. LUR 3 illig th bads Sptmbr 6