Modr Smcoductor Dvcs for Itgratd rcuts haptr. lctros ad Hols Smcoductors or a bad ctrd at k=0, th -k rlatoshp ar th mmum s usually parabolc: m = k * m* d / dk d / dk gatv gatv ffctv mass Wdr small d / dk havy mass -k rlatoshp for parabolc bad wth sotropc ffctv mass 3-D: m x * = m y * = m z * = m* kx k y kz m* k k x y kx m* m* m* quato for sphr k spac Sharpr larg d / dk lght mass a k dagram ad b sphrcal costat-rgy surfac for GaAs Th radus of th sphr stads for rgy ad th surfac of th sphr s sam rgy, whch s calld costat rgy surfac.
Modr Smcoductor Dvcs for Itgratd rcuts haptr. lctros ad Hols Smcoductors Th ral -k dagram of S s mor complcatd drct smcoductor. Th bottom of ad top of appar for dffrt valus of k. b llpsodal costat-rgy surfacs th coducto bad. Thr ar 6 quvalt mma alog [00] drcto a k dagram of S larg d / dk small lght hol d / dk havy hol Rad subscto.5.;how to masur th ffctv mass? -k rlatoshp for parabolc bad wth asotropc ffctv mass 3-D: m x * m y * m z *. I S, m x * = m y * m z *. k k m m m x y kz * * * x y z k k x y kx * * * mx my mz quato for llpsod k spac Th costat rgy surfac s ot sphr, but llpsod.
Modr Smcoductor Dvcs for Itgratd rcuts Dsty of lctro Stats haptr. lctros ad Hols Smcoductors It s usful to thk of a rgy bad as a collcto of dscrt rgy stats. If w cout th umbr of stats a small rag of rgy, Δ, w ca fd th dsty of stats: umbr of stats D volum 8m * m* 3 h : dsty of lctro stats umbr of stats pr ut volum ad ut rgy a rgy bad as a collcto of dscrt rgy stats. b D s th dsty of th rgy stats. coducto bad dsty of stats valc bad dsty of stats D 8m m, 3 h D 8m m, h p p 3 Hom work: Drv th dsty of stat s fucto.
Modr Smcoductor Dvcs for Itgratd rcuts haptr. lctros ad Hols Smcoductors Statstcal Laws Maxwll-Boltzma probablty fucto Th partcls ar dstgushabl o lmt to th umbr of partcls allowd ach rgy stat Gas molculs a cotar at low prssur Bos-st fucto Th partcls ar dstgushabl o lmt to th umbr of partcls prmttd ach quatum stat Photos 3 rm-drac fucto Th partcls ar dstgushabl Oly o partcl s prmttd ach quatum stat lctros a sold crystal
Modr Smcoductor Dvcs for Itgratd rcuts haptr. lctros ad Hols Smcoductors Th rm-drac Dstrbuto ucto ad th rm rgy rm-drac Probablty ucto, f th probablty that a quatum stat at th rgy wll b occupd by a lctro = rato of flld to total quatum stats f / k : Boltzma costat =8.67 x 0-5 /K T : Tmpratur Klv K : rm rgy At T = 0 K or > : f xp 0 or < : f xp
Modr Smcoductor Dvcs for Itgratd rcuts At T > 0 K, xpotal dcay wth crasg rgy: Boltzma approxmato haptr. lctros ad Hols Smcoductors If =, f = / Thr s som probablty f that stat abov ar occupd ad thr s a corrspodg probablty - f that stat blow ar ot occupd. 3 If 3 f f / / Most stat at rgy 3 or mor abov wll b mpty. 4 If - 3 f [ / ] / f / Probablty of a stat ot bg occupd. = Probablty of a stat bg occupd by a hol Dcay xpotally zro wth dcrasg rgy. Th rm fucto dagram
Modr Smcoductor Dvcs for Itgratd rcuts haptr. lctros ad Hols Smcoductors Probablty of bg uoccupd by lctro, -f = Probablty of bg occupd by a hol Probablty of bg occupd by a lctro, f Th lctro ad hol probablts ar compltly symmtrcal, always gvg wh addd to ach othr.
Modr Smcoductor Dvcs for Itgratd rcuts lctro ad Hol octratos haptr. lctros ad Hols Smcoductors Dstrbuto of carrrs = Probablty of occupacy Dsty of stat = f D Total umbr of lctros B coducto bad at qulbrum = Top of coducto bad f D d Du to xpotal dcay of f wth larg Total umbr of hols B valc bad at qulbrum = p D f d alc bad bottom Schmatc bad dagram, dsty of stats, rm Drac dstrbuto, ad carrr dstrbutos vrsus rgy
Modr Smcoductor Dvcs for Itgratd rcuts haptr. lctros ad Hols Smcoductors qulbrum Dstrbuto of arrrs Wh s postod th mddl of th bad gap lctros dstrbuto = hol dstrbuto,.., f = - f Itrsc Wh s postod th uppr half of th bad gap lctros dstrbuto >> hol dstrbuto,.., f >> - f -typ Wh s postod th uppr half of th bad gap lctros dstrbuto << hol dstrbuto,.., f << - f P-typ
Modr Smcoductor Dvcs for Itgratd rcuts haptr. lctros ad Hols Smcoductors -typ door Itrsc P-typ accptor
Modr Smcoductor Dvcs for Itgratd rcuts arrr octrato Assumg odgrat approxmato 3, haptr. lctros ad Hols Smcoductors f / / Topof coductobad f D d 8m h 3 m / d 8m h m Itroducg a w varabl, x / / / 3 d 0 Th, th tgrato s th form of gamma fucto, x x dx 0 c /, m [ ] h 3/ ffctv dsty of stats at coducto bad dg Smlarly for hol, p /, m [ h p ] 3/ ffctv dsty of stats at valc bad dg
haptr. lctros ad Hols Smcoductors Modr Smcoductor Dvcs for Itgratd rcuts δ- fucto wth magtud of / 0 f d f 0 f d f or trsc matral, f / p / g p / g / ad from = p / / g p g m m l 4 3 l ffctv Dsty of Stats Mass Acto Law
Modr Smcoductor Dvcs for Itgratd rcuts haptr. lctros ad Hols Smcoductors or xtrsc matral, p / / / g Dgratd ad o-dgratd S p mass acto law always tru for odgrat / / / / p / p / p If dopg coctrato s small o tracto btw door lctros -typ dscrt door rgy stat o-dgrat smcoductors If dopg coctrato s larg Door lctros bg to tract wth ach othr dscrt door rgy wll splt to a bad of rgs If d ~ c, th bad of door stats may ovrlap th bottom of th coducto bad Locato of rm lvl vs. dopat coctrato S at 300 ad 400 K. dcrass as movs farthr blow, ad vc vrsa; p dcrass as movs farthr abov. Wh s about 0 m ~ from or havly dopd smcoductor, > ~ 0 9 cm -3, th Boltzma approxmato s o logr vald. Dgrat
Modr Smcoductor Dvcs for Itgratd rcuts Gral Thory of ad p haptr. lctros ad Hols Smcoductors Assumptos: uformly dopd smcoductor ad odgrat p = full ozato of th dopat atom shallow mpurts rom charg utralty ad mass acto law, p d a = 0 ad p = / d a = 0 d a = 0 Solv ths quadratc quato for th fr lctro coctrato,, ad tak oly th plus root d a [ d a ] /. Itrsc smcoductor a = 0, d = 0 Smlarly for hol coctrato, p, p a d [ a d ] / = ad p =. d a >>.., -typ = d a ad p = / If, furthrmor, d >> a, th = d ad p = / d
Modr Smcoductor Dvcs for Itgratd rcuts haptr. lctros ad Hols Smcoductors 3. a d >>.., P-typ p = a d ad = /p If, furthrmor, a >> d, th p = a ad = / a 4. >> d a Ths ca b happd at vry hgh tmpratur v for dopd smcoductor = p = All smcoductor bcom trsc at vry hgh tmpratur. 5. ompsato Both doors ad accptors ar prst a smcoductor ad d ad a ar comparabl ad ozro. If d > a -typ If d < a P-typ If d = a xact compsato d d d a a a d, ff d a p p a, ff a d
Modr Smcoductor Dvcs for Itgratd rcuts haptr. lctros ad Hols Smcoductors Tmpratur Dpdc of arrr octratos xtrsc Ths rgo should b wd to hav good dvc prformac ad tmpratur charactrstcs. Wd badgap matral small s prfrrd gral T KT 3 c g / v g / xampl or door g / >> d d domat ~ d >> d ry small arato of carrr coctrato a -typ smcoductor ovr a wd rag of tmpratur ~ d < d d ~0 at 0 K d Icrasg tmpratur
Modr Smcoductor Dvcs for Itgratd rcuts haptr. lctros ad Hols Smcoductors Itrsc arrr octratos G, S, ad GaAs vs. T