The real E-k diagram of Si is more complicated (indirect semiconductor). The bottom of E C and top of E V appear for different values of k.

Transcription

1 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.

2 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.

3 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.

4 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

5 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

6 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

7 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.

8 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

9 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

10 Modr Smcoductor Dvcs for Itgratd rcuts haptr. lctros ad Hols Smcoductors -typ door Itrsc P-typ accptor

11 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

12 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

13 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

14 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

15 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

16 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

17 Modr Smcoductor Dvcs for Itgratd rcuts haptr. lctros ad Hols Smcoductors Itrsc arrr octratos G, S, ad GaAs vs. T