Lecture contents. Density of states Distribution function Statistic of carriers. Intrinsic Extrinsic with no compensation Compensation
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1 Ltur otts Dsity of stats Distributio futio Statisti of arrirs Itrisi trisi with o ompsatio ompsatio S 68 Ltur #7
2 Dsity of stats Problm: alulat umbr of stats pr uit rgy pr uit volum V() Larg 3D bo (L is larg, is larg) with or-vo Karma ouary oitios: ( L, y, z) (, y, z) L z L L y V L r ltro approimatio: ir y z L L y L z y z y 3 L L L z y z y z umbr of stats: D : L D : S 3D : V 3 S 68 Ltur #7
3 Dsity of stats 3 V 3 V m m V m ( ) V I th itrval to + umbr of stats : I th itrval to + umbr of stats pr uit volum (spi ilu): 3D : 4 L m ( ) V 3 D : L ( ) m D : L ( ) m V S 68 Ltur #7
4 Dsity of stats a imsioality 4 rom Sigh, 3 S 68 Ltur #7
5 Dsity of stats i 3D a DOS fftiv mass 5 fftiv mass sity of stats Gas 3D sity of stats 3 m* ( ) V 3 m outio ba DOS mass i G poit: m * * os m outio ba DOS mass i iirt gap smioutors ( gray of th vally): m 3 mm3 * 3 * * * os * * 3/ * 3/ 3 Val ba DOS mass : m os m hh m lh S 68 Ltur #7
6 Gral: (), f() a arrir otratios 6 Itrisi () -typ f(). p-typ Dsity of stats, () rmi-dira istributio futio, f() arrir otratios a p top f S 68 Ltur #7
7 illig th mpty bas: Distributio futio 7 ltro otratio at th rgy (Dsity of stats) (istributio futio): ( ) ( ) f ( ) ltros (rmi partils) follow rmi-dira statistis. rmi-dira istributio futio: f D ( ) I th o-grat as (ltro rgis ar far from ): oltzma istributio futio may b us: f ( ) S 68 Ltur #7
8 S 68 Ltur #7 8 Statistis of arrirs: Gral ltro otratio i th rgy rag to + los to th outio ba miimum: otal ltro otratio i th outio ba p ) ( 3 * 3 m m p 3 * * 3 p m 3 * p m Gral quatio for 3D arrir otratio (fftiv sity of stats) (rmi itgral of ½ orr): ) / ( ( ) ( ) ( ) f ltro otratio at th rgy (Dsity of stats) (istributio futio): Gral quatio for D arrir otratio (fftiv sity of stats) (rmi itgral of zro orr): * l p s m DOS fftiv DOS
9 Statistis of arrirs: Gral, 3D 9 h sam is tru for hols i th val ba (3D): p * mh 3 p V ) / ( p V ) / ( V fftiv sity of stats of ltros (or hols) O-to-o orrspo otratio of mobil (ba) arrirs rmi lvl positio S 68 Ltur #7
10 S 68 Ltur #7 Statistis of arrirs: o-grat systm If all th.. rgis ar far from rmi lvl: >> (> 3 ) : ) / ( V V p p ) ( p p Gral quatio: p p V V p otratio of ba arrirs o-grat systm: Gral as: ) / ( Law of mass atio: p i g V p Dfiitio of itrisi arrir otratio
11 Statistis of arrirs: Itrisi smioutors I itrisi smioutors harg osrvatio (o ioiz impuritis): = p = i Itrisi arrir otratio p p V V Itrisi rmi lvl: Usful rlatios btw rmi lvl a otratio of arrirs i trisi smioutors: S 68 Ltur #7
12 arrirs i itrisi smioutors rmi lvl positio i itrisi smioutor S 68 Ltur #7
13 ig impuritis: trisi smioutors 3 Simpl impurity with two harg stats,.g. simpl oor: + +, otal oor otratio: otratio of utral (fill with ltro) a ioiz oors: has a gray fator g g f D gf ( D ) f D ( ) Ratio of utral to harg oors: g= for simpl oors a g=4 for simpl aptors Ioizatio ratio for oors a aptors: a a g gf gf D D g (ograt) g with a g a S 68 Ltur #7
14 S 68 Ltur #7 4 trisi smioutors: o ompsatio a p I trisi smioutors harg utrality oitio ilus ioiz impuritis ( ista of = p i itrisi smioutors): Wh impurity of o typ (say oors) ar prst: h gral quatio for rmi lvl (s to b solv for grat smioutors) : i o-grat as ( ioizatio rgy): or rmi lvl positio (o-grat) usig What happs with rmi lvl if smioutors otais impuritis? p if p ; g / g 4 g with p 4 l or
15 trisi smioutors: o ompsatio 5 t high tmpraturs, for 4 t low tmpraturs, for rmi lvl a otratio 4, g or l g rmi lvl positio i -G (uompsat) S 68 Ltur #7
16 S 68 Ltur #7 6 trisi smioutors with ompsatio a p Lt s osir o-grat as with utrality oitio: quatio for rmi lvl for o-grat as : Or Solvig this quarati quatio: rmi lvl positio (o-grat) usig What happs with rmi lvl if smioutors otais both oors a aptors? g ) ( 4 g p 4 l p with
17 t high tmpraturs, for trisi smioutors with ompsatio What is th auray of assumptio? or -typ matrial: 4 p t high tmpratur: a i a 7 or p-typ matrial: t low tmpraturs, for, rmi lvl l g a otratio g S 68 Ltur #7
18 Dop smioutors: mpratur p 8 arrir otratio vs. tmpratur urv has 3 istit rgios (4 rgios i ompsat smioutor) ypial p for Si S 68 Ltur #7
19 arrir otratio at low tmpraturs 9 rmi lvl positio a ltro otratio i -G 6 m m m m S 68 Ltur #7
20 S 68 Ltur #7 Strog gray, i.. rmi lvl lis i th outio (or val) ba: arrir otratio: Substitutig rmi futio by stp futio (goo for ) ially: Whih is similar to simpl mtal Strog o-gray or p ) / ( p
21 Summary of arrir statistis i smioutors Importa of opig: Itrisi Si Uop Dop w/ 5 s atoms/m 3 Rsistivity 5 -m 5 -m 5 orr of magitu rsistivity hag u to i 5 millio impuritis! -typ 5 s atoms/m 3 i 5 Si atoms/m 3 p-typ li to Java applts ltroi proprtis ar trmly ssitiv to impuritis, fts, fils, strsss rmi lvl trmis stati arrir otratios Gral quatios a b simplifi i o-grat a strogly grat ass S 68 Ltur #7
22 Hall fft: arrir harg a otratio trmiatio Lortz for o a movig partil: v Stay stat: bala of fors i y irtio: osir o typ of arrirs (.g. >> p) urrt sity i -irtio: Hall offiit (Hall a rift mobilitis osir qual): v y J v z R p y H J z for for typ p typ S 68 Ltur #7
23 D Statistis: Quatum Wlls 3 Shröigr quatio ur fftiv mass approimatio (with vlop futio ): ltros (or hols) ar fr alog a y: V ( z) m* i i y y f (z) W rtur to a problm of a partil i a wll m* V ( z) f ( z) f ( z) Solutio: iit umbr of isrt rgy lvls Siusoial wavfutio isi th wll potial ay i th barrir rom Sigh, 3 S 68 Ltur #7
24 Quatum Wlls 4 ah lvl i D wll orrspos to a D ba with paraboli isprsio () isprsio orrspos to a fftiv mass i -y pla Hol D bas split aorig to thir fftiv masss (light a havy hols) or bttr auray for hols us Koh-Luttigr val ba strutur rom Sigh, 3 S 68 Ltur #7
25 D statistis: Dsity of stats i Quatum Wlls 5 Dsity of stats i D ba (a sigl subba i th itrval to + umbr of stats pr uit volum ) ( ) m* rom Sigh, 3 D arrir otratio i a sigl subba [m - ] * m s l p S 68 Ltur #7
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