Lecture contents. Semiconductor statistics. NNSE508 / NENG452 Lecture #12

Transcription

1 Ltur otts Sioutor statistis S58 / G45 Ltur #

2 illig th pty bas: Distributio futio ltro otratio at th rgy (Dsity of stats) (istributio futio): ( ) ( ) f ( ) Pauli lusio Priipl: o two ltros (frios) a hav itial quatu ubrs. ltros follow ri-dira statistis. ri-dira istributio futio: f D ( ) I th o-grat as (ltro rgis ar far fro ): oltza istributio futio ay b us: f ( ) S58 / G45 Ltur #

3 S58 / G45 Ltur # 3 Statistis of arrirs: Gral ltro otratio i th rgy rag to + los to th outio ba iiu: otal ltro otratio i th outio ba p ) ( 3 * 3 p 3 * * 3 p 3 * p Gral quatio for 3D arrir otratio (fftiv sity of stats) (ri 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) (ri itgral of zro orr): * l p s DOS fftiv DOS

4 Statistis of arrirs: Gral 4 h sa is tru for hols i th val ba: p * h 3 p V ) / ( p V ) / ( V fftiv sity of stats of ltros (or hols) O-to-o orrspo otratio of obil (ba) arrirs ri lvl positio S58 / G45 Ltur #

5 S58 / G45 Ltur # 5 Statistis of arrirs: o-grat syst If all th.. rgis ar far fro ri lvl: >> (> 3 ) : ) / ( V V p p ) ( p p Gral quatio: p p V V p otratio of ba arrirs o-grat syst: Gral as: ) / ( p i g V p Dfiitio of itrisi arrir otratio

6 arrirs i itrisi sioutors 6 ri lvl positio i itrisi sioutor S58 / G45 Ltur #

7 fftiv ass approiatio 7 Sall prturbatio of prioiity: shallow ipuritis, ost of ha-a struturs, tral fors O-ltro Shröigr quatio with wa a slow varyig prturbatio V i (fftiv ass approiatio): p V r V ( r) ( r) ( r) i i A as usual buil a solutio as a wav pat of loh wavfutios : loh wav pat: ( r ) ( r ) u ( r ) ( r) ( ), ir u ( r) Dpig o sig of th prturbatio, th topost or botto-ost stat splits fro th ba : With isios i ral a -spa r a Larg prturbatio of prioiity othr bas to b osir: p ipuritis V i S58 / G45 Ltur #

8 S58 / G45 Ltur # 8 apl of MA: Hyrog-li ipurity (oor) Solutio for rgy : r r V i ) ( Hyrog-li ipurity = shallow ipurity Shröigr quatio for Hyrog ato with fftiv ass a sr oulob pottial: ) ( ) ( ) ( * r r r p 4 * * Ry a r a r p ) ( 3 vlop futio of th grou stat : * a or oors i GaAs (*=.7: a =.6 ): fftiv Ry* : Ry* =6.6 V, a = 9 A ro Yu a oroa, 3

9 Wh fftiv ass approiatio is appliabl? 9 With of th ba MP wors fi wh prturbatio of rystal prioi pottial is sall i valu a larg i sal: Pottials i vi struturs ltrostati pottials Htrostruturs Shallow ipuritis = Hyrog-li ipuritis: thos with siilar ovalt raius that o ot isturb prioiity sigifiatly a hav siilar val stats V ( r) V ( r R) prt r a Wh MA wors poorly: Ipuritis a fts with strog isturba of rystal prioiity = p lvls or p ipuritis: vaais, group I, VII i III-V s, -tals, low raius. ro Mils, 973 S58 / G45 Ltur #

10 Doors Shallow oors: - usually group V lts i Si a G (P, As, Sb) - group IV lts o group III sublatti i III-V s (Si, S i GaAs) - group VI lts o group V sublatti i III-V s (S,, S i GaAs) S58 / G45 Ltur #

11 Aptors Shallow aptors: - usually group III lts i Si a G (, Al, Ga, I) - group II lts o group III sublatti i III-V s (, Mg, Z i GaAs) - group IV lts o group V sublatti i III-V s (, Si, G i GaAs) Group IV ipuritis i III-V s ar oft aphotri. + S58 / G45 Ltur #

12 Aig ipuritis: trisi sioutors Sipl ipurity with two harg stats,.g. sipl 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 sipl oors a g=4 for sipl aptors Ioizatio ratio for oors a aptors: gf gf g D D g a a g a a S58 / G45 Ltur #

13 trisi sioutors: o opsatio 3 What happs with ri lvl if sioutors otais ipuritis? I trisi sioutors harg utrality oitio ilus ioiz ipuritis ( ista of = p i itrisi sioutors): a p Wh ipurity of o typ (say oors) ar prst: p ; if p h gral quatio for ri lvl (s to b solv for grat sioutors) : A i o-grat as ( ioizatio rgy): / g g or with g 4 ri lvl positio (o-grat) usig p 4 l S58 / G45 Ltur #

14 trisi sioutors: o opsatio 4 At high tpraturs ( > ), for 4 At low tpraturs, for ri lvl a otratio 4, g or l g ri lvl positio i -G (uopsat) arrirs ar frzig out S58 / G45 Ltur #

15 S58 / G45 Ltur # 5 trisi sioutors with opsatio (rsults for a o-grat as) At high tpraturs, for What is th auray of assuptio? or -typ atrial: At low tpraturs, for ri lvl a otratio A A g l A A, A A g A A 4 A A a p i a

16 Dop sioutors: pratur p 6 arrir otratio vs. tpratur urv has 3 istit rgios (4 rgios i opsat sioutor) ypial p for Si S58 / G45 Ltur #

17 S58 / G45 Ltur # 7 Strog gray, i.. ri lvl lis i th outio (or val) ba: arrir otratio: Substitutig ri futio by stp futio (goo for ) ially: Whih is siilar to sipl tal Strog o-gray: tals agai or p ) / ( p

18 Quatu ofit: Quatu Wlls 8 Shröigr quatio ur fftiv ass approiatio (with vlop futio ): ltros (or hols) ar fr alog a y: V ( z) * i i y y f (z) W rtur to a probl of a partil i a wll * V ( z) f ( z) f ( z) Solutio: iit ubr of isrt rgy lvls Siusoial wavfutio isi th wll potial ay i th barrir ro Sigh, 3 S58 / G45 Ltur #

19 Quatu Wlls 9 ah lvl i D wll orrspos to a D ba with paraboli isprsio () isprsio orrspos to a fftiv ass i -y pla Hol D bas split aorig to thir fftiv asss (light a havy hols) or bttr auray for hols us Koh-Luttigr val ba strutur ro Sigh, 3 S58 / G45 Ltur #

20 Dsity of stats i Quatu Wlls Dsity of stats i D ba (a sigl subba i th itrval to + ubr of stats pr uit volu ) ( ) ro Sigh, 3 D arrir otratio i a sigl subba [ - ] * s l p S58 / G45 Ltur #

21 Suary of arrir statistis i sioutors Itrisi Iporta of opig: Si Uop Dop w/ 5 As atos/ 3 Rsistivity orr of agitu rsistivity hag u to i 5 illio ipuritis! -typ p-typ 5 As atos/ 3 i 5 Si atos/ 3 ltroi proprtis ar trly ssitiv to ipuritis, fts, fils, strsss ri lvl tris stati arrir otratios Gral quatios a b siplifi i o-grat a strogly grat ass li to Java applts S58 / G45 Ltur #

22 Ltur rap I sioutors, statistis grally appli to paraboli bas DOS ps o isioality of th syst a urvatur (fftiv ass tsor) is th priary proprty, othr fftiv asss (* DOS, * outivity ) ar riv fro it Gral () =DOS Distr.u., total through ri itgral wo ass approiat iffrtly: o-grat a grat sioutor Dopig us to otrol arrir sity = ri lvl positio S58 / G45 Ltur #