Complementi di Fisica Lecture 6

Transcription

1 Comlemei di Fisica Lecure 6 Livio Laceri Uiversià di Triese Triese, 15/

2 Course Oulie - Remider The hysics of semicoducor devices: a iroducio Basic roeries; eergy bads, desiy of saes Equilibrium carrier coceraio ( irisic, erisic ) Carrier rasor heomea Drif ad Diffusio Geeraio ad Recombiaio Coiuiy equaios Quaum Mechaics: a iroducio Advaced semicoducor fudameals 15/ L.Laceri - Comlemei di Fisica - Lecure 6 2

3 Lecure 5 - oulie Drif ad diffusio a equilibrium (buil-i field): a eamle Coiuiy equaios ad Geeraio-Recombiaio Recall coiuiy equaio: charge coservaio (ad Mawell equaios ) Coiuiy equaios, searaely for elecros ad holes, i a semicoducor (1-d) Coiuiy equaios for mioriy carriers i low-ijecio codiio Mioriy carrier lifeime: overview Direc, idirec ad surface recombiaio codiios Homework: Derivaios of e recombiaio ad lifeimes; Three imora secial cases high field effecs 15/ L.Laceri - Comlemei di Fisica - Lecure 6 3

4 Drif ad diffusio A eamle

5 J V No-uiform doig: buil-i field -ye doig, hermal equilibrium (o eeral el.field alied!): q µ q µ q µ Ɛ is he buil-i elecric field: 1 d 1 dn A Ɛ Vh Vh d N d h kt q Ɛ Ɛ Ɛ d q D d kt q µ q V h 1 A d d d d 0 hermal volage equivale 0 () () diffusio N A Small sace charge ubalace Ɛ() Buil-i field drif 15/ L.Laceri - Comlemei di Fisica - Lecure 6 5 holes elecros holes elecros comesaes diffusio

6 Coiuiy equaios

7 Summary of From: The Feyma Lecures o Physics, vol.ii 15/ L.Laceri - Comlemei di Fisica - Lecure 6 7

8 Coservaio of charge: coiuiy equaios Ay e flow of charge mus come from some suly! S r r r d J ˆ ds JdV d r r J J V J y y J z ρ dv dq d The flu of a curre from a closed surface is equal o he decrease of he charge iside he surface ρ is he e charge desiy (egaive ad osiive, algebraic sum) Le us cosider elecros ad holes, searaely, i a semicoducor, i a simle oe-dimesioal case z V ρ 15/ L.Laceri - Comlemei di Fisica - Lecure 6 8

9 Elecros ad holes Cosiderig egaive ad osiive charge desiies searaely, i is ecessary o iclude also: Geeraio rae G : G umber of free carriers geeraed (searaig elecros from holes) er secod ad er ui volume G is usually a fucio of he available eergy (emeraure, ec.) Recombiaio rae R: R umber of free carriers disaearig due o recombiaio er secod ad er ui volume R is usually roorioal o he roduc of coceraios of carriers ad recombiaio ceers ad o a caure coefficie defied as c v h σ, where v h is he hermal velociy ad σ is he recombiaio rocess cross-secio Ne recombiaio effec: U R - G 15/ L.Laceri - Comlemei di Fisica - Lecure 6 9

10 Coiuiy for elecros Eeral volage V Volume eleme Ad How fas does he umber of elecros chage i A d? 1 ρ q Subsiuig: J J Ad q ( ) A J ( d) J ( d) J ( ) d... q A Ne carriers er secod hrough he walls geeraio - recombiaio ( G R ) Ad ad dividig by A d see e age 15/ L.Laceri - Comlemei di Fisica - Lecure 6 10

11 Coiuiy for elecros ad holes Oe-dimesioal Three-dimesioal ( ) ( ) ( ) ( ) R G J q R G J q R G J q R G J q r r r r Oe-dimesioal, uder low-ijecio codiios, for mioriy carriers: 15/ L.Laceri - Comlemei di Fisica - Lecure 6 11 G D E E G D E E τ µ µ τ µ µ Recombiaio: This is he ricky ar! mioriy carrier ecess mioriy carrier lifeime Elecros: i -ye holes: i -ye Simly subsiue JJ(drif)J(diffusio) (elecric field)

12 Recombiaio: a hi recombiaio is ofe domiaed by idirec rocesses hrough recombiaio ceers (direc recombiaio egligible for Si) For isace, i a -ye semicoducor, uder lowijecio codiios: for he mioriy-carriers (holes!) ecess-recombiaio, he boleeck is hole caure, ha deermies he hole lifeime τ Oce caured, he hole recombies quickly, sice here are may elecros available U τ v h v σ N h 1 σ N ( ) 0 15/ L.Laceri - Comlemei di Fisica - Lecure 6 12

13 Pay aeio: ermiology Caure, emissio : From he oi of view of he ra! I aricular (figure, e slide): (a) elecro caure (3) (b) elecro emissio (2) (c) hole caure (4) (d) hole emissio (1) 15/ L.Laceri - Comlemei di Fisica - Lecure 6 13

14 more recombiaio for low-ijecio -ye semicoducor: elecro-lifeime domiaed by elecro caure (3) i emy RG ceers U τ U τ v v v h h h v σ N σ N h 1 σ N 1 σ N ( ) ( ) 0 -ye semicoducor: hole-lifeime domiaed by hole caure (4) i full RG ceers 0 15/ L.Laceri - Comlemei di Fisica - Lecure 6 14

15 Lecure 6 - summary The coiuiy equaios are he mai ools for simulaig semicoducor devices, comlemeed by 2 Gauss law r E r To udersad he key recombiaio mechaisms ad see some racical eamles, homework for e ime: readig of Direc recombiaio (2.4.1), idirec recombiaio (2.4.2), surface recombiaio (2.4.3) Alicaios: seady sae ijecio from oe side (2.5.1), mioriy carriers a he surface (2.5.2), Hayes-Shockley eerime (2.5.3) High field effecs (2.6), Hall effec (2.1.3) ρ ε 0 V ρ ε 0 15/ L.Laceri - Comlemei di Fisica - Lecure 6 15

16 Lecure 6 Iems o be udersood Some iems ha require more hough: Coiuiy equaios: eamles, alicaios Geeraio rae: orders of magiude, differe rocesses Recombiaio rae: orders of magiude, differe rocesses Wha haes if he low-ijecio aroimaios are o ossible? Wha abou curres ad coiuiy equaio for he majoriy carriers? Whe are hey releva? 15/ L.Laceri - Comlemei di Fisica - Lecure 6 16

17 Lecure 6 - Glossary buil-i field coiuiy equaios geeraio rae recombiaio rae low-ijecio mioriy carriers lifeime 15/ L.Laceri - Comlemei di Fisica - Lecure 6 17

18 Lecure 6 - eercises Eercise 6.1: Calculae he elecro ad hole coceraio uder seady-sae illumiaio i a -ye silico wih G L cm -3 s -1, N D cm -3, ad τ τ 10 µs. Eercise 6.2: A -ye silico samle has arseic aoms/cm 3, bulk recombiaio ceers/cm 3, ad surface recombiaio ceers/cm 2. (a) Fid he bulk mioriy carrier lifeime, he diffusio legh, ad he surface recombiaio velociy uder lowijecio codiios. The values of σ ad σ s are ad cm 2, resecively. (b) If he samle is illumiaed wih uiformly absorbed ligh ha creaes elecro-hole airs/(cm 2 s), wha is he hole coceraio a he surface? Eercise 6.3: The oal curre i a semicoducor is cosa ad is comosed of elecro drif curre ad hole diffusio curre. The elecro coceraio is cosa ad equal o cm -3. The hole coceraio is give by ()10 15 e(-/l) cm -3 (>0), where L 12µm. The hole diffusio coefficie is D 12cm2/s ad he elecro mobiliy is µ 1000cm 2 /(Vs). The oal curre desiy is J 4.8 A/cm 2. Calculae (a) he hole diffusio curre desiy as a fucio of, (b) he elecro curre desiy versus, ad (c) he elecric field versus. 15/ L.Laceri - Comlemei di Fisica - Lecure 6 18