Lecture contents. Transport, scattering Generation/recombination. E c. E t. E v. NNSE508 / NENG452 Lecture #13. Band-to-band recombination

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1 Lctur cotts Trasort, scattrig Gratio/rcobiatio E E c E t E v Bad-to-bad rcobiatio Tra-assistd (SRH) rcobiatio ad gratio Augr rcobiatio

2 Elctro trasort: Gral cosidratios How fr carrirs ract o xtral lctric fild? Motio i ral sac = thral otio + drift + scattrig Ma fr ath: v th Motio i wavvctor sac: isid a bad vally (ulss itrvally scattrig is ivolvd) Currt dsity is roortioal to drift vlocity of carrirs J v d A c

3 Elctro trasort: hoological aroach 3 Kitic rgy r lctro i 3D (fro kitic thory of gass) v 3 k B T Thral vlocity v th 3kBT * 7 c s v d Forc o a fr lctro icluds lctric ad frictio with otu rlaxatio ti : lctric fild) ( I th stady stat drift vlocity is roortioal to th fild ( drift obility): v d * dv dt * d * v d c * Vs Ad currt dsity (s coductivity) givs Oh s law: J v s s * For sicoductor cotaiig both lctro ad hols: s h - 3D:(Ω c) s D:(Ω/sq.) -

4 Scattrig chaiss 4 Mathiss s rul for rlaxatio ti:... ac o ii i I low lctric filds ioizd iuritis acoustic hoos Dforatio ottial Pizolctric I high lctric filds otical hoos itrvally scattrig At high coctratios carrir-carrir scattrig To udrstad how ths chaiss affct obility o ds to cosidr ddc (E)

5 Coductivity ffctiv ass 5 For o-dgrat coductio bad iiu (Goit): * G Idirct coductio bad iiu: ffctiv ass will dd o dirctio! For xal: Si alog [] dirctio: J Jx Jy Jz 6 l 6 t Coductivity ffctiv ass of Si [] lctros:: * c 3 l t Valc bad axiu: d to add u cotributios of light ad havy hols: Coductivity ffctiv ass of hols lh J Jlh J hh lh lh hh lh * 3 v lh hh lh hh h hh 3 hh Effctiv asss ad low fild obilitis at roo tratur:

6 Tratur ddc of obility Rlaxatio ti dds o rgy, ad thrfor obility dds o tratur Mobility i -GaAs 6 Mobility i -Si Mobility i -Si LO hoo scattrig Itravally acoustic hoo scattrig + (TA+LO) itrvaly hoo scattrig Fro Yu ad Cordoa, 3, ad Shur, 3

7 High fild lctro trasort 7 Hot lctros trasfr rgy ito thral vibratios of th crystal lattic (hoos). Such vibratios ca b odld as haroic oscillatios with a crtai frqucy, w h. Th rgy lvls of a haroic oscillator ar quidistat with th rgy diffrc btw th lvls qual to E h w h Two tys of hoos scattr lctros diffrtly: acoustical (slow) ad otical (fast): Hc, rocss for a hot lctro ca b rrstd as follows. Th lctro acclrats i th lctric fild util it gais ough rgy to xcit otical hoo: E E ac w ac kt ot wot 4 V

8 Itrvally scattrig i high lctric fild 8 GaAs has a L vally just.9 V highr tha G -vally At high filds lctros ca b trasfrrd ito L-vallys. Th coductivity will dd o coctratios ad obilitis of lctros i both vallys: s Modl for ddc of drift vlocity o lctric fild i GaAs N G G N L L Fro Shur, 3

9 Diffusio ad drift of carrirs If th carrir coctratio is chagig fro oit-to oit, th carrirs will rdistribut to qualiz th coctratio = Diffusio Diffusio is dscribd by th first Fick s law: Elctro diffusio currt Hol diffusio currt If lctric fild (wak) is alid: Total lctro currt: Total hol currt: (Flux of articls)= -(Diffusio cof.) x (gradit of coctratio) J J J J D D D D o D 9 k B T lays th rol of a ffctiv fild Diffusio cofficit ad diffusio currt hav ss oly if coctratio chag is sall o th a fr ath: Substitutig th gradit with th ffctiv fild: k T B

10 Equilibriu of drift-diffusio: Eisti rlatio Drift flux Diffusio flux At quilibriu coditio (o total currt) diffusio currt is qual to drift currt: D E C E F For odgrat sicoductor: N C x Carrir coctratio is chagig i th lctric fild, followig th ottial j(r): EF E k T B C ( x) j( x) x kbt Gradit of lctro coctratio is : Ad substitutig, obtai Eisti rlatio: j( x) k T B k T B k T B D *

11 Cotiuity quatio Th cotiuity quatio allows us to calculat th carrir distributio i th rsc of gratio ad rcobiatio GR, Th rat of chag of th carrirs btw x ad x + dx is dtrid by fluxs ad gratio/rcobiatio of carrirs: 3 c s ( x, t) J( x) J ( x dx) Adx A G( x, t) R( x, t) Adx t q q E C J( x ) J( x dx) R G E V x x dx x J( x) dx q x Cotiuity quatio for lctros: Ad siilar for hols ( x, t) divj ( x, t) G ( x, t) R ( x, t) t q ( x, t) divj ( x, t) G ( x, t) R( x, t) t q

12 Gratio/Rcobiatio I th silst odl t rcobiatio rat (itrisic = rsos of th sicoductor) is roortioal to th EXCESS carrir dsity (, quilibriu coctratios, s - iority carrir liftis): U R G 3 c s U R G U ight b gativ t gratio E Rcobiatio chaiss Bad-to-bad rcobiatio E t Tra-assistd (SHR) rcobiatio ad gratio Augr rcobiatio E c E v At throdyaic quilibriu, th gratio rat ad th rcobiatio rat ar qual: U Aftr gratio, xcss coctratio dcays xotially dcays: U () t G tt E E hoto =h>e g Gratio chaiss P E c E v Extrisic gratio rat, ot atrial rsos E-h airs gratio by light Ioizatio du to rgtic articl/hoto

13 Bad-to-bad trasitios i dirct-badga ad idirct-badga sicoductors 3 High issio robability Short radiativ lifti high radiativ quatu fficicy, fficit LED, lasrs GaAs, IGaAs, GaN, IP, IGaSb Low issio robability du to d ordr (hoo-assistd) rocss (otu cosrvatio) Log radiativ lifti low radiativ quatu fficicy Si, G, diaod, GaP, AlAs

14 Otical absortio 4 Br-Labrt absortio law I(x) =I x(-ax) Sigificat absortio at E hoto = h > E g Idirct badga low ar dg absortio, gradual sctral slo (otu cosrvatio) Dirct badga shar dg Absortio sctru dds o dsity of sats i th bads ad trasitio robabilitis. Fro Colig, 5

15 Otical absortio ad o-quilibriu carrirs 5 Itsity [W/c =J/s-c ] of light is xotially dcrasig du to absortio I(x) =I x(-ax) Absorbd itsity i a layr with a thickss d: I I I a abs d I cas of uifor gratio (low absortio) ad <<<, ca b always alid to a thi slic Nubr of gratd -h airs is qual to th ubr of hoto absorbd. Wh gratio is uifor: Dsity of both lctros ad hols icras udr bad-tbad otical xcitatio by th sa aout Highr rgy hotos grat hot carris that rlax towards th bad xtra G/R is tyically slowr ( s) tha rgy rlaxatio du to scattrig ( s) G Extrisic gratio rat I I ad abs Allows to trat lctros ad hols statistically iddt i.. itroduc sarat quasi-fri lvls for lctros ad hols. ad Iot Iot G a d h h D = D

16 Radiativ rcobiatio 6 Radiativ rcobiatio rat R r ad radiativ lif ti r : d dt r R r d dt Radiativ rcobiatio rat R r (B- radiativ or Bi-olcular rcobiatio cofficit) cosists of quilibriu ad o-quilibriu arts: R r Cosidrig otical xcitatio: D = D B B D)( D ) BD ( D B i Nw o-quilibriu R r (=U r ) Might dd o Two cass: High ijctio rat (biolcular rcobiatio):, D R r B D ad r BD Low ijctio: D r B i a dod atrial o of th coctratios doiats

17 Radiativ rcobiatio 7 Radiativ lifti for low ijctio D Miority carrir lifti For xal for -atrial lifti of ijctd lctros will b B r Fro Yu ad Cordoa, 3

18 Shockly-Hall-Rd (SHR) Rcobiatio 8 No-radiativ rcobiatio through rcobiatio ctrs N t : Shockly-Hall-Rd (SHR) rocsss Cosidr lctro G/R: t - ctrs occuid with lctros ( f Fri distributio fuctio) lifti: At quilibriu, lctro issio : R v s N f E _ th t t t G N f E t t v s N _ th t Ei Et v _ ths i x kt For iority carrirs (.g. lctros i -ty) Cobiig lctros ad hols U SHR i Et Ei Ei Et i x i x kt kt for ctrs whr th rcobiatio rat is highst (i.. for ) U SHR U SHR U R G U R G i i i i E t E i

19 Othr rcobiatio chaiss Surfac rcobiatio i quasi-utral -ty rgio, i 9 With surfac rcobiatio vlocity s [c/s] U s surf, Uifor (hoto)gratio: x s L L D( x) D( x) G s L G L x Augr rcobiatio cotribut at high carrir coctratios: R A C C C 3 I gral i cas of xtral gratio G : whr lifti is a rsult of various rocsss: oradiativ through rcobiatio ctrs, radiativ, Augr rcobiatio: Lifti at high xcitatio coditios: d dt G A r A B C

20 Lctur rca Ohic bhavior rsults fro scattrig Drift-diffusio J D Rcobiatio rat U R G Cotiuity quatio (D) J t q x G R Miority carrir lifti