Carrer Acto uder Perturbato Eulbrum: o curret ad o formato ca be represeted. Ferm-level s flat! Perturbato s ecessary to artfcally ecode formato perturbed states: electrc feld (drft), cocetrato gradet (dffuso), temperature gradet (thermal dffuso) ad excess carrers (R-G), ad radato (photoelectrc R-G). Ferm-level s ot flat! Drft curret: whe the carrers semcoductor s uder eulbrum, all radom moto k (phase) space s exactly cacelled: o curret flowg ad the system has a costat Ferm eergy However, whe there s a electrc feld F, the forward moto s preferred amog all radomzg scatterg: a skew dstrbuto Electros fucto of k space wll result et curret. dv dk wth all sort F m h ef eergy k y k y dt dt below E F eft F kt () k(0) move all h drectos. k x k x v0 Ferm sphere at t0 Ferm sphere at t Chapter
Quas Ferm Levels Whe we dsturb our system from eulbrum, our mathematcal treatmet wll employ aother cocept Quas Ferm level. Smlar to the Ferm-level used descrbg the occupato probablty eulbrum, uas Ferm-level s used to descrbe the carrer occupato probablty uder o-eulbrum codto. uder eulbrum, p ; uder o-eulbrum p E F splts to E FN ad E FP where E FN ad E FP follow the carrer relatoshp: EFN E c ( EFN Ec )/ kbt ( EFN E )/ kbt Nc F/ Nce e f Ec EFN 3kBT π kt B Ev E FP ( Ev EFP) / kbt ( Ev EFP) / kbt p Nv F/ Nve e f EFP Ev 3kBT π kt B Chapter
Drft Velocty ad Carrer Moblty J p drft p v p p μ p F wth v p μ p F J drft - v μ F wth v -μ F The moblty μ s the ut of cm /Vs ad s the same as the Ohm s law. Actual pcture s: accelerato, scatterg J drft (p μ p + μ )F A L J drft (/ρ) A L F ρ L I (A/ρ L) VR V μ + pμ p Moblty aga takes to accout all QM scatterg processes E < τ > F E a v a< τ > E μe * * m m l where < τ > s the mea tme betwee collso ν Chapter th 3 A
Carrer Scatterg The major scatterg evets semcoductors: phoo scatterg (lattce vbrato) mpurty scatterg (ozed ad eutral mpurty) surface scatterg (termato of perodc potetal) Matthesse s rule: μ total μ phoo + μ mpurty + μ surface +... μ μ mpurty σ Moblty reducto μ total Dopat freeze-out μ phoo T T Chapter 4
Hgh-feld Moblty: Velocty Saturato Velocty Saturato: pheomeologcal expressos, sce the physcal effects behd the velocty saturato s too complex to gve good physcal fuctos drectly optcal phoo scatterg s stroger whe the carrer has larger ketc eergy. May popular semcoductors have velocty saturato aroud 0 7 cm/s (/3,000 of speed of lght) μ μ 0 / ( + (μ 0 F/v sat ) β ) /β, where v sat s the saturato velocty coducto moblty s derved from the effectve mass of coducto ad the mea free tme of collso. For S, μ 360 cm /Vs ad μ p 460 cm /Vs at room temperature for trsc samples v drft v sat μ F Chapter 5 F
Dffuso - Fck s Frst Law F F C(l) C(0) C(-l) X0 F: Flux; ½ represets eual probablty to move to left or rght C(x,t): cocetrato V th : thermal velocty l: mea free path F C ( l ) vth F C () l vth F F [ ] F vth C( l) C( l) Cxt (, ) Cxt (, ) vth C(0) l C(0) l x + x Cxt (,) Cxt (,) vl th D x x Taylor seres D s dffuso coeffcet Chapter 6
Dffuso (Perturbato from Carrer Numbers) Carrer dffuso J pdff - D p p; J dff D D, D p are dffuso coeffcet wth uts of cm /s Thermal dffuso J ptdff - p D Tp T; J Tdff D T T; to tell the type of carrer coducto, ether thermal or Halleffect duced curret has to be used. Carrer Curret Euatos: J p J pdrft + J pdff + J ptdff pμ p F - D p p - p D Tp T J J drft + J dff + J Tdff μ F + D + D T T If oly homogeeous lattce temperature s cosdered, the eulbrum codto AND Boltzma statstcs (odegeerate cases) wll result the Eeste relatoshp betwee μ ad D Chapter 7
Este Relato < τ > μ m D ν l th D μ * * m ν * th l mν th From K.E. a oe-dmesoal case, * mν th kt Therefore, D μ kt Chapter 8
Este Relato The electrc feld F ca be cosdered as bad bedg (except whe there s heterojucto): F / E / E C / E V For homogeeous T, eulbrum -D, o-degeerate dopg e ( E E )/ k T F B d de de dx k T dx dx B F D μ Dp μ p kt B d de de de J F D F dx dx dx dx F F N 0 μ + μ μ μ I eulbrum de F /dx 0; Ferm-level s flat! Out of eulbrum de F /dx 0 def E EF EFp 0 J μ ψf 0 ψf dx Chapter 9 F
Electro-Hole Par Geerato-Recombato Fck slaw for carrer coservato (or the cotuty euato): v J + G R p(x)v t d A p(x+dx) v d A p v J p + Gp Rp t G p R The geerato-recombato evets semcoductors ca be categorzed by bad-to-bad, bad-to-trap ad trap-to-trap. The reured coservato of total mometum ad eergy wll be completed by teracto wth phoos (thermal), photos (lght) or aother carrer (mpact ozato ad Auger recombato) Remember that traps are localzed allowable states wth the badgap for electros ad/or holes. Sce trap states are local space, they caot cause drft curret drectly, but ther charge states wll affect the Posso euato ad cotuty euatos Chapter 0
Category of Geerato-Recombato Geerato- Recombato bad-to-bad bad-to-trap trap-to-trap thermal optcal carrer (elec. capture) (elec. capture) (chage locato) (trap hoppg/tuelg) (chage eergy) (mpact ozato) Photos have large eergy but small mometum, ot effectve for drect badgap Phoos have small eergy (kt/) but large mometum, ot effectve for.ev badgap Trap asssted geeratorecombato s most effcet at mdgap (Au, Cu, M, Cr, Fe S) Chapter
Shockley-Read-Hall (SRH) processes Two thermal bad-to-trap processes that complete a bad-to-bad electro-hole par geerato or recombato are called SRH (Shockley-Read-Hall) processes, whch s most mportat S. p p cnp cnp ep p cnp t t p p T 0 p T p T T p T C E T 0 Capture rate s proportoal to the umber of carrers (p), umber of traps (N T ) ad capture coeffcet (c p ), whch has the ut of a cross secto (cm ) the thermal velocty (cm/s). I eulbrum, geerato has to cacel recombato (detaled balace). Off eulbrum, the et geerato-recombato s the: p t R G c p N T ( p p0) c A more geeral expresso preservg the low-jecto lmt: p N T 0 Δp Δp τ p Detal balace t R G Δ τ p t R G t R G p τ ( + ) + τ ( p + p p ) e ( E T E ) / kt p e ( E E T ) / kt Chapter
Sx Shockley Euatos of States Most classcal devce models are derved from these euatos Frst put together by Wllam Shockley for semcoductor classcal devce aalyss (o QM effects such as tuelg yet) ρ ψ ( ψ) F ε Posso E. curvature + + ε ε s J μ F + D p p p 0 J pμ F D p v J + G R t p v J + G R t p t t t ( + p N ) D NA T p p p T Chapter 3 R G R G Elec. Curret E. Hole Curret E. Elec. Cotuty E. Hole Cotuty E. Trap Charge E.
Posso Euato ad Bad Bedg Majorty charge dstrbuto ρ(x) s perturbed. E E c E F E E v x E E c E F E E v x e Ne c ( E E )/ k T F B ( E E )/ k T c F B I -D, E ψ E : eletrostatc (electro) eergy + + p N + N + ( N ) ψ ( ) dx s d x ( ) ψ ε ε ε ε D A T D 0 s 0 ε ε ( ψ( x) ψf )/ kbt ( e ND) Chapter s 0 4
The Debye Legth from the Posso Euato d ψ ( x) ( ψ( x) ψf )/ kbt ( e ND) dx εsε0 d ( ψ( x) ψ( )) ( ( x) ( ))/ kbt ( ( ) F )/ kbt ( ) ψ ψ ψ ψ e e ND dx εsε0 d Δψ ND ( )/ kbt ( ) e Δψ dx εsε0 L D s the Debye legth. It takes L D to resolve the potetal perturbato from the chage et majorty charge. For example, L D 0.04μm for N D 0 6 cm -3 S. Chapter 5 E E c E F E E v x e Ne c ( E E )/ k T F B ( E E )/ k T c F B Assume ( Ψ )/k B T << Ψ << k B T/ uas-eutralty codto Expad the expoetal fucto a Taylor seres ad stop after the secod term. d Δψ N N Δ Δψ dx kt kt D D ( ψ ) εsε0 B εsε0 B x Δψ exp LD LD εsε0kt B N D
Delectrc Relaxato Tme I -D -type materals, there s a small charge desty dsturbace ad we ow wat to fd out about how log t takes to reestablsh a steady state Cotuty E. Δ J t x J F+ D x Curret E. μ Posso E. F x Δ ε ε s 0 F ρ Ψ << kbt/ uas-eutralty codto (t) exp(-t/ρ ε s ε 0 ), where ρ ε s ε 0 s called the delectrc relaxato tme, or the majorty-carrer respose tme, ad s typcally < 0 - s. Chapter 6
Dffuso Legth ad Morty Carrer Lfetme I -D p-type materals, the morty carrers ( 0 /N A ) t J x ( ) τ 0 J μ F + D If J s eglgble (o carrer jecto), the morty carrer decays the order of τ (morty carrer lfetme), 0-4 to 0-9 s depedg o the ualty of the slco crystal. I steady state (o tme varato, costat jecto) ad the drft curret s small for morty carrers exp(-x/l) wth L(D τ ) / beg the dffuso legth (usually o the order of 0-00μm S). x Chapter 7
Tme ad Legth Scale Semcoductor For electro: Debye Legth Delectrc Relaxato Tme τ Morty Carrer Lfetme τ Dffuso Legth L D L εsε0kt B N Dτ D ρ ε ε D s 0 ~ m majorty carrer ~ ps > μs > μm morty carrer L Dτ D D I both majorty ad morty carrer cases, the tmes ad legths gve the umbers for how fast a devato from the carrer eulbrum wll be eualzed ad over whch dstaces small devatos are felt. Chapter 8