3. Carriers. 3.1 Carriers in semiconductors. 3.2 Equilibrium Carriers. 3.3 Excess Carriers

Transcription

1 3. Carrers v.18.aug 3.1 Carrers semcoductors tyes (electros, holes), roertes (charge, mass, m eergy, coducto) trsc / extrsc morty / majorty 3. qulbrum Carrers ad grahcal / aalytcal solutos mass acto law 3.3 xcess Carrers geerato recombato steady state 3. Carrers

2 lectros & holes trsc extrsc 3.1 Carrers semcoductors Carrers Metals (Cu: 8.51 cm -3 ) ve charge oly: sea of free electros Carrers Semcoductors (S: 1 1 cm -3 K: o carrers, o coducto > K: ve ad +ve charges thermal exctato across G electro ( ) CB ad hole () VB ( ) electro-hole ar (HP) Carrers ( ) roertes: charge mass mmum eergy bad dagram mmum eergy crcut ( ) ( ) CB - VB h + e C V Thermal vbratos of atoms ca break bods ad thereby create electrohole ars. 3. Carrers 1385

3 lectros & holes trsc extrsc Charge Mass Mmum eergy hole 1 hole CB: HP/cm 3 << 3x1 19 states/cm 3 free ( ) - each carres a egatve charge = q VB: flled bad, movemet of (wavevector k j ) s ecoutered by (-k j ) et curret desty = N J ( q) oe removed, et curret desty N (N C ) v J ( q) v ( q) v j qv ostve charge ostve velocty lectros (CB) have egatve charge Holes (VB) have ostve charge j -k j -k k k j k artcle-wave mv k Hole: - a mssg electro a valece bod - a emty electroc state the VB - behaves (resods to electrc feld) as f t were a ostvely charged artcle 3. Carrers

4 lectros & holes trsc extrsc Charge Mass Mmum eergy Free lectros sace e - mometum s = mv = 1 mv Partcle: -v Wave: -k 1 m k m k d dk k sace m electro rest mass m e = kg crystals lectros must take to accout the shae of eergy bads 3-D k-sace e - solds are ot free, eergy bads are close to mma ad maxma effectve mass: CB: ostve m* VB: egatve m* (!) m* d Holes = mssg VB electros ostve m* dk m* of electros S:.6m e GaAs:.6m e m* of holes S:.69m e GaAs:.57m e lectros ad holes have ostve mass lectros are always lghter tha holes (= better resod to electrc feld) 3. Carrers

5 lectros & holes trsc extrsc Charge Mass Mmum eergy ( semcoductors) Physcs: carrers thermalzato (wth lattce, crystals) k = ħk = K = (K+P) CB m. eergy for both electros & holes k Water & bubble VB lectros sk to the bottom of the CB ( C ) Holes float to the to of the VB ( V ) 3. Carrers

6 lectros & holes trsc extrsc Charge Mass Mmum eergy ( semcoductors uder bas [ a crcut]) Sace (x) ergy-mometum V(x) lectro ergy lectrostatc P(x) = -ev x x F c d F v A -Tye Semcoductor B V 3. Carrers

7 metals (electros CB)... sulators V? V(x) x x FO mty levels B lectros FO - ev ergy bad dagram B - ev 3. Carrers

8 lectros & holes trsc extrsc 3.1. Itrsc Semcoductors erfect semcoductor crystal, o murtes, o lattce K: o charge carrers > K: HPs are the oly charge carrers HP geerato eergy (thermal, otcal) requred G ostos (wavefuctos) of e - ad h + are sread out over several lattce sacgs y covers 1s of atoms ( ) ( ) ( ) 3. Carrers

9 lectros & holes trsc extrsc Itrsc carrers lectro cocetrato: Hole cocetrato: HP: Par Steady state: rate of HP geerato (G) = recombato (R) G( T) R( T) R(T) deeds o equlbrum cocetratos of e - ad h + : R R G ut: electros/cm 3 ut: holes/cm 3 ut: cm 3 Cocetratos of - S atoms S crystal: 51 cm 3 - e or h S crystal: 1 1 cm 3 Subscrts - : equlbrum (vs steady state) - : trsc (vs extrsc) 3. Carrers

10 ( ) lectros & holes trsc extrsc xtrsc Semcoductors Carrers = Itrsc + xtrsc thermally geerated murtes urosely troduced to semcoductors (dog) Dog ether electros or holes domate P, C D V ( ) = dlute substtutoal sold solutos (m) (solute+solvet) dfferet valecy P atom has 5 outer electros [Ne] 3s 3 3 P S matrx requres 4 e - (3s, 3 ) for comlete bodg Thermal eergy eables outer e - to overcome Coulombc bdg (to the P atom) e - doated to the lattce as a whole -tye dog ( ) B Itrsc: udoed xtrsc: doed C A V Dogs - -tye: creased egatve carrer - -tye: creased ostve carrer B has 3 valece e - (oe bod s comlete) B S matrx the comlete bod ca be trasferred to other atoms (aalogy) e - hos from a adjacet bod h + doated to the lattce as a whole -tye dog 3. Carrers

11 lectros & holes trsc extrsc Doats The org of the door level ( D ) -S III IV V B C N Al S P Ga Ge As I S Sb ( ) ( ) 1 3 P S P D C V colum-v murtes (P, As, Sb) troduce D ear C where all e - are doated to CB(T > 5 K) Colum V atoms are called room tem, ) ( colum-iii murtes (B, Al, I) troduce A ear V whch accet e - from VB (T > 5 K) Colum III atoms are called room tem., ) ( K 5 K K 5 K -S -S 3. Carrers

12 lectros & holes trsc extrsc Iozato eergy ( I ) tycally 5% of g Majorty & Morty carrers tye of charge (egatve / ostve) - door: C D - accetor: A V -tye: >> -tye: >> tye of materal ( / ) ergy (ev) C 1.1 V C Sb P As T C Pt Au O Slco I I.5 B Al Ga I Pd S Se S Te S O ( ) Cu ( ) c F v CB c F v c F v VB (trsc) -tye -tye (a) tyes of (b) materal (c) ergy bad dagrams for (a) trsc (b) -tye ad (c) -tye semcoductors. I all cases, = ergy (ev) Gallum Arsede V Be Mg Z Cd C Cu Fe Cr 3. Carrers

13 carrer cocetrato trsc extrsc comesated 3..1 Carrer Cocetratos (, ) Am: to determe electrcal roertes, also (T) Need to determe carrer cocetratos (,) of: Majorty Carrers: from dog Morty Carrers: ot obvous qulbrum carrer cocetratos (, ) semcoductor statstcs: dstrbuto of carrers over avalable eergy states Total umber of carrers deeds o: Subscrt meas equlbrum codto desty of states (umber of avalable states / ev / cm 3 ): N() robablty of occuacy (Ferm dstrbuto fucto): f() summato of the above at all avalable eerges f ( ) N ( ) d C Ferm-Drac statstcs Quatum mechacs ad Paul s excluso rcle Summato over etre coducto bad Drude model (see later C4) e 3. Carrers e e To determe (T), eed to kow: (T), (,T) check ut! hyscal meag: (cf. vol. coc.of atoms, 1 cm -3 ) V 1 f ( ) N( ) d h

14 ( ) N() carrer cocetrato trsc extrsc comesated Grahcal soluto Aalytcal soluto Mass acto law Ferm-Drac dstrbuto fucto Probablty that a avalable eergy state at ad at temerature T wll be occued by electro f ( ) ( ) 1 e Ferm eergy: F 1 F kt Drac Nobel 1933 Ferm Nobel 1938 f() 1 f ( ) f ) N( ) d, N( ) d N N C ( Desty of states* c v ( ) ( ) N( ) ; V V V C ( forbdde ga ) N() C f() K f() 1-f() V C F (ev) 5 K F (ev) VB CB *states = each eergy level 3. Carrers

15 carrer cocetrato trsc extrsc comesated Grahcal soluto Aalytcal soluto Mass acto law 1 f ( ) f ) N( ) d, N( ) d C ( V ( ) states ev cm 3 ( ) # state N() V f() trsc F C 1-f() # cm 3 d ( ) # ev cm 3 () kt/ () ote: ut N() = / ev / cm 3 N C = / cm 3 3. Carrers

16 carrer cocetrato trsc extrsc comesated Grahcal soluto Aalytcal soluto Mass acto law 1 f ( ) f ) N( ) d, N( ) d C ( V C F V ( ) ( ) ( ) CB VB -tye (trsc) -tye () () N() 1-f() f() e- ad h+ dstrbuto ca merely be dcated by F (wthout havg to draw the whole dstrbuto fucto) ergy dfferece gves a measure of carrer cocetrato 3. Carrers uer half mdga lower half

17 carrer cocetrato trsc extrsc comesated Grahcal soluto Aalytcal soluto Mass acto law 1 f ( ) f ) N( ) d, N( ) d C ( V lectro cocetrato ( ) f ( C ) N C N C [ ex C kt F ] N C = effectve desty of C reresets all electroc states coducto bad ote: C > F as F C, N C * 3/ m kt h Hole cocetrato ( ) 1 f ( ) ote: F > V V N V N V ex as F V F kt, V N V = effectve desty of V reresets all electroc states valece bad N V m h * * * m kt 3 / m N C (cm 3 ) N V (cm 3 ) S 1.9m e 1.15m e GaAs.63m e.48m e Carrers

18 carrer cocetrato trsc extrsc comesated Grahcal soluto Aalytcal soluto Mass acto law 1 f ( ) f ) N( ) d, N( ) d C ( V xtrsc (doed): F osto = deeds o dog N N C V ex ex C kt F kt F V xtrsc ad Itrsc carrer cocetratos are related through the mass acto law Itrsc (udoed): F osto mdga = & e Notes: 1. F (or ) = mdga f N C = N = F = = N N C V ex e ex F kt kt F C kt kt V The roduct of ad s: sce G kt NC NVe C For 3 K, = 1 1 cm -3 N N V e G kt 3. Carrers

19 7 C (cm -3 ) (T) T (C) 1 Ge S GaAs 3 K ,/T (K -1 ) NC N 3/ kt ( T ) h V * * 4 G / kt m m 3/ 3 K 3 K Ge: S: GaAs: s temerature deedet (exoetal) trsc semcoductor s rarely used (Devces are mostly extrsc) 3. Carrers

20 trsc temerature extrsc ozato (T) lectro cocetrato (1 16 cm -3 ) 3 1 S, N D = 1 16 cm -3 (cotrollable, see C1: dog) ozato extrsc total (temerature deedet) T T (K) D CB VB CB VB CB VB o s temerature deedet, yet cotrollable extrsc semcoductor s always used 3. Carrers 1385

21 carrer cocetrato trsc extrsc comesated 3..4 Comesated Semcoductors (cotas both doors ad accetors) Foud (*) dodes & trasstors, es. BJT (emtter & base) * * xamle: -tye wth N D > N A ) from d CB ) from VB a ) created VB v) CB recombes wth VB v) reeat )-v) for all -- comesato v) et cocetrato CB ~ N D N A ~ N D N A Sace charge eutralty: mmoble (boded) electros moble holes moble electros Nd N a moble? 3. Carrers

22 Summary Carrer Cocetratos of Semcoductors uder qulbrum Cocetratos of - Itrsc Semcoductors: - xtrsc Semcoductors: -tye -tye electros holes = >> << - Relatosh betwee doat cocetratos (N D, N A ) & carrer cocetratos (, ) Nd N a -tye: N D >, N A =, >> N D -tye: N A >, N D =, >> N A 3. Carrers 1385

23 Geerato Recombato Steady State 3.3 xcess Carrers - carrers: electros, hotos - hosts: elemetal (S), comoud (III-V) semcoductors - meda: wres (Cu), fbers (SO, lastc) sgal Ifo Trasfer Badwdth: -MHz -Mbs lectro, hole trasort semcoductor devces -,t -f T,f max -SNR,NF excess Charge jecto across juctos Otcal exctato d,d carrers, ose qulbrum 3. Carrers 1385

24 Geerato Recombato Steady State Geerato Otcal exctato Requremet for otcal absorto Overvew hc/l G h C V ( ) ( ) t Drect & Idrect semcoductors ( ) ( ) G h < G o absorto (ths s why NaCl crystal s clear ad glass s trasaret.) lectros ad holes geerated are excess (D,d) carrers; out of balace wth the evromet. They must recombe; ror to that they cotrbute to coductvty ( hoto ). Both tyes absorb, but drect semcoductor absorb much more effcetly (hgher ). C V L k x [111] y [1] X CB x VB y GaAs ( ) C V L c a [111] [1] X 3. Carrers b d CB b c d VB S ( ) Drect trasto: x, y, c, d, a (effcet) Idrect trasto: b (effcet) ffcecy quatfed by

25 Geerato Recombato Steady State hoto Absorto coeffcet (cm -1 ) Absorto xermet: quatfcato of lght absorto by a materal xermetal Setu Iut (sgle l) testy I hotos/cm -s Beer s law for otcal absorto: Lght testy, di( x) I( x) I( x) Ie dx trasmtted x = l I t x I e l radom rocess I moochromator l x = samle L x x = L detector ( ) I A I A I A I A B I B B I B L x ( ) 3. Carrers

26 Peetrato deth (m) Geerato Recombato Steady State hoto - Semcoductor absorbs hotos wth eerges G - swee l, measure I t ca lot (I t, l), equvalet to (, ) - oset of absorto at G - ear badedge absorto dcate drect (shar) or drect (gradual) semcoductor Absorto Coeffcet (cm -1 ) Wavelegth (m) K ISb GaAs IAs IP GaP Ge S / Peetrato deth deth where most (1/e) hotos are absorbed by a materal ergy (ev) 3. Carrers

27 Geerato Recombato Steady State hoto Photocoductvty hoto otcal absorto D, D Coductvty () hotocoductvty (c.f. dark coductvty) xcess carrers recombe to retur to equlbrum dark hoto q q q q + Illumato D D eqlb. excess carrers + A tye semcoductor the dark. = o << o B Illumato wth h > g creates excess holes: = o +D D C I dark after llumato. xcess holes are dsaearg by recombato. follows mass acto law? Illumato of a -tye semcoductor results excess electro 3. Carrers ad hole cocetratos. After the llumato, the recombato1385 7

28 hoto Alcato examles: Lght-deedet resstor (LDR) or hotocoductve cell Chage resstace whe exosed to lght (d,d) r R most sestve to lght wth h G ot absored h G h G absored at surface does ot cotrbute to bulk coductvty Other alcatos: ght lghts ON/OFF at dusk/daw, automatc traffc lght cameras measure llumato levels BTS/lbrary couters (toll booths), burglar alarms sgallg systems (Tx: lghtbeam, Rx: hotocoductve cells) G (CdS) =.4 ev D 3. Carrers

29 Geerato Recombato Steady State LD Recombato of excess carrers semcoductors: drect: hotos (oly chage s requred) drect: hoos (chage ad are requred) CB electros (moble) fall to VB (boded) D released as lght troducto of curret electrolumescece lumescece carrers excted by hoto absorto hotolumescece hgh eergy e - bombardmet cathodolumescece CRT fast (1-8 s) Y Drect bad-to-bad recombato rocess fluorescece? Ar + Hg electrc dscharge vsble + UV N e.g. ZS colour TV volve tras --> slow (a few sec, or few m) hoshorescece (TV scree) 3. Carrers

30 Drect Idrect Geerato Recombato Steady State xcess carrers are quckly removed from drect semcoductors by radatve recombato (RR). We wat to kow how quck. Ths how quck lmts swtchg seed of otcal commucato systems. RR rocess s sotaeous (rob. of e - ad h + recombe s costat tme) exoetal decay of excess carrers quatfy ths exoetal : Net rate of chage e - cocetrato coducto bad s d() D d() D Before (lght) d( t) d( t) After (dark) d( t) dt geerato (G) dd( t) ( dt ( t) ( t) d( t) d( t) recombato (R) Drect Bad Ga g -k ) d( t) d h C V VB ( t) (a) GaAs -k CB c Photo v (from slde #9) k r VB (c) S wth a reco 3. Carrers

31 Solve: Aroxmato: dd t) ( dt ( ) d( t) d ( t 1. extrsc materal -tye o >> o -tye o >> o For -tye ( >> ) For -tye ( >> ) dd( t) dt d( t) De * d( t) * t 1 t De t / t ). low-level jecto d << ( + ) dd( t) d( t) dt d( t) De t 1 t De t / t For drect recombato, HPs are ahlated ars excess majorty ad morty carrers decay at exactly the same rate: d( t) d( t) t Note: more accurate soluto recombato lfetme: ช วช ว ตการผสมกล บ (wthout arox. 1) morty carrer lfetme: ช วช ว ตพาหะข างน อย 1 t t mea free tme: เวลาเสร เฉล ย (ext chater) for S, = 1 * -1 cm 3 /s (cf slde #4) Beer s law for otcal absorto 3. Carrers

32 t S, 3 K t Lfetme τ ad dffuso legth L of holes -tye S vs. door desty t Lfetme τ ad dffuso legth L of electros -tye S vs. accetor desty t t 1 source: htt:// for S, = 1-1 cm 3 /s 3. Carrers

33 t : what t meas at materal level GaAs 1 15 * 1-3 ** GaAs doed wth N A = 1 15 cm -3 ( o ) = 1 6 cm -3 = / = 1-3 cm -3 ( >> aroxmato vald) Gve that 1 14 HP/cm 3 t =.e. D = D (@ t = ) = 1 14 (=.1 ) t = t = 1 s oly 1% low-level Note: % chage majorty carrer () s small * morty carrer () s large ** 3. Carrers

34 related to, R, I, V t : what t meas at devcelevel Imlcato: t bs traset traset majorty equlbrum steady-state equlbrum morty 3. Carrers

35 Drect Idrect Geerato Recombato Steady State Colum IV materals (S, Ge) are drect rob. of drect recombato of e - ad h + s v. small very weak badga lght ca be gve off, but eed a very sestve strumet CB Recombato evets vastly occur va recombato levels ( r ) wth the bad ga Drect Bad Ga g Imurtes or lattce defects c Photo v C CB Fudametal mechasms related to r ( ) ( ) ( ) ( ) -k VB (a) GaAs k -k VB k vb (b) S C r V VB -k CB r c Phoos v VB k (c) S wth a recombato ceter Tra electro Release electro Tra hole Release hole Idrect recombato = (ก) (ค) Coservatos of - ergy - Mometum 3. Carrers

36 Drect Idrect Geerato Recombato Steady State ergy level the forbdde ga ca result 3 rocesses deedg o relatve osto the badga. Close to md-ga ( ): 1. r < F : (start flled) tra hole (a), the electro (b). r > F : (start emty) tra electro (b), the hole (a) Close to bad-edge ( C or V ): 3.1 close to CB : mostly tra/release CB electros 3. close to VB : mostly tra/release VB holes CB c v VB CB c v VB r r r Recombato ceter t Trag ceter (b) r? t r : Recombato ceters t : Trag ceter (regardless of exctato mechasms) (a) Recombato t (a) t Phoos (b) Trag 3. Carrers

37 Geerato Recombato Steady State Drect Idrect S C doors D t tras (electros) r recombato ceters t tras (holes) V accetors A same dagram as slde #1.e. t 1 N N t t t cosh kt / cosh t kt N t (Au atoms/cm 3 ) t (s) Carrers

38 Geerato Recombato Steady State Steady State Carrer Cocetrato Steady state = exctato source ON ad costat -- equlbrum dsturbed, all rocesses are costat ad balaced by oosg rocesses F s oly meagful whe o excess carrers are reset (.e. equlbrum) Ferm level -- equlbrum: Wth exctato, lots of e/h at the same tme, the same semcoductor! Q) How to rereset ths stuato terms of eergy dagram? A) wrte steady state cocetratos the same form usg quas-ferm levels F ad F ( F ) ex o otcal exctato kt o exteral exctato excet for temerature (costat tem., the dark, o - or B-feld) qulbrum ( F ) Steady State (F, F ) ( F ) ex kt 3. Carrers C F V ( F ) ex kt ( F ) ex kt F F F ( F ) ex kt CB VB

39 qulbrum: Steady State ractce: otcal exctato (steady lght shoe o samle) theory:.e. g (from slde #9) ( T ) d; g( T) g ( steady-state recombato ad o trag o low-level exctato (gore d ) d d)( ( ) d) ( d d) : g( T ) go d d g o ) ( d d t (smlar to #9) Practcal Notes Otcal ower: W = J/s = ev/s Power desty (volume): W/cm 3 = ev/cm 3 -s Power desty (areal): W/cm = ev/cm -s Quatum effcecy: (assume 1%, S@3K) 1.1 ev 1 HP d d g o t xctato s usually gve terms of how may HP/cm 3 are geerated for every t secod (HP/cm 3 -s) 3. Carrers

40 & H trsc extrsc q, m*, Coclusos 1. Carrers semcoductors tyes (+,-), roertes (q, m*, ) trsc: org (thermal), cocetrato (T) carrer coc trsc extrsc comesated G R SS V 1 f ( ) f ) N( ) d ; N( ) d C ( hoto t, r, t F, F extrsc: org (thermal + dog), cocetrato (T), tyes (, ), classfcato (m, maj). qulbrum carrers (, ) Ferm-Drac dstrbuto fucto f(), Ferm eergy F, DOS N(), effectve DOS N C, N V determato by grahcal/aalytcal techques mass-acto law: = 3. xcess carrers (D, D, d(t), d(t)) Geerato: mechasms (otcal absorto, electrcal jecto), results (hotocoductvty hoto ) Recombato: o mechasms drect (lumescet) & drect (heat) semcoductors ( r, t ) o lfetmet Steady state o reresetato bad dagram (F, F ) ad breakdow of mass-acto law N A or N D 3. Carrers

41 3.1 A S crystal s doed such that C F =. ev, fd ) the robablty that C wll be occued by a electro, ) electro cocetrato. N C = cm A GaAs crystal s doed such that F V =. ev, fd ) the robablty that V wll be occued by a hole, ) hole cocetrato. N V = cm The S crystal 3.1 s to be doed wth As. Determe the As cocetrato that wll make F = C. 3.4 A S samle s doed wth 1 17 As atoms/cm 3. =?. For 3K: G = 1.1 ev, = 1 1 cm -3. Draw bad dagram wth C, V, F ad I. 3. Carrers

42 ต.ย. 3.7 Materal: S: = 1 17 cm -3, = 1 1 cm -3 Recombato: t = t = 3 s Otcal exctato: Geerato rate g o : 1 16 HPs/cm 3 er 1 s (a) fd: d, d,, (b) draw bad dagrams w, w/o exctato ต.ย. 3.8 Materal egeerg: doed wth Au to reduce t to 1 s, lot (t) ad (t) ด F slghtly above F small chage majorty carer coc. F large shft from F large crease morty carrer coc. Devatos of F ad F from F dcate how far steady state values (, ) dffer from equlbrum values (, ) F - F measures devato from equlbrum equlbrum F = F = F.31 ต.ย. 3.7 GaAs S tye drect drect t s s 3. Carrers