ECE606: Solid State Devices Lecture 9 Recombiatio Processes ad Rates Gerhard Klimeck gekco@urdue.edu Outlie ) No-equilibrium systems ) Recombiatio geeratio evets 3) Steady-state ad trasiet resose ) Motivatio of R-G formula 5) Coclusio Ref. Chater 5,. 3-6
Curret Flow Through Semicoductors Deeds o chemical comositio, crystal structure, temerature, doig, etc. V I I G V q v A Carrier Desity velocity Quatum Mechaics + Equilibrium Statistical Mechaics Ecasulated ito cocets of effective masses ad occuatio factors (Ch. -) Trasort with scatterig, o-equilibrium Statistical Mechaics Ecasulated ito drift-diffusio equatio with recombiatio-geeratio (Ch. 5 & 6) 3 No-equilibrium Systems Chater 6 Chater 5 V I vs. How does the system go BACK to equilibrium?
Idirect vs. Direct Badga The to & bottom of bads do ot alig at same wavevector k for idirect badga material 5 Direct Bad-to-bad Recombiatio I real sace I eergy sace Photo Photo e ad h must have same wavelegth i,000,000 ecouters Direct trasistio direct ga material GaAs, IP, ISb (3D) Lasers, LEDs, etc. 6
Direct Excitoic Recombiatio I eergy sace Mostly i D systems Requires strog coulomb iteractios Photo (wavelegth reduced from bulk) I real sace CNT, IP, ID-systems Trasistors, Lasers, Solar cells, etc. 7 Idirect Recombiatio (Tra-assisted) Phoo Tra eeds to be mid-ga to be effective. Cu or Au i Si States close to Ec or Ev do ot hel efficietly. Ge, Si,. Trasistors, Solar cells, etc. Cu i Si is extremely efficiet i rovidig recombiatio Cu i Si was avoided comletely for may years Itel itroduced Cu for itercoects secial recautios so Cu does ot eter the Si. 8
Idirect Recombiatio (Tra-assisted) Phoo Tra eeds to be mid-ga to be effective. Cu or Au i Si States close to Ec or Ev do ot hel efficietly. Ge, Si,. Recombiatio rate is i geeral slower tha direct via hoto. Why? Trasistors, Solar cells, etc. electro ad oe hole eed to localize AND fid a tra. less likely evet. Cu i Si is extremely efficiet i rovidig recombiatio Cu i Si was avoided comletely for may years Itel itroduced Cu for itercoects secial recautios so Cu does ot eter the Si. 9 Imact Ioizatio A Geeratio Mechaism Requires very high electric field 3 Iverse of the Auger recombiatio Si, Ge, IP Lasers, Trasistors, etc. 0
Auger Recombiatio 3 Phoo (heat) Requires very high electro desity 3 IP, GaAs, Lasers, etc. Photo Eergy ad Wavevector E hoto k hoto π a E V + E ħk + ħk V hoto hoto E C ħk C k hoto π λ i µ m π. / i ev E hoto π π << a 5 0 µ m Photo has large eergy for excitatio through badga, but its wavevector is egligible comared to size of BZ
Phoo Eergy ad Wavevector E V + E ħk + ħk V hoo hoo E C ħk v soud ~ 0 3 m/s << v light c ~ 0 6 m/s λ soud >> λ light C k hoo π π λ ħ υ / E soud hoo π π a 5 0 µ m Phoo has large wavevector comarable to BZ, but egligible eergy comared to badga 3 Localized Tras ad Wavevector a π π k tra ~ a 5 0 µ m Tra rovides the wavevector ecessary for idirect trasitio
Outlie ) No-equilibrium systems ) Recombiatio geeratio evets 3) Steady-state ad trasiet resose ) Derivatio of R-G formula 5) Coclusio Ref. Chater 5,. 3-6 5 Equilibrium, Steady state, Trasiet Equilibrium Device (,) Steady state time Trasiet Eviromet (,) time 6
Detailed Balace: Simle Exlaatio Equilibrium is a very active lace I a tyical semicoductor device there are e7 to e0 electros i the coductio bad. All electros carry charge ad are occuyig their resective DOS. Fermi-Dirac distributio demads exloratio of allowed states I equilibrium each rocess is balaced by its couter rocess > Detailed Balace > Exterally it looks as if othig is haeig. This is differet from steady state!!! 7 Detailed Balace: Simle Exlaatio Mexico USA The rates of exchage of eole (articles) betwee every air of coutries (eergy levels) is balaced. Hece the ame Detailed Balace. Detailed balace is the roerty of equilibrium Chia 3 3 Idia The oulatio of each of the coutries (eergy levels) remais costat uder detailed balace. The cocet of detailed balace is owerful, because it ca be used for may thigs (e.g. reduce the umber of ukow rate costats by half, ad derive article distributios like Fermi- Dirac, Bose-Eistei distributios, etc.) 9 i & 9 out All umbers are eole/uit time. Equilibrium is a very active lace Fermi-Dirac distributio demads exloratio of allowed states 8
Disturbig the detailed balace requires o-equilibrium coditios (eeds eergy). Uidirectioal forces (red lies) ca create such No-equilibrium coditios. The rates of exchage of eole (articles) betwee every air of coutries (eergy levels) is NOT balaced, but the sum of all arrival ad deartures to all coutries is zero. The flux at steady state is balaced overall, but the flux is NOT the same as i detailed balace (e.g. i ad out i SS vs. 9 i ad 9 out for Detailed Balace, for examle). The oulatio of a coutry (eergy level) remais costat with time after steady state is reached. Steady-state Resose Mexico Chia 3 6 Idia USA 5 Oe ca use the requiremet that et flux at steady state be zero to calculate steady state oulatio of a coutry (Eq. 5.) i ad out from USA 9 Detailed Balace, Trasiet, Steady-state Mexico USA Mexico 3 USA Mexico 3 USA Chia 3 3 3 5 Chia 5 Chia 6 5 Idia Idia Idia 9 i 9 out Poulatio coserved Equilibrium Detailed balace Forced uidirectioal coectios (red lies) disturbs equilibrium (e.g. 0 i/ out at time t local oulatios ot coserved, but global oulatio is. Trasiet oulatios i out Poulatio stabilized Steady State But NOT Equilibrium 0
Outlie ) No-equilibrium systems ) Recombiatio geeratio evets 3) Steady-state ad trasiet resose ) Motivatio of R-G formula 5) Coclusio Ref. Chater 5,. 3-6 Idirect Recombiatio (Tra-assisted) Phoo Ge, Si,. Trasistors, Solar cells, etc.
total tras N electrofilled tras T Physical view of Carrier Cature/Recombiatio + Tras have destroyed oe electro-hole air No chage i T ad T (3) After hole cature T emty tras T electro Crystal / atoms hole with vibratios () Before a cature () After electro cature 3 Carrier Cature Coefficiets υ th t * 3 m υ th kt 7 υ 0 th cm s d Volume T RelArea dt TotalArea t d A υtht T σ dt A t c T c σ υ th
Cature Cross-sectio σ π r 0 Z cature model e 5 0 8 e 0 cm 6-6 0 5 h e3 7 0 h Cascade model for cature 5 Coclusios ) There are wide variety of geeratio-recombiatio evets that allow restoratio of equilibrium oce the stimulus is removed. ) Direct recombiatio is hoto-assisted, idirect recombiatio hoo assisted. 3) Cocets of equilibrium, steady state, ad trasiet dyamics should be clearly uderstood. 6
Outlie ) Derivatio of SRH formula ) Alicatio of SRH formula for secial cases 3) Direct ad Auger recombiatio ) Coclusio Ref. ADF, Chater 5,. -5 7 Sub-rocesses of SRH Recombiatio () (3) () () ()+(3): oe electro reduced from Coductio-bad & oe-hole reduced from valece-bad ()+(): oe hole created i valece bad ad oe electro created i coductio bad 8
SRH Recombiatio Physical icture Equivalet icture () () () () (3) () (3) () ()+(3): oe electro reduced from C-bad & oe-hole reduced from valece-bad ()+(): oe hole created i valece bad & oe electro created i coductio bad 9 Chages i electro ad hole Desities () () (3) () t, c T + et ( fc ) t 3, c T + T e f υ 30
Detailed Balace i Equilibrium () () ( ) f c Assume o-degeerate t, e c 0 T 0 + et 0 0 + e c 0 T 0 T 0 0 T 0 c T 0 c ( ) 0 c 0 T 0 T 0 (3) () t c +e T T 3, 0 c 0T 0 + T 0e e c 0 T 0 T 0 c ( ) 0 c 0 T 0 T 0 3 Exressios for ( ) ad ( ) () () (3) () 0 T 0 T 0 0 T 0 0 T 0 T 0 T 0 0 T 0 T 0 0 0 i 3
Exressios for ( ) ad ( ) Tra is like a door! NT T 0 NT ( f00 ) β ( E ) + T EF gde + g ex + g ex g ex f00 f00 + g ex ( f ) 00 f00 0 T 0 T 0 0 N T ( NT f00 ) ( f ) 00 f 00 i E T g ex / g ex + g ex + g ex i i β ( E E ) β ( E E ) e g e g e D F i T F D β ( E E ) T i i i D ( E E ) g e β i T 33 Dyamics of Tra Poulatio () (3) () () T + t t, t 3, c T et c T +e T c ( ) c ( ) T T T T 3
Steady-state Tra Poulatio () (3) () () t T 0 c ( ) c ( T T ) T T c N + c N c ( ) T T T T T c ( + ) + c ( + ) 35 Net Rate of Recombiatio-Geeratio () (3) () () d R c ( T T ) dt i c N ( + ) + ( + ) T cnt 36
Outlie ) Derivatio of SRH formula ) Alicatio of SRH formula for secial cases 3) Direct ad Auger recombiatio ) Coclusio 37 Case : Low-level Ijectio i -tye R i ( + ) + ( + ) ( 0 + )( 0 + ) i ( + + ) + ( + + ) 0 0 ( 0 + 0 ) + ( + + ) + ( + + ) 0 0 ( 0 ) ( ) 0 0 0 0 0 + 0 + Lots of holes, few electros > ideedet of holes 38
Case : High-level Ijectio R i ( + ) + ( + ) ( 0 + )( 0 + ) i ( + + ) + ( + + ) 0 0 ( 0 + 0 ) + ( + + ) + ( + + ) 0 0 ( + ) ( + ) e.g. orgaic solar cells 0 0 0 + 0 + Lots of holes, lots of electros > deedet o both relaxatios 39 High/Low Level Ijectio R high ( + ) 0 0 R low 0 0 which oe is larger ad why? 0
R Case 3: Geeratio i Deletio Regio Deletio regio i PN diode: 0 i ( + ) + ( + ) i ( ) + ( ) NEGATIVE Recombiatio > Geeratio 0 << i > geeratio to create, Equilibrium restoratio! Outlie ) Derivatio of SRH formula ) Alicatio of SRH formula for secial cases 3) Direct ad Auger recombiatio ) Coclusio
( ) R B i ( ) 0 0 Bad-to-bad Recombiatio B is a material roerty Direct recombiatio at low-level ijectio ( )( ) R B + + B, 0 0 0 i 0 Direct geeratio i deletio regio ( i ) R B B i 3 Auger Recombiatio electro & hole ( i ) ( ) R c + c c,c ~ i 9 6 0 cm /sec Auger recombiatio at low-level ijectio R ( ) ( ) N 0 0 A cn A auger auger cn A
Effective Carrier Lifetime Light R RSRH + Rdirect + RAuger t eff ( t) ( t 0) e + + ( c N BN c N ) + + eff T D,auger D SRH direct Auger ( T D,auger D ) c N + BN + c N 5 Effective Carrier Lifetime with all Processes N D c N eff,auger D ( c N BN c N ) + + eff T D,auger D Elec. Dev. Lett., (8), 99. 6
Coclusio SRH is a imortat recombiatio mechaism i imortat semicoductors like Si ad Ge. SRH formula is comlicated, therefore simlificatio for secial cases are ofte desired. Direct bad-to-bad ad Auger recombiatio ca also be described with simle heomeological formula. These exressios for recombiatio evets have bee widely validated by measuremets. 7