ECE606: Solid State Devices Lecture 8

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1 ECE66: Solid State evices Lecture 8 Gerhard Klimeck gekco@urdue.edu Remider:»Basic cocets of doors ad accetors»statistics of doors ad accetor levels»itrisic carrier cocetratio Temerature deedece of carrier cocetratio Presetatio Outlie Multile doig, co-doig, ad heavy-doig Coclusio Referece: Vol. 6, Ch. 4

2 oor Atoms i H-aalogy + r r 3 oor Atoms i Real ad Eergy Sace r ~s mev E m * 4 host ( Ks,hostħ) 4πε mq 4 4 ( πεħ) q * m 3. 6 m host m m * host K s,host K s,host E T E /β~k B T~5meV at T3K 4

3 How to Read the Table Summary A bulk material must be charge eutral over all A dv Further if the material is satially homogeous A e Ae + ( EF E ) / kbt ( EA E F ) / kbt + e + 4e ( EF E V ) / kbt ( EC E F ) / kbt A V

4 Remider»Basic cocets of doors ad accetors»statistics of doors ad accetor levels»itrisic carrier cocetratio Temerature deedece of carrier cocetratio Presetatio Outlie Multile doig, co-doig, ad heavy-doig Coclusio Referece: Vol. 6, Ch. 4 Carrier-desity with Uiform oig A bulk material must be charge eutral over all A dv Further if the doig is satially homogeous A F itegral vs. F fuctio? ( ) A V F β EF EV A F β ( EC EF ) + β ( EF E ) β ( EA EF ) π π + e + 4e e e E F A + EF B A E F )/ kbt + e + 4e ( EF E V )/ kbt ( E C )/ kbt V A ( E )/ k T ( E (arox.) Oce you kow E F, you ca calculate,, +, A-.

5 Itrisic Cocetratio A e A Ce + ( EF E )/ kbt ( EA E F ) / kbt + e + 4e ( EF E V ) / kbt ( EC E F )/ kbt V β ( E E ) β ( E E ) c F + v F C V e e EG V EF Ei + l β C Remider»Basic cocets of doors ad accetors»statistics of doors ad accetor levels»itrisic carrier cocetratio Temerature deedece of carrier cocetratio Presetatio Outlie Multile doig, co-doig, ad heavy-doig Coclusio Referece: Vol. 6, Ch. 4

6 Carrier esity with oors I satially homogeous field-free regio A Assume -tye doig Ce + ( EF E ) / kbt ( EA E F ) / kbt + e + 4e ( EF E V ) / kbt ( EC E F ) / kbt V e A (will lot i ext slide) Temerature-deedet Cocetratio Freeze out Extrisic Itrisic i Temerature

7 Physical Iterretatio Freeze out Extrisic Itrisic i Temerature Electro Cocetratio with oors C F C Ce e e β ( E E ) β E β E C F + + β F E e ( E ) β ( Ec E ) + e + C ξ

8 Electro cocetratio with oors + + e e + F B + e ( EF E V ) / kbt ( EC E F ) / kbt V C ( E E ) / k T i i + + o aroximatio so far. ξ High oor desity/freeze out T i i + ξ ξ ξ ξ C ξ C e β ( E E ) Freeze out Extrisic Temerature Itrisic i ξ 4 + ξ

9 Extrisic T C ( EC E ) / kt ξ e ξ 4 + ξ ξ 4 + ξ Freeze out Extrisic Temerature Itrisic i Electro cocetratio equals door desity hole cocetratio by x i Itrisic T for < ( EF E )/ E E kbt + e + F i + + i ξ + Freeze out Extrisic Temerature Itrisic i i i

10 Extrisic/Itrisic T For i + + i 4 Freeze out Extrisic Itrisic For i + + i 4 i Temerature i What will hae if you use silico circuits at very high temeratures? Badga determies the itrisic carrier desity. etermiatio of Fermi-level β ( E ) c E F Ce EF EC + l β C + + e V E F Ce + ( EF E )/ kbt + e ( E E )/ k T ( E )/ k T F V B C B

11 Remider»Basic cocets of doors ad accetors»statistics of doors ad accetor levels»itrisic carrier cocetratio Temerature deedece of carrier cocetratio Presetatio Outlie Multile doig, co-doig, ad heavy-doig Coclusio Referece: Vol. 6, Ch. 4 Multile oor Levels Multile levels of same door + + A ( E F E )/ kbt + e ( E F + E )/ kbt e ( EA + 4 EF )/ kbt e Codoig e + e + 4e A ( E F E )/ kbt ( E F E )/ kbt ( EA E F )/ kbt

12 Heavy oig Effects: Badtail States E k E k? Heavy oig Effects: Hoig Coductio E E Badga arrowig k K? e C V β E e.g. Base of HBTs * G Bad trasort vs. hoig-trasort e.g. a-silico, OLE

13 Arragemet of Atoms Poly-crystallie Thi Film Trasistors Amorhous Oxides Crystallie Poly-crystallie material Isotroic badga ad icrease i scatterig

Charge eutrality coditio ad law of mass-actio allows calculatio of Fermi-level ad all carrier cocetratio.

14 Bad-structure ad Periodicity PRB, 4, 58, 97 Edagawa, PRL,,39, 8 Periodicity is sufficiet, but ot ecessary for badga. May amorhous material show full isotroic badga Coclusios. Charge eutrality coditio ad law of mass-actio allows calculatio of Fermi-level ad all carrier cocetratio.. For semicoductors with field, charge eutrality will ot hold ad we will eed to use Poisso equatio. 3. Heavig doig effects lay a imortat role i carrier trasort.

15 Outlie ) o-equilibrium systems ) Recombiatio geeratio evets 3) Steady-state ad trasiet resose 4) erivatio of R-G formula 5) Coclusio Ref. Chater 5, Curret Flow Through Semicoductors eeds o chemical comositio, crystal structure, temerature, doig, etc. V I I G V q v A Carrier esity velocity Quatum Mechaics + Equilibrium Statistical Mechaics Ecasulated ito cocets of effective masses ad occuatio factors (Ch. -4) Trasort with scatterig, o-equilibrium Statistical Mechaics Ecasulated ito drift-diffusio equatio with recombiatio-geeratio (Ch. 5 & 6)

16 o-equilibrium Systems Chater 6 Chater 5 I vs. V How does the system go BACK to equilibrium? 3 irect Bad-to-bad Recombiatio I real sace I eergy sace Photo Photo e ad h must have same wavelegth i,, ecouters irect trasistio direct ga material GaAs, IP, ISb (3) Lasers, LEs, etc.

17 irect Excitoic Recombiatio I eergy sace Mostly i systems Requires strog coulomb iteractios Photo (wavelegth reduced from bulk) I real sace CT, IP, I-systems Trasistors, Lasers, Solar cells, etc. Idirect Recombiatio (Tra-assisted) Phoo Tra eeds to be mid-ga to be effective. Cu or Au i Si Ge, Si,. Trasistors, Solar cells, etc.

18 Auger Recombiatio 3 Phoo (heat) Requires very high electro desity IP, GaAs, Lasers, etc. Imact Ioizatio A Geeratio Mechaism 3 4 Si, Ge, IP Lasers, Trasistors, etc.

19 Idirect vs. irect Badga The to & bottom of bads do ot alig at same wavevector k for idirect badga material 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 4 5 µ m Photo has large eergy for excitatio through badga, but its wavevector is egligible comared to size of BZ

20 Phoo Eergy ad Wavevector E V + E ħk + ħk V hoo hoo E C ħk v soud ~ 3 m/s << v light c ~ 6 m/s λ soud >> λ light C k hoo π π λ ħ υ / E soud hoo π π a 4 5 µ m Phoo has large wavevector comarable to BZ, but egligible eergy comared to badga Localized Tras ad Wavevector a π π k tra ~ 4 a 5 µ m Tra rovides the wavevector ecessary for idirect trasitio

21 Outlie ) o-equilibrium systems ) Recombiatio geeratio evets 3) Steady-state ad trasiet resose 4) erivatio of R-G formula 5) Coclusio Ref. Chater 5, Equilibrium, Steady state, Trasiet Equilibrium evice (,) Steady state time Trasiet Eviromet (,) time 4

22 etailed Balace: Simle Exlaatio Mexico USA The rates of exchage of eole (articles) betwee every air of coutries (eergy levels) is balaced. Hece the ame etailed Balace. etailed balace is the roerty of equilibrium Chia 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- irac, Bose-Eistei distributios, etc.) 9 i & 9 out All umbers are eole/uit time. Equilibrium is a very active lace Fermi-irac distributio demads exloratio of allowed states isturbig the detailed balace requires o-equilibrium coditios (eeds eergy). Uidirectioal forces (red lies) ca create such o-equilibrium coditios. The rates of exchage of eole (articles) betwee every air of coutries (eergy levels) is OT 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 OT the same as i detailed balace (e.g. i ad out i SS vs. 9 i ad 9 out for etailed Balace, for examle). The oulatio of a coutry (eergy level) remais costat with time after steady state is reached. Steady-state Resose Mexico Chia Idia USA 5 4 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

23 etailed Balace, Trasiet, Steady-state Mexico USA Mexico 3 USA Mexico 3 USA Chia Chia 5 4 Chia Idia Idia Idia 9 i 9 out Poulatio coserved Equilibrium etailed balace Forced uidirectioal coectios (red lies) disturbs equilibrium (e.g. i/ out at time t local oulatios ot coserved, but global oulatio is. Trasiet oulatios i out Poulatio stabilized Steady State But OT Equilibrium 45 Outlie ) o-equilibrium systems ) Recombiatio geeratio evets 3) Steady-state ad trasiet resose 4) erivatio of R-G formula 5) Coclusio Ref. Chater 5,

24 Idirect Recombiatio (Tra-assisted) Phoo Ge, Si,. Trasistors, Solar cells, etc. total tras electrofilled tras T Physical view of Carrier Cature/Recombiatio + Tras have destroyed oe electro-hole air o 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

25 Carrier Cature Coefficiets υ th t * 3 m υ th kt 7 υ 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 Z cature model e 5 8 e cm h e3 7 4 h Cascade model for cature

26 Coclusios ) There are wide variety of geeratio-recombiatio evets that allow restoratio of equilibrium oce the stimulus is removed. ) irect recombiatio is hoto-assisted, idirect recombiatio hoo assisted. 3) Cocets of equilibrium, steady state, ad trasiet dyamics should be clearly uderstood. Outlie ) erivatio of SRH formula ) Alicatio of SRH formula for secial cases 3) irect ad Auger recombiatio 4) Coclusio Ref. AF, Chater 5,. 4-54

27 Sub-rocesses of SRH Recombiatio () (3) () (4) ()+(3): oe electro reduced from Coductio-bad & oe-hole reduced from valece-bad ()+(4): oe hole created i valece bad ad oe electro created i coductio bad 53 SRH Recombiatio Physical icture Equivalet icture () () () () (3) (4) (3) (4) ()+(3): oe electro reduced from C-bad & oe-hole reduced from valece-bad ()+(4): oe hole created i valece bad & oe electro created i coductio bad

28 Chages i electro ad hole esities () () (3) (4) t, c T + et ( fc ) t 3, 4 c T + T e f υ etailed Balace i Equilibrium ( ) t, e () () f c Assume o-degeerate c T + et + e c T T T c T c ( ) c T T (3) (4) t c +e T T 3, 4 c T + T e e c T T c ( ) c T T

29 Exressios for ( ) ad ( ) () () (3) (4) T T T T T T T T i Exressios for ( ) ad ( ) Tra is like a door! T T T ( f ) β ( E ) + T EF ge E T f f + g ex + g ex g ex f f g ex / g ex + g ex f + g ex + g ex ( ) T T T ( T f ) ( f ) i i i β ( E E ) β ( E E ) e g e g e F i T F β ( E E ) T i i i ( E E ) g e β i T

30 yamics of Tra Poulatio () (3) () (4) T + t t, t 3, 4 c T et c T +e T c ( ) c ( ) T T T T Steady-state Tra Poulatio () (3) () (4) t T c ( ) c ( T T ) T T c + c c ( ) T T T T T c ( + ) + c ( + )

31 et Rate of Recombiatio-Geeratio () (3) () (4) τ d R c ( T T ) dt i c ( + ) + ( + ) T ct τ Outlie ) erivatio of SRH formula ) Alicatio of SRH formula for secial cases 3) irect ad Auger recombiatio 4) Coclusio

32 Case : Low-level Ijectio i -tye R τ τ τ τ i ( + ) + τ ( + ) ( + )( + ) i ( + + ) + τ ( + + ) ( + ) + ( + + ) + τ ( + + ) ( ) ( ) τ + + Lots of holes, few electros > ideedet of holes Case : High-level Ijectio R τ τ τ i ( + ) + τ ( + ) ( + )( + ) i ( + + ) + τ ( + + ) ( + ) + ( + + ) + τ ( + + ) ( τ + τ ) ( τ + τ ) e.g. orgaic solar cells + + Lots of holes, lots of electros > deedet o both relaxatios

33 High/Low Level Ijectio R high ( τ + τ ) R low τ which oe is larger ad why? R τ τ Case 3: Geeratio i eletio Regio eletio regio i P diode: i ( + ) + τ ( + ) i ( ) + τ ( ) EGATIVE Recombiatio > Geeratio << i > geeratio to create, Equilibrium restoratio! 66

34 Outlie ) erivatio of SRH formula ) Alicatio of SRH formula for secial cases 3) irect ad Auger recombiatio 4) Coclusio ( ) R B i Bad-to-bad Recombiatio B is a material roerty irect recombiatio at low-level ijectio ( ) ( )( ) R B + + B i irect geeratio i deletio regio, ( i ) R B B i

35 Auger Recombiatio electro & hole ( i ) ( ) R c + c c,c ~ i 9 6 cm /sec Auger recombiatio at low-level ijectio R ( ) ( ) A c A auger τ auger c A τ Effective Carrier Lifetime Light R RSRH + Rdirect + RAuger t τ eff ( t) ( t ) e + + τ τ τ τ ( c B c ) + + eff T,auger SRH direct Auger ( T,auger ) c + B + c

36 Effective Carrier Lifetime with all Processes τ c eff,auger τ ( c B c ) + + eff T,auger Elec. ev. Lett., (8), 99. 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. irect 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.

ECE606: Solid State Devices Lecture 9 Recombination Processes and Rates

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