α I I R α N I F I F i C i E R Cx R Ex C Cx i B R Bx

Transcription

1 Bipolar transistor, continud Ebrs-Moll modl α I I R α N I F E i E i C C R Ex I F I R R Cx C Cx i B R Bx B E B C Figur 1: Dashd lin indicats th "intrinsic" portion of th dvic, xcluding "parasitic" xtrinsic lmnts. Th modl for th diod blocks can b furthr spcifid to includ th intrnal junction capacitanc.

2 Bipolar transistor, continud Small-signal quivalnt circuit of an abrupt junction HBT C E α B i E α C E i E CC i C C R Ex R E R Cx R B C Cx i B R Bx B Ε V Figur 2: This modl is good for "ballistic" propagation of carrirs across th bas. For diffusiv propagation, th intrinsic portion must b adjustd, A. A. Grinbrg and S. Luryi, IEEE Trans. Elctron Dvics ED-40, pp (1993).

3 Bipolar transistor, continud Elmnts of small-signal analysis All variabls A (t) ar considrd varying harmonically in a small rang about a dc point: A (t) A 0 I (t) I 0 V (t) V 0 + δ A i ω t + δ I i ω t + δ V i ω t Many altrnativ notations,.g., i and v instad of δi and δv. Th δa s ar complx quantitits, may b position-dpndnt filds, δa (x ). Th rlationship btwn diffrnt δa s,.g. btwn δv and δi (gnralizd impdancs or admittancs) dpnd on th chosn dc point. 2 -PORT δ I 1 δ I 2 δv 1 δv 2 common trminal Figur 3: Gnral two-port. Transformrs ar 2-ports ("passiv"). From th small-signal point of viw, transistors ar two-port amplifirs. Admittanc matrix: δi 1 δi 2 y 11 y 12 y 21 y 22 δv 1 δv 2

4 Bipolar transistor, continud Admittanc matrix: δ I 1 δ I 2 y 11 y 12 y 21 y 22 δ V 1 δ V 2 For xampl: y 11 d I 1 d V 1 V 2, input admittanc Impdanc matrix: δ V 1 δ V 2 z 11 z 12 z 21 z 22 δ I 1 δ I 2 For xampl: z 22 d V 2 d I 2 I 1, output impdanc Hybrid matrix (h-paramtrs): δ V 1 δ I 2 h 11 h 12 h 21 h 22 δ I 1 δ V 2 For xampl: h 21 d I 2 d I 1 V 2, forward currnt gain Dfinition of ths paramtrs ssntially involvs spcification of th boundary condition at on or anothr port. Thus h 22 y 22 h 21 is th output admittanc for opn-circuit input port. is th output admittanc for short-circuit input port. is th forward currnt gain for short-circuit output port, tc. Each st of paramtrs ( z-paramtrs, y-paramtrs, h-paramtrs) is complt in th sns that it can b usd to driv th othr sts unambiguously.

5 Bipolar transistor, continud Common trminal configurations δ I 1 δ I 2 δv 1 δv 2 common trminal δ I 1 δ I 2 δv 1 E B C δv 2 common bas δ I 2 δ I 2 δ I 1 C δ V 2 δ I 1 E δ V 2 δ V 1 B δ V 1 B E C common mittr common collctor Figur 4: Diffrnt common-trminal configurations giv paramtrs. Thus, th short-circuit currnt gains ar ris to vry diffrnt h 21 b β h 21 α and hnc h 21 b 1 h 21 b h 21

6 Bipolar transistor, continud Indfinit paramtrs All of th paramtr sts corrsponding to diffrnt configurations (common- bas, common-mittr, common-collctor) ar drivabl from on anothr. A convnint trick (works bst in y-paramtr rprsntation) is to disrgard th common rfrnc and trat th third trminal as an additional port: δ I 1 δ I 2 δv 1 δv 2 δ I 3 δv 3 Figur 5: indfinit matrix: δ I 1 δ I 2 δ I 3 y 11 y 21 y 31 y 12 y 22 y 32 y 13 y 23 y 33 δ V 2 δ V 3 δ V 1 From Kirhhoff s Law and th fact that th matrix should work for arbitrary st of { δ V i } it follows that th sum of all columns (or rows) in th indfinit matrix is zro. Thus, if w assum a short circuit at ports 1 and 2, th fact that th sum of all currnts must b zro implis that th Σ of y -paramtrs in th third column vanishs, and so on. To prov that th Σ must vanish in ach row, w not that if all thr { δ V i } ar qual no ac currnt can flow at any port. It is xcdingly simpl to transform from on common-trminal configuration to anothr. Thus, if w know th y-matrix in common bas configuration, th corrsponding common-mittr matrix is: b y 11 b y 21 y 31 b y 12 b y 22 y 32 y 13 y 23 y 33 y 11 y 21 y 31 y 12 y 22 y 32 y 13 y 23 y 33 y 11 y 21 y 31 y 12 y 11 y 21 y 13 y 12 y 22

7 Bipolar transistor, continud Powr gain dfinitions: Powr gain G is th ratio of powr dlivrd to th load to powr input into th ntwork. It dpnds on both th input and th load circuits. Maximum availabl gain (MAG) is th maximum gain achivabl from a particular transistor without xtrnal fdback. MAG quals th valu of forward gain G which rsults whn both th input and th output ar simultanously matchd in an optimum way. For xampl, ralization of MAG rquirs that th load rsistanc b matchd to th output rsistanc R (z 22 ). Unilatral gain U is th maximum availabl powr gain of a dvic aftr it has bn mad unilatral by adding a losslss rciprocal fdback circuit. This mans that th losslss ntwork around th amplifir (inductancs and capacitancs) is adjustd so as to st th rvrs powr gain to zro. Unilatral gain is indpndnt of common-lad configuration! Th unilatral gain U can b calculatd from any of th following quivalnt xprssions: U z 21 z 12 2 ; 4 [ R (z 11 ) R (z 22 ) R (z 12 ) R (z 21 ) ] y 21 y 12 2 ; 4 [ R (y 11 ) R (y 22 ) R (y 12 ) R (y 21 ) ] h 21 + h 12 2, 4 [ R (h 11 ) R (h 22 ) + Im (h 12 ) Im (h 21 ) ] whr z i j, y ij, and h ij ar th impdanc, th admittanc, paramtrs of a transistor, rspctivly, for any configuration. and th hybrid This rmarkabl rsult (Mason s thorm) is th main rason for th wid-sprad us of U. S S. J. Mason, " Powr gain in fdback amplifirs", IRE Trans. Circuit Thory CT-1, pp (1954).

8 Bipolar transistor, continud Small-signal modl of an abrupt junction HBT C E α B i E α C E i E CC i C C R Ex R E R Cx R B C Cx i B R Bx B Ε V Figur 6: Small-signal analysis of this simpl modl, including frquncy dpndnc of th powr gain in both ballistic and diffusiv rgims, has bn carrid out by Grinbrg and Luryi (1993). A. A. Grinbrg and S. Luryi, " Cohrnt transistor", IEEE Trans. Elctron Dvics ED-40, pp (1993). A. A. Grinbrg and S. Luryi, " Dynamic Early ffct in htrojunction bipolar transistors", IEEE Elctron Dvic Ltt. EDL-14, pp (1993). Quasi-static (Ebrs-Moll-lik) modl of abrupt-junction HBT can b found in A. A. Grinbrg and S. Luryi, " On th thrmionic-diffusion thory of minority transport in htrostructur bipolar transistors", IEEE Trans. Elctron Dvics ED-40, pp (1993).

9 Cohrnt transistor aftr Grinbrg & Luryi, Bas transport factor α ( ω ) Ι c / Ι α i ωτ spirals clockwis with incrasing frquncy In th usual dtrmination of short circuit currnt gain f T dgradation of th magnitud α plays littl (if any) rol V BE For us it is crucial! "cohrnt" bas transport whn α > 0.5 for ωτ > 1 Im α ~ ~ Ballistic HBT V BC (b) collimatd and mononrgtic bam α ( ω ) i ωτ kt 2 kt 3 kt "Cohrnt" transport >> kt Rα

10 Cohrnt transistor aftr Grinbrg & Luryi, 1993 β Currnt Gain [db] ωτ Cohrnt transistor α ( ω ) - i ωτ β α 1 α 1 2 sin ωτ/2 f T Frquncy ωτ ~ 1 ωτ ( f < f ) T n (z) Minority carrir dnsity n (z) i ω ( t - z/v ) λ 2π v / ω Rsonanc whn W m λ B quivalnt to ωτ 2π m Landau dampd 0 distanc z W B

11 Cohrnt transistor aftr Grinbrg & Luryi, 1993 Exampl: partial cohrnc β 2 Common Emittr Currnt Gain [db] db/dc 5 kt β ~ 1 ωτ f T β f X ~ 2 ω τ 40 db/dc 2 2 Extndd frquncy cutoff f X f T τ τ Dimnsionlss Frquncy ωτ Cohrnt transistor is limitd by th disprsion τ rathr by τ itslf

12 Cohrnt transistor aftr Grinbrg & Luryi, 1993 Partial cohrnc bas transit tim is a random variabl with distribution ρ 8 i ωτ α (ω) ρ (τ) d τ 0 charactristic function of ρ ρ τ.g., for a normal distribution α (ω) 1 2 ω 2 τ 2 i _ ωτ _ τ τ for thrmal distr. τ _ ~ kt 2 τ β α max whn Im (α) 0 and R (α) > 0 1 α _ ω τ m ~ 2π m 2 β m ~ ω τ 2 2 m pak frquncis ω τ ~ m 1 xtndd frquncy cutoff

13 Cohrnt transistor aftr Grinbrg & Luryi, 1993 Inclusion of collctor transit β 2 30 α α α B C Currnt gain (db) τ B 0 10 kt τ C 1 ps τ B 2 ps Frquncy (GHz) α B α C φ i sin θ θ ωτ B θ 1 2 ωτ C φ i θ ω W B V B 1 2 W B ω W C W C V Sat 1.0 Im ( α Bα C ) R ( α B α C ) Cohrnt transistor has currnt gain at frquncis whr th transistor which has no bas dlay at all is compltly dampd

14 Cohrnt transistor aftr Grinbrg & Luryi, 1993 What is spcial about bas transit? why is th phas gaind in constant-vlocity collctor transit not as good as that gaind in bas transit? α B α C i sin θ θ φ i θ v Shockly-Ramo thorm (Shockly, 1938). I j v Ej d v I v/d

15 Cohrnt transistor aftr Grinbrg & Luryi, 1993 Powr gain intrinsic limit 50 5 Gain [db] Frquncy (GHz) unilatral gain U U β 2 10 kt 2 α α B C 4 ω 2 C 2 C R B r22 r 22 φ τ B ωτ B 2 ps θ τ C Output Rsistanc r 22 [ k Ω ] ωτ C 1 ps r 22 cos ( φ ) _ cos ( φ + θ ) ωc C θ α B

16 Cohrnt transistor aftr Grinbrg & Luryi, 1993 Cohrnt transistor loadd with parasitics C E α B i E α C E i E CC i C C R Ex R E R Cx R B can charactriz th parasitics by a dlay τ x transistor will b activ at xtndd frquncis, providd i B R Bx B C Cx α sin ( φ + θ 2 B ) + ωτ x < 0 i.., providd transistor is not ovrdampd by th parasitics Exampl: C 0 x U R φ α α B C ω 2 C 2 C (R B + R Bx ) R φ + R X _ cos ( φ ) cos ( φ + θ ) ωc C θ α B τ x R x C C whr R x combination of parasitic rsistancs

17 Cohrnt transistor aftr Grinbrg & Luryi, 1993 Exampl: CT loadd with parasitics 0.5 µ m L E kt W B W C R Ex C E R E R B C C R Bx C Cx Gain (db) U R Cx β 2 R E R B R Bx R Ex R Cx 5 Ω. µ m 25 Ω. µ m 25 Ω. µ m 20 Ω. µ m W B W C 20 Ω. µ m C Cx C C 0.5 f F/ µ m C E µ m 0.1 µ m 1 f F/ µ m f F/ µ m Frquncy (GHz) ρ B Ω. cm T 4.2 K Limitations Ultra-high frquncis, cryognic tmpraturs ω τ x < 1 parasitics impurity scattring cohrnc kt << < E opt ballistics bas cannot b too long ( W B < 0.2 µ m)

18 Cohrnt by othr mans aftr Luryi t al., 1993 Cohrnc by othr mans Rcall: all w nd is slow spiralling in of α nd α ( ω ) Ι c / Ι α i ωτ α > 0.5 ~ for φ ωτ ~ π Gradd-gap HBT V BE E G V BC α φ 2 2r i φ whr r τ diff τ drift E G 2 kt

19 Cohrnt by othr mans aftr Luryi t al., 1993 Cohrnc by Diffusion W B W C α B 1 cosh [2i φ] 1/2 φ << π φ ω τ φ 2 3 i φ to within cubic trms j α j φ 2 j 3 i φ j W j α Π α j φ Σ φ j N φ j i φ 2 3N φ assuming no rturn (larg nough ) j For larg nough N gain phas φ without sacrificing magnitud α > π/2 at φ if N > φ 2 3 ln 2 For φ π nd N > 5

20 Cohrnt by othr mans aftr Luryi t al., 1993 "Stppd up" diffusion Im α W >> λsc (diff) W << λsc (ball) Th fact that th magintud of α dviats from unity quadratically in phas φ is tru for any transport mchanismm >> kt (cohrnt) R α For diffusion: α j φ 2 j 3 i φ Im α j Im α R α φ Σ φ j α Π α j N φ j i φ 2 3N φ R α Essntial condition: no rturn at th stp Stps largr than optical phonon nrgy

21 Cohrnt by othr mans aftr Luryi t al., Currnt Gain β 2 [db] 10 0 N Dimnsionlss Frquncy, 2 ω W /2D f T shifts by a factor of N du to nhancmnt of ffctiv diffusion vlocity Rsonant pak in currnt gain appars only for unralistic valus of N > 19

22 Cohrnt by othr mans aftr Luryi t al., 1993 Exmplary AlGaAs HBT loadd with parasitics 40 T 300 K Gain, db U N 5 h N Frquncy, GHz

23 Cohrnt by othr mans aftr Luryi t al., 1993 Comparison Stp-bas Gradd-gap α φ 2 3N i φ α φ 2 2r i φ whr N < E G E opt whr r τ diff τ drift E G 2 kt For N 5 in AlGaAs nd E G > 180 mv ~ Sam ffct rquirs r 7.5 nd E G 15 kt 380 mv Considr W 2000 A and D 40 cm 2 / Vs Pak in U corrsponding to th φ π rsonanc will occur at th frquncy f 500 GHz f 750 GHz

24 Cohrnt by othr mans aftr Luryi t al., 1993 Summary Anatoly A. Grinbrg & SL IEEE TED 40, pp (August 93) Cohrnt ballistic transistor Natur of gain roll-off with frquncy in collisionlss bas transport It is possibl to supprss Landau damping in a cryognic HBT with abrupt junction Phasshift in cohrnt bas propagation can b usd to obtain activ transistor bhavior abov convntional cutoff frquncis Currnt gain at f >> f T Powr gain at f >> f max Rol of parasitics SL, AAG & Vra B. Gorfinkl APL 63, pp (Sptmbr 13, 1993) Oscillation frquncis up to 1 THz ar prhaps fasibl at low tmpraturs; "slow" opration is not possibl... vry stringnt rquirmnts on th parasitics. Cohrnt drift-diffusion transistor It is possibl to slow down and obtain cohrnt ffcts at room tmpratur in a gradd-gap HBT Stp-bas approach appars prfrabl Rduction of th parasitic bas-collctor capacitanc is ncssary... Nd: top collctor HBT tchnology