BJT - Mode of Operations

Transcription

1 JT - Mode of Operations JTs can be modeled by two back-to-back diodes. N+ P N- N+ JTs are operated in four modes. HO #6: LN JT M Models Page 1

2 1) Forward active / normal junction forward biased junction reversed biased here, β F 2) Reverse active, Mode of Operations where β F forward gain + V nverse aturation ut-off Normal junction reverse biased and junction forward biased here, reverse gain, β R / 1 3) aturation region and are forward biased 4) ut-off region and are reversed biased +V HO #6: LN JT M Models Page 2

3 asic JT Model asic JT model can be derived considering two back-to-back diodes as npn-jt. When: 1) - junction is forward biased: forward current, F flows through - diode. F a R R n p α F F R n α F F flows in the collector here, α F forward gain / if V is +ve 2) - junction is reversed biased: reverse current, R flows through - diode. α R R flows in the emitter here, α R reverse gain / if V is +ve HO #6: LN JT M Models Page 3

4 M1 JT Model: njection Version The basic bers-moll model (M1): α R R α F F O O F Terminal currents: O F + α R R (1) α F F R (2) F α R R α F F + R (1 α F ) F + (1 α R ) R (1 α F ) F + (1 α R ) R (3) R HO #6: LN JT M Models Page 4

5 M1 JT Model: njection Version We know from current flow analysis: Here F R n n e e qv qv saturation current; V - voltage - saturation current; V - voltage The terminal currents from (1), (2), (4), (5): α F e e qv qv (4) (5) qv 1 + α R e 1 (6) qv 1 e 1 (7) HO #6: LN JT M Models Page 5

6 M1 JT Model: njection Version From reciprocity property: α F α R Therefore, α Again, F e e qv qv α e e qv 1 1 β F forward current gain α F /(1 α F ) β R reverse current gain α R /(1 α R ) R qv (8) (9) ince f(n i 2 ) f(t) T T T ) T k g 1 1 T T ref ( ) ( ref e (10) ref 3 HO #6: LN JT M Models Page 6

7 M1 JT Model: njection Version Model equations: Where ( T ) e α F ( T ) e qv α F β F /(1 + β F ) α R β R /(1 + β R ) qv 1 + ( T ) e ( T ) 1 e α 3 R qv qv 1 1 g T k T Tref ( T ) ( Tref ) e (11) Tref Total five model parameters: β F, β R,, T ref, and g can be used to describe basic JT device characteristics without parasitics. 1 1 (8') (9') HO #6: LN JT M Models Page 7

8 M1 JT Model: Transport Version Model equations (8) and (9) can be written as: 1 + α F 1 + α R Where the reference source currents: e e qv qv 1 1 O α F α R /α F (12) (13) O HO #6: LN JT M Models Page 8 O /α R

9 M1 JT Model: Nonlinear Hybrid-π From transform model (12) and (13), we get: 1 1 ( ) T α F β (14) F 1 ( ) 1 T α (15) R β R Where the reference source current is: qv qv T e 1 e 1 The diode currents are: qv e 1 β F β F qv e 1 β β R R (16) (17) (18) HO #6: LN JT M Models Page 9

10 M1 JT Model: Nonlinear Hybrid-π The model: T - O O /β F The terminal currents: O /β R T β F T β R + β F β R (19) (20) (21) HO #6: LN JT M Models Page 10

11 M1 JT mall-ignal Model: Linear Hybrid-π T - g m v O /β F O /β R Non-linear hybrid-π Transconductance : nput resistance : Output resistance : i v i v i v For M1 basic JT model, r r g π µ m r µ Q ( V Q O, 1 V A ) O β F g i q m open circuit r π v + + v O i r µ Linear hybrid-π i O HO #6: LN JT M Models Page 11

12 M1 JT Model: Deficiencies M1 JT model: advantages dc model fewest model parameters - β F, β R,, T ref, g can predict dc characteristics for most applications deficiencies no ohmic bulk resistors to the terminals (r' e, r' b, r' c ) no charge storage in the devices ( D, D, j, j, sub ). r' e r' b r' c j j HO #6: LN JT M Models Page 12

13 M2 Model to mprove D haracterization Addition of three ohmic resistors to the basic M1 model. N P N /β R /β F T npn-jt M1 model nclude 3-ohmic resistors to M1 r' b ' ' r' c M1 model ' r' e HO #6: LN JT M Models Page 13

14 ffect of Ohmic ulk Resistors - r' c ollector series resistance, r' c : typical value ~ 200 ohm decreases the slope of vs. V characteristics in the saturation region of operation improves modeling of dc device characteristics. M1 ffect of r' c V HO #6: LN JT M Models Page 14

15 ffect of Ohmic ulk Resistors - r' e and r' b mitter series resistance, r' e : typical value 5 ohm due to polysilicon contact reduces junction potential by a factor of r' e V r' e ( + )r' e (1 + β F )r' e r' e is equivalent to a base resistance of (1 + β F )r' e effects and of the device. ase series resistance, r' b : effects small-signal and transient response difficult to measure accurately due to the dependence on r' e and operating point. ln(, ) V r' b + r' e V HO #6: LN JT M Models Page 15

16 M2 Model: Modeling harge torage ffect Addition of capacitors in M1 model to account for charge storage effects in JTs: r' c sub D r' b ' D j ' M1 model j ' D -junction diffusion capacitance. D -junction diffusion capacitance. j -junction capacitors. r' e j -junction capacitors. sub -substrate junction capacitance. HO #6: LN JT M Models Page 16

17 omplete M2 JT Model The complete model to account for terminal resistors and charge storage effects in JTs. D r' b D ' r' c j j ' /β R T /β F sub Where β F β F β β R R e e qv qv 1 1 r' e ' Here V V internal voltage internal voltage HO #6: LN JT M Models Page 17

18 M2 Model: Junction apacitors j and j model the incremental fixed charges stored in the and space charge layers of JT due to applied bias V '' and V '', respectively. From PN-junction theory (HO #3, page 19) we get: j0 j ( V ) m (22) V 1 φ j0 -junction capacitance at V '' 0 φ built-in potential (/q)ln(n A N D /n i2 ) j 0 j ( V ) m (23) V 1 φ j0 -junction capacitance at V '' 0 φ built-in potential (/q)ln(n A N D /n i2 ) HO #6: LN JT M Models Page 18

19 M2 Model: Diffusion harge, Q D Q Q n p (0) Q Q n p o W Total minority carrier charge due to forward current : Q D Q + Q + Q + Q (τ + τ + τ + τ ) where τ F τ F total forward delay time consisting of emitter delay τ, space charge layer transit time τ, base transit time τ, and space charge layer transit time τ. HO #6: LN JT M Models Page 19

20 ase Transit Time, τ n the absence of ε-fields in the base (N A constant, low level injection), then the injected e- concentration varies linearly across the base. n p Therefore, the total e- charge in the base is simply given by: Q (1/2)qn p W A (24) where A emitter area ~ 0 The transit time across the base is: τ Q / (25) ince 0, qa D n (dn/dx) qa D n n p /W where D n average e- diffusivity in the base region. HO #6: LN JT M Models Page 20

21 ase Transit Time ubstituting for Q and in (25), we get, τ 2 2 W D n (26) xample: f W 1 µm and the base is lightly doped so that D n 38 cm 2 /sec, then from (26) τ 132 psec. f the base doping is graded (typically in transistors), an aiding ε-field speeds up the carriers and τ is reduced by at least 2 times. Also, under high level injection, to maintain base neutrality, the hole concentration in the base and has a gradient similar to the e- gradient. This sets up an ε-field which also speeds up the e-. Usually, τ is not the dominant frequency limitation in modern JTs. HO #6: LN JT M Models Page 21

22 M2 Model: Diffusion harge, Q D Q R Q R QR Q W Total minority carrier charge due to reverse current : Q D Q + Q R + Q R + Q R (τ + τ + τ R + τ ) where τ R τ R total reverse delay time consisting of collector delay τ, reverse space charge layer transit time τ, reverse base transit time τ R, and reverse -space charge layer transit time τ. HO #6: LN JT M Models Page 22

23 M2 Model: Diffusion apacitors D and D The diffusion charges are modeled by two non-linear capacitors D and D given by: D D ( V ) ( V ) Q V Q V D D τ F V τ R V (27) (28) D small signal dq dv D V 0 d dv V 0 ( τ ) F g mf τ F (29) D small signal dq dv D V 0 d dv V 0 ( τ R ) gmrτ R (30) The model parameters: τ F and τ R. HO #6: LN JT M Models Page 23

24 M2 Model: Model Parameters ubstrate apacitance ( sub ) is considered as a constant in M3 model. the complete set of model parameters is given by: five M1/D - three bulk ohmic resistors - β F, β R,, T ref, g r' c, r' e, r' b three -junction capacitors - three -junction capacitors - two minority-carrier delay time - one -substrate capacitor - j0, φ, m j0, φ, m τ F,τ R sub HO #6: LN JT M Models Page 24

25 M2 Model: Discussions Advantages of M2 model is the improvement in D and A device characterization over M1 model by addition of: bulk ohmic resistors charge storage in the devices The limitations of M2 model are: base-width modulation variation of β with current level distributed collector capacitance variation of device parameters with temperature high current effect on τ F. New model parameters are added to M2 JT model to improve device characterization M3 JT model. HO #6: LN JT M Models Page 25

26 M3 Model: ase Width Modulation N+ x P x N- deb dbc W(V ) x 0 x W ase-width modulation describes the change in the quasi-neutral base-region W due to a change in the -junction voltage, V. n the normal active mode of JT operation: -junction is forward biased -junction is reversed biased depletion width, X d f(v ) base-width, W changes significantly with V. HO #6: LN JT M Models Page 26

27 ase Width Modulation by V arly ffect As the reverse bias across the base-collector junction increases: -junction depletion-layer width increases W decreases injected minority carrier gradient, (dn/dx) in W as V. ase-width modulation is modeled by a parameter, V A forward arly voltage. ffect of base-width modulation: 5 > 4 > 3 > 2 > f(v, V A ) 3 β F f(v, V A ) τ F f(v, V A ) V A V HO #6: LN JT M Models Page 27

28 Let us assume ase Width Modulation uniformly doped base-region linear region of operation W f(v ) Using Taylor s series expansion, we can show that W ( V ) W (0) + V dw dv V + W ( V ) W (0) 1 W (0) We define: V A dv d V dw dv 0 V ( Here W (0) W ( V 0) ) 0 (31) ( V + V ) (32) d d d dv HO #6: LN JT M Models Page 28

29 ase Width Modulation Now, d qa Dn n N AW ( V 2 i ) e qv V dw W (0) d 0 W (0) dw V 0 (33) From (32) and (33): V A dv dw V W (0) (34) ombining (31) and (34) we get: V + W ( V ) W (0) 1 (35) V A HO #6: LN JT M Models Page 29

30 ase Width Modulation - Model parameters 2 qa n ni (0) D N AW ( V 0) W (0) ( V ) (0) W ( V ) V ( V ) (0) 1 VA imilarly, we can show that: β ( V F τ ( V ) ) V β (0) 1 V V τ (0) 1 + V A A (0) V 1+ V (36) (37) (38) HO #6: LN JT M Models Page 30 2 A

31 ase Width Modulation The base-width modulation model parameter, V A : does not change the equivalent circuits changes the terminal current equations over M1/M2 models. The expression for current source: T (0) qv qv e 1 e 1 V 1 + VA (0) qv (0) e qv 1 + e 1 β (0) β (0) F R V A is measured from the slope of vs. V characteristic in the active region of JT operation. slope (d /dv ) V constant (d /dv ) V constant (39) (40) HO #6: LN JT M Models Page 31

32 M3 Model: β dc Variation with urrent xperimental β dc vs. data can be divided into three regions: β F β FM Region Region Region V 0 log( ) β dc vs. plot does not clearly show the dependence of: β dc on various component of in Region- β dc on various factors causing degradation in in Region-. vs. V and vs. V plots are required to explain the observed β dc vs. characteristics. HO #6: LN JT M Models Page 32

33 β dc Variation with urrent ln(, ) Region- β F For the simplicity of analysis, we assume: V 0 (neglect the effect of base-width modulation, V A ) V 0 neglect ohmic-bulk resistors (r' e, r' b, r' c ) ln β ln( ) FM Region- Region- V V V '' V V '' normal active mode i.e., forward current gain analysis only. HO #6: LN JT M Models Page 33

34 β dc Variation with urrent Region- β F β FM (maximum gain) constant M1 model V 0, we get Region- (0) e (0) e β F qv qv 1 1 (41) (42) β dc as due to its additional components from carrier recombination at the surface, (surface) carrier recombination in the space-charge layer, (-scl) surface channels, (channel). ln β ln(, ) ln( ) FM Region- β F Region- V 0 Region- V HO #6: LN JT M Models Page 34

35 β dc Variation with urrent: Low Region Thus, (total) (total) (surface) + (0) 2 e q n V ( scl) + 1 (channel) Ι in (43) can be represented by an additional non-ideal diode in the basic M1 model with diode current given by: (0) q 2 e n V 1 Here we have two new parameters for modeling low region: n low-current forward emission coefficient (~ 2) 2 models various components of in low (~ 10 3 ). (0) qv e q 1 + (0) 1 2 e n V β FM (43) (44) HO #6: LN JT M Models Page 35

36 β dc Variation with urrent: Low Region f V > 0 in the inverse region of JT operation, two additional parameters can be used to model the components of : n low-current inverse emission coefficient (~ 2) 4 components of in in the inverse region (~ 10 3 ). Ι in the inverse region is represented by an additional non-ideal -diode in the basic M1 model with diode current given by: q ( inverse - region) 4 (0) e n V 1 Therefore, the general expression for is given by: (0) e β FM (0) + e β RM qv qv (0) e (0) e q n V qn V c 1 1 (45) HO #6: LN JT M Models Page 36

37 β dc Versus : Modeling Low Region Non-linear hybrid JT model to model β vs. at low current level. (0) q n V 1 4 e r' c ' sub D j /β R r' b ' T D j /β F (0) q n V 1 2 e r' e ' HO #6: LN JT M Models Page 37

38 β dc Variation with urrent: High Region Region- at high level injection, the minority carrier concentration into base region is significant with respect to majority carrier concentration. from the condition of quasi-neutrality in the base: ρ(base) q[n A (x) + n(x)] under these conditions it can be shown that: the high-level injection is modeled with an additional model parameter θ in M3 model. for high level V 0 is given by: (high level) e qv 2 (0) qv e qv θ 2 e (46) HO #6: LN JT M Models Page 38

39 β dc Variation with urrent: Model Parameters ln ln( 2 (0)) ln() (0) θ slope 1 ln( (0)) (0) ln β FM slope 1/2 β F slope 1 slope 1/n V 0 M3-Model parameters to describe β F - data: qv / Region- 2 n Region- β FM Region- θ The parameter set { 2, n, β FM, θ} is obtained from above plot. n the inverse region a set { 4, n, β RM, θ R } is defined at V 0. HO #6: LN JT M Models Page 39

40 β dc Versus : ffect of Ohmic Resistors The series resistances (r' e, r' b, r' c ): do not effect theoretical analysis and model equations effect measured device characteristics. To account for the ohmic resistors in the model equations, we change: measured V with the internal V '' given by V '' V + ( r' b + r' e ) measured V with the internal V '' given by V '' V ( r' c + r' b ) HO #6: LN JT M Models Page 40

41 M3 Model: mproved harge torage Model mproved j - r' b model: j is distributed over r b ' split up on either side of r b ' additional model parameter RATO is used to model distributed j 0 < RATO < 1. j r' b ' r' c ' Rest of M3 Model (1 RATO) j (V '' ) r' c ' (RATO) j (V '' ) r' b ' Rest of M3 Model HO #6: LN JT M Models Page 41

42 M3 Model: Variation of τ F with At high level injection, τ F as due to an increase in τ. At high current level, the increase in τ is due to: a reduction in the low-level aiding field effect in drift transistors an effective base-widening effect (Kirk-effect) two-dimensional spreading effect. n the normal active region, it can be shown that: τ 2 2 dqd 1 L F ( ) τ FL (0) 1 + 1, for ac 0 d 4 W 0 where L emitter width W base width 0 at which τ F starts to increase. (47) HO #6: LN JT M Models Page 42

43 M3 Model: Variation of τ F with Two additional model parameters: 0 (L /W) The model parameters are obtained by curve fitting τ F ac vs. plots to q. (47). τ F ac τ FL (0) 0 HO #6: LN JT M Models Page 43

44 M3 Model: Temperature Dependence Physics-based temperature variation: inherent temperature dependence for parameters τ F, j, and j. τ ( T ) τ F Analytical temperature variation: additional temperature coefficients for parameters β F, r' b, r' c. Where FL ( T ref W ( T ) ) ( ref ) W T Par target parameter T 1 first-order temperature coefficient T 2 second-order temperature coefficient. 2 T T ref 1.5 ε { 1+ ( T T ) m( γ γ φ )} (48) jx ( T ) jx ( Tref ) ref 2 T T (49) Par( T ) Par( T { 1+ T ( T T ) + T ( T T 2 } ref ) 1 ref 2 ref ) (50) HO #6: LN JT M Models Page 44

45 Additional model parameters base-width modulation: V A β roll-off: M3 Model: ummary { 2, n, β FM, θ} (inverse region: { 4, n, β RM, θ R }) distributed j : RATO, 0, L /W temperature coefficients for β F, r' b, r' c : T 1 T 2. HO #6: LN JT M Models Page 45

46 Home Work 3: Due April 21, ) An npn-jt is used as an open-collector diode as shown in Figure below: (a) use the injection version of M1 JT model to derive an expression for as a function of V. (b) use q derived in part (a) to calculate V for 1 ma. Given that: α F 0.98 α R A T 300 K + V 2) The npn-jt in Problem #1 is used as a shorted base-collector diode as shown in Figure below: (a) use the injection version of M1 model to derive an expression for as a function of V (b) use q derived in part (a) to calculate V if -1 ma. + V HO #6: LN JT M Models Page 46

47 Home Work 3: Due April 21, ) The M1 model discussed in class can be simplified for different modes of operation. As an example, in the forward active mode, R is negligibly small and therefore, the model can be simplified for hand calculation. Use the injection version of M1 model to: (a) show the simplified model in the normal active mode of an npn-jt operation. (b) show the simplified model in the saturation mode of an npn-jt operation. learly state any assumptions you make. 4) Following the procedure discussed in class, derive the models for lateral pnp-jts: (a) sketch the basic M1 model (b) write q for terminal currents (c) include bulk-ohmic resistors and charge storage elements in M1 model to generate lateral pnp-jt M2 model. 5) Develop and sketch small-signal linear hybrid-π M2 model for npn-jts. Derive the model equations. HO #6: LN JT M Models Page 47