Device Physics: The Bipolar Transistor

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1 Monolithic Amplifier Circuits: Device Physics: The Bipolar Transistor Chapter 4 Jón Tómas Guðmundsson 2. Week Fall

2 Introduction In analog design the transistors are not simply switches Second-order effects impact their performance With each generation these effects become more significant The goal is to develop a circuit model for each device by formulating its operation to do this a good understanding of the underlying principles is necessary 2

3 Bipolar Junction Transistors emitter + p type base n type collector p type The bipolar junction transistor (BJT) is a semiconductor device containing three adjoining alternately doped regions The central region is known as base and the outer regions emitter and collector The emitter is much more doped than the collector 3

4 Bipolar Junction Transistors pnp bipolar junction transistor circuit symbol, voltage and current polarities keep the device in the active region 4

5 Bipolar Junction Transistors npn bipolar junction transistor circuit symbol, voltage and current polarities keep the device in the active region 5

6 Bipolar Junction Transistors The pnp transistor is basically two very closely spaced p-n-junctions The base region is 1 µm thick, so the junctions are very closely spaced and interact with each other Thus the transistor is capable of current and voltage gain 6

7 Bipolar Junction Transistors The bipolar junction transistor has four regions of operation The most common region of operation is the active region E-B-junctions are forward biased C-B-junctions are reverse biased gives the largest signal gain almost all linear amplifiers operate in the active region The saturation region is when E-B-junctions are forward biased C-B-junctions are forward biased 7

8 Bipolar Junction Transistors The cutoff region E-B-junctions are reverse biased C-B-junctions are reverse biased represents an off state for the transistor as a switch The inverted region is when E-B-junctions are reverse biased C-B-junctions are forward biased 8

9 Bipolar Junction Transistors The four operating regions of the bipolar junction transistor 9

10 Bipolar Junction Transistors In circuit applications the transistor typically has a common terminal between input and output 10

11 Bipolar Junction Transistors This gives three amplifier types common base (CB) common emitter (CE) common collector (CC) (emitter follower) 11

12 Bipolar Junction Transistors Common emitter is the most popular The common emitter amplifier has output variables v EC and i C input variables v EB and i B if two of the voltages (or currents) are known the third is also known (Kirchhoff s laws) 12

13 Bipolar Junction Transistors The current components within the transistor The holes injected from the emitter to base are I Ep I E = I Ep +I En 13

14 Bipolar Junction Transistors The holes injected from the emitter that reach the collector junction are I Cp The current due to thermally generated electrons near the junction are I Cn I C = I Cp +I Cn I B = I E I C = I B1 +I B2 I B3 14

15 Bipolar Junction Transistors The base transport factor is defined as the ratio of the hole current diffusing into the collector to the hole current injected at the E-B junction α T = I Cp I Ep It is preferred that α T is unity, but due to recombination in base α T is slightly less than unity The emitter injection efficiency γ = I Ep I E = I Ep I En +I Ep and measures the injected hole current compared to the total emitter current 15

16 Bipolar Junction Transistors As the emitter is more heavily doped γ 1 and I En 0 The ratio I C /I E in the active region is defined as the dc alpha α dc = I C I E = I Cp +I Cn I Ep +I En If the E-B junction is forward biased I Cp I Cn and α dc = I Cp I Ep +I En which can be written as α dc = I [ ] [ ] Cp 1 I Ep = α T I Ep 1+I En /I Ep I Ep +I En or α dc = γα T 16

17 Bipolar Junction Transistors Beta is defined as or β dc = I C I B = I C I E I C β dc = I C/I E 1 I C /I E = α dc 1 α dc Note that as α dc 1 then β dc 17

18 Bipolar Junction Transistors The emitter current crossing the E-B depletion region is evaluated as two minority carrier diffusion currents I E = I Ep (0)+I En (0 ) or I E = qad B d p B dx d n E qad E x=0 18 dx x =0

19 Bipolar Junction Transistors The collector current crossing the C-B junction is evaluated as I C = I Cp (W)+I Cn (0 ) or I C = qad B d p B dx d n C + qad C x=w dx 19 x =0

20 Bipolar Junction Transistors For holes in the base region or D B d 2 p B (x) dx 2 d 2 p B (x) dx 2 = p B(x) τ B = p B(x) L 2 B where L B = D B τ B is the minority carrier diffusion length The solution is p B (x) = C 1 exp(x/l B )+C 2 exp( x/l B ) Similar equations can be found for n E (x ) and n C (x ) 20

21 Bipolar Junction Transistors The junction voltage V EB and V CE determine the minority carrier concentrations at the edges of the depletion region give the boundary conditions 21

22 Bipolar Junction Transistors If there is no recombination of holes in the base τ B = then which has solution of the form d 2 p B (x) dx 2 = 0 p B (x) = C 1 x+c 2 The two boundary conditions yield p B (0) = p B0 (exp(qv EB /kt) 1) = C 1 0+C 2 p B (W) = p B0 (exp(qv CB /kt) 1) = C 1 W +C 2 [ ] pb (0) p B (W) p B (x) = p B (0) x W 22

23 Bipolar Junction Transistors Thus the slope of p B (x) is [ ] pb (0) p B (W) W 23

24 Bipolar Junction Transistors Similarly n E (x ) = C 1 exp( x /L E )+C 2 exp(x /L E ) and with the boundary conditions n E (x ) = n E0 (exp(qv EB /kt) 1)exp( x /L E ) Thus the electron current is I En (0 ) = qad E d n E (x ) dx x =0 = qad E L E n E0 (exp(qv EB /kt) 1) 24

25 Bipolar Junction Transistors Simlarly the hole current is I Ep (0) = qad B d p B (x) dx which gives = qad E x=0 W [ p B(0) p B (W)] I Ep (0) = qad B W p B0[(exp(qV EB /kt) 1) (exp(qv CB /kt) 1)] The total emitter current is the sum of the hole and electron currents [ DE n E0 I E = qa + D ] Bp B0 (exp(qv EB /kt) 1) W L E qa D Bp B0 W (exp(qv CB/kT) 1) 25

26 Bipolar Junction Transistors The total collector current is the sum of the hole and electron currents [ DC n C0 I C = qa + D ] Bp B0 (exp(qv CB /kt) 1) W L C The base current is then +qa D Bp B0 W (exp(qv EB/kT) 1) I B = qa D En E0 L E (exp(qv EB /kt) 1)+qA D Cn C0 L C (exp(qv CB /kt) 1) 26

27 Bipolar Junction Transistors Ebers Moll The currents are now given different names and I F = I F0 (exp(qv EB /kt) 1) [ DE n E0 = qa + D ] Bp B0 (exp(qv EB /kt) 1) L E W }{{} I F0 α R I R = α R I R0 (exp(qv CB /kt) 1) = qa D Bp B0 W (exp(qv CB/kT) 1) 27

28 Bipolar Junction Transistors Ebers Moll The emitter current can be written as I E = I F α R I R Similarly the collector current can be split up into and I R = I R0 (exp(qv CB /kt) 1) [ DC n C0 = qa + D ] Bp B0 (exp(qv CB /kt) 1) L C W }{{} I R0 α F I F = α F I F0 (exp(qv EB /kt) 1) = qa D Bp B0 W (exp(qv EB/kT) 1) 28

29 Bipolar Junction Transistors Ebers Moll The emitter current can be written as I E = I F α R I R The collector current is then written as With I C = α F I F I R I B = I E I C = (1 α F )I F +(1 α R )I R These equations are known as the Ebers - Moll equations for an ideal pnp bipolar junction transistor 29

30 Bipolar Junction Transistors Ebers Moll The Ebers - Moll circuit for an ideal pnp bipolar junction transistor The complete set of input-output I V characteristics can be calculated from these equations 30

31 Bipolar Junction Transistors Ebers Moll Note that α F I F0 = α R I R0 = I S If β F = α F 1 α F and β R = α R 1 α R then only three numbers are necessary for the Ebers-Moll equations to be completely specified, β F, β R, and I S (or α F, α R, and I S ) and all other parameters can be calculated The direct connection between the doping densities, base width, lifetime, etc. and the Ebers - Moll equations makes them particulary useful in integrated circuit analysis 31

32 BJT - Small-signal models Analog circuits are often operated with signal levels that are small compared to the bias voltages and currents Small signal models are then derived to calculate the circuit gain and terminal impedances We recall I C = I S exp V BE V T The transconductance is defined as g m = di C dv BE = di Sexp(V BE /V T ) dv BE Note that g m = 38 ms for I C = 1 ma at 25 C. = I S V BE exp V BE V T = I C V BE 32

33 BJT - Small-signal models A change in the base-emitter voltage V BE causes a change in the minority carrier charge in the base Q e 33

34 BJT - Small-signal models The minority carriers are supplied by the base lead so the device has input capacitance (npn) where C b = Q h V BE = τ F g m = τ F I C V T τ F = W2 B 2D n is the base transit time in the forward direction Typical values are τ F = ps for npn τ F = 1 40 ps for pnp 34

35 BJT - Small-signal models In the forward active region the base current is related to the collector current and or I B = d di C β 0 = I C I B = I B = I C β F ( IC [ d β F di C ) I C ( )] 1 IC where β 0 is the small signal current gain of the transistor β F 35

36 BJT - Small-signal models Typical values of β 0 are close to those of β F If β F is constant then β F = β 0 A single value of β is often assumed for a transistor and then used in both ac and dc calculations 36

37 BJT - Small-signal models Small-signal input inpedance of the device r π = V BE I B = V BE I C β 0 = β 0 g m The small-signal input shunt resistance for a bipolar transistor depends on the current gain and is inversly proportional to I C Small changes V CE in V CE lead to changes I C in I C I C = I C V CE V CE or V CE = V A = r 0 I C I C where V A is the Early voltage 37

38 BJT - Small-signal models r 0 is the small-signal output resistance of the transistor r 0 = 1 g m V A V T The large-signal characteristics of the transistor when the Early effect is included is ( )( VBE I C = I S exp 1+ V ) CE V T V T 38

39 BJT - Small-signal models Note that r 0 is inversly proportional to I C and thus r 0 can be related to g m r 0 = 1 where ηg m η = kt qv A So if V A = 100 V then η = at 25 C 39

40 BJT - Small-signal models Combination of the above small-signal elements yields the small-signal model often referred to as the hybrid-π model This is the basic model that arises directly from essential processes in the device 40

41 BJT - Small-signal models Technological limitations in the fabrication of a transistor give rise to a number of parasitic elements that have to be taken into account as well 41

42 BJT - Small-signal models An increase V CE in V CE leads to a decrease I B in I B modelled by Typically I B1 < 0.1I B so r µ = V CE I B1 = V CE I C I C I B1 = r 0 I C I B1 r µ 2β 0 r 0 5β 0 r 0 All pn junctions have voltage dependent capacitance C je is for the base-emitter junction C µ is for the base-collector junction C cs is for the collector-substrate junction 42

43 BJT - Small-signal models The base-collector junction capacitance C µ = C µ0 (1 Vψ0 ) n where V is forward bias and n Furthermore there are resistive parasitics r b is series resistance in base r c is series resistance in collector r ex is series resistance in emitter The capacitance C π contains the base-charging capacitance C b and the emitter-base depletion layer capacitance C je C π = C b +C je 43

44 BJT - Frequency Response The addition of the resistive and capacitive parasitics to the basic small-signal circuit gives the complete small-signal equivalent circuit = Example

45 BJT - Frequency Response The parasitic circuit elements influence the high-frequency gain of the transistor The capability of a transistor to handle high frequency is specified by the frequency where the magnitude of the shor-circuit sommon-emitter gain falls to unity This frequency is the transition frequency f T To determine f T we neglect r ex and r µ and redraw the complete small signal equivalent circuit If r c is assumed small then r o and C cs have no influence 45

46 BJT - Frequency Response Then the small signal voltage v 1 is v 1 = r π 1+r π (C π +C µ )s i i 46

47 BJT - Frequency Response If the current fed forward through C µ is neglected we can write and thus i o i o g m v 1 g m r π 1+r π (C π +C µ )s i i and the high frequency current gain i o i i (jω) β 0 1+ β 0(C π +C µ ) g m jω = β(jω) At high frequencies the imagniary part of the denominator dominates and the equation simplifies to β(jω) g m jω(c π +C µ ) 47

48 BJT - Frequency Response When the gain is unity β(jω) = 1 we find and ω = ω T = f T = 1 2π g m C π +C µ g m C π +C µ 48

49 BJT - Frequency Response The frequency response of a bipolar junction transistor β(jω) 49

50 BJT - Frequency Response The frequency ω β is defined as the frequency where β(jω) = β 0 2 or has fallen 3 db down from the low frequency value That is ω β = 1 β 0 50 g m C π +C µ

51 BJT - Frequency Response A time constant τ T is associated with ω T or = Example 4.2 τ T = 1 ω T τ T = C π +C µ g m = C je +C b +C µ g m = τ F + C je +C µ g m 51

52 Further reading This discussion is based on sections 1.1, 1.4, and 2.6 in Neudeck (1983). It is worth looking at the original paper by Ebers and Moll (1954). A detailed discussion on the small-signal models and frequency response is found in sections 1.3 and 1.4 of Gray et al. (2001). References Ebers, J. J. and J. L. Moll (1954). Large-signal behavior of junction transistors. Proceedings of the IRE 42(12), Gray, P. R., P. J. Hurst, S. H. Lewis, and R. G. Meyer (2001). Analysis and Design of Analog Integrated Circuits (4 ed.). New York: John Wiley & Sons. Neudeck, G. W. (1983). The Bipolar Junction Transistor. Reading, Massachusets: Addison-Wesley. 52

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