ELEC 3908, Physical Electronics, Lecture 19 BJT Base Resistance and Small Signal Modelling
Lecture Outline Lecture 17 derived static (dc) injection model to predict dc currents from terminal voltages This lecture begins by considering the resistance associated with current flow through the base region base resistance Now consider small signal operation and derive equivalent circuits for low and high frequency operation Will also use the high frequency equivalent circuit to define the transit frequency, a common figure of merit for bipolar transistors Page 19-2
Physical Origin of Base Resistance So far BJT analysis has been 1D, i.e. no lateral effects considered More accurate modelling of structure requires consideration of base current flow: From contact, I B must flow through base material to edge of active region Base current flow to emitter occurs in a distributed way along path into active device Page 19-3
Extrinsic Base Resistance The extrinsic (or external) base region is that between the active area and the base contact region, assume width is s B Resistance of this region can be determined from basic equation for resistance from resistivity - if ρ Bx is the resistivity of the extrinsic base region then ρl ρbxsb rbx = = A lw E B Page 19-4
Intrinsic Base Resistance - Geometry The intrinsic (or internal) base region is the volume of the active device area enclosed by the neutral base width and the emitter width and length Modelling of this resistance is more complicated because the current flow is not 1 dimensional - base current enters the side and leaves the top (to be injected into the emitter in forward active operation) Page 19-5
Intrinsic Base Resistance - Modelling Assume linear distribution of i B The lhs voltage due to the distributed voltage drop through the region is b 1 E Veff = Rxixdx b ( ) ( ) E The resistance from the lhs edge to a point x is Bix Rx ( )= ρ Wl The linearly distributed current is 0 B E ix ( )= ib 1 x b E Page 19-6
Intrinsic Base Resistance - Modelling Substituting the expressions for R(x) and i(x) into the V eff expression gives V eff = 1 b E b E 0 ρbix Wl B E i B 1 x b E dx Performing the integral and substituting limits gives the result r bb Veff Bib = = 1 ρ i 6 Wl B The effective intrinsic base resistance is therefore 1/6 of the value obtained if i B was flowing through the region E B E Page 19-7
Example 19.1: Base Resistance Calculation Calculate the intrinsic and extrinsic base resistances for the structure and potentials shown below. Assume the internal and external base dopings are identical, and that the base separation s B is 1.5 μm. The emitter width is 1.0 μm and the length is 10.0 μm. Page 19-8
Example 19.1: Solution Since the intrinsic and extrinsic base regions are identically doped, their resistivities will be the same, given by ρ Bx 1 1 = ρbi = = 19 17 qpμ 16. 10 10 480 p The external base resistance is therefore (note that the neutral base width was calculated in an earlier example) r Bx 4 ρbxsb 013. 15. 10 = = Wl 14. 10 10 10 B E 4 4 And the intrinsic base resistance is therefore r Bi 4 1 ρ BibE 1 013. 1 10 = = 6 Wl 6 14. 10 10 10 B E 4 4 = 013. Ωcm = 139 Ω = 155. Ω Page 19-9
Low Frequency Small Signal Parameters The transconductance g m, the rate of change of collector current with base emitter voltage, is given by g m di dv C BE d dv BE qa D n WB E nb Bo To account for the back injection component of base current in forward active operation, another conductance g π, the rate of change of base current with base-emitter voltage, is used, where e qv BE kt = q kt I C g π di dv B BE d dv BE qa D W p E pe Eo E e qv BE kt q kt I q kt I IB = B = C = 123 { IC g m 1 β F g β m F Page 19-10
Low Frequency Small Signal Equivalent Circuit Now assemble the base resistance and conductance parameters into an equivalent circuit valid for small signal operation Include a controlled current source to model the dependence of collector current variation on base current variation, and an output resistance to model the Early effect Result is a low frequency hybrid-π small signal equivalent circuit Page 19-11
Example 19.2: Conductance Parameter Calculations Calculate the parameters r π and g m for the device of example 19.1. Page 19-12
Example 19.2: Solution Using the value of I C calculated from the injection model equation 1 4 3 g m = 228. 10 = 88. 10 mhos 0026. From the value of g m found above and the forward active current gain already calculated in an earlier example g π g m = = β F 88. 10 204 3 5 = 4. 3 10 mhos = 233. kω Page 19-13
Depletion Capacitance in the BJT Structure Each pn-junction in the bipolar structure has an associated depletion capacitance C depbc is the absolute depletion capacitance (F) of the base collector junction, C depbe is the absolute capacitance (F) of the base-emitter junction Recall from diode discussion (lecture 13) ˆ ˆ ε C ( 0) Si dep Cdep ( VD ) = = W V 1 V V ( ) D D bi Page 19-14
BJT Depletion Capacitance Models Write models for each junction of the form of the original pn-junction model equation C$ ( V ) depbe BE = C$ ( 0) depbe ( 1 V V ) BE bibe z BE C$ ( 0) depbe = ε W BE Si ( 0) C$ ( V ) depbc BC = C$ ( 0) depbc ( 1 V V ) BC bibc z BC C$ ( 0) depbc = ε W BC Si ( 0) The total (absolute) capacitances are then the per unit area terms multiplied by the emitter area C = C$ A C = C$ A depbc depbc E depbe depbe E Page 19-15
Physical Origin of Base Diffusion Capacitance Recall that in general, capacitance is associated with the requirement to source or sink charge when changing a terminal s potential Because the stored charge in the base must be changed if V BE is changed, a capacitance will be created between the base and emitter terminals This is termed the base diffusion capacitance Page 19-16
Base Diffusion Capacitance Model The base diffusion capacitance C π (in F, not a per unit area term) can be modeled in terms of the base transit time τ B, the time taken to cross the neutral base region C The transit time is given in terms of physical parameters for a constant doped base by = g π τ B m τ B = W 2D 2 B nb Page 19-17
High Frequency Small Signal Equivalent Circuit If the capacitances are added to the low frequency equivalent circuit, the high frequency hybrid-π equivalent circuit is obtained Note that the base-emitter capacitances appear inside the base resistances, since they are part of the basic internal transistor structure Page 19-18
Example 19.3: Capacitance Calculations Calculate the high frequency hybrid-π equivalent circuit capacitances for the device in example 19.2. Page 19-19
Example 19.3: Solution The first step in determining the depletion capacitances is to find the zero bias depletion widths. Applying the formula for W at zero bias to each junction (using previously calculated built in potentials) gives W BE 2ε Si 1 1 ( 0) = + VbiBE = 11. 10 q N N AB DE 5 cm W BC 2ε Si 1 1 ( 0) = + VbiBC = 45. 10 q N N The zero bias capacitances are therefore AB DC 5 cm C$ ( 0) depbe ε Si. $ ε Si = = 94 10 Fcm CdepBC( 0) = = 23. 10 W ( 0) W ( 0) BE 8 2 8 2 BC Fcm Page 19-20
Example 19.3: Solution (con t) The capacitances at bias are therefore (note that the grading coefficient is 1/2 for uniformly doped junctions) C$ ( 08. ) = depbe C$ ( 07. ) = depbc 94. 10 ( 1 08. 093. ) 23. 10 ( 1+ 07. 074. ) = 25. 10 = 17. 10 The absolute depletion capacitances are therefore C C depbe depbc 8 05. 8 05. ( ) ( ) 7 8 Fcm 2 Fcm = 25. 10 1 10 10 10 = 25. 10 7 4 4 14 = 17. 10 1 10 10 10 = 17. 10 8 4 4 15 2 F F Page 19-21
Example 19.3: Solution (con t) The base transit time is found from τ B W 2D ( 14. 10 ) 2 4 2 B 10 = = = nb 2349. 28. 10 sec The base diffusion capacitance is therefore found using the transit time and the previously calculated transconductance as C 10 3 12 = g = 28. 10 88. 10 = 25. 10 F π τ B m Page 19-22
Transit Frequency Measurement Configuration One widely used figure of merit in bipolar performance is the transit frequency f T, the frequency at which the current gain with the collector and emitter short circuited becomes unity Page 19-23
Transit Frequency Analysis To determine f T, the high frequency small signal hybrid-p circuit can be used The effect of the collector-emitter short circuit is to short the output conductance r o The phasor collector current will be given by (note point a is ground) ( ω ) I = g V jωc V = V g j C C m b e depbc b e b e m depbc Page 19-24
Transit Frequency Analysis (con t) The base current phasor will be given by I ( g jωc jωc ) = V + + B b e π π depbc The ratio of collector current phasor to base current phasor is therefore I I C B = gm jωcdepbc 1 g + jω C + C + C jω C C g τ π ( + + ) ( ) ( ) π depbe depbc depbe depbc m B Page 19-25
Transit Frequency Analysis (con t) The transit frequency is then defined by I I C 1 1 = 2πf C C g T (( + ) + τ ) B f = f T depbe depbc m B And hence f T is given in terms of physical parameters as f T = 2π 1 (( C C ) g ) depbc + depbe m + τb Faster transit time, reduced capacitance and increased transconductance all improve the transit frequency Page 19-26
Example 19.4: Transit Frequency Calculation Calculate the transit frequency for the device of example 19.3. Page 19-27
Example 19.4: Solution The required values have all been calculated previously, so the result is obtained by substituting the values f T = 1 ( ) ( 14 15 3 10 + + ) 2π 25. 10 17. 10 88. 10 28. 10 8 = 56. 10 Hz This example transistor therefore has a transit frequency of 0.56 GHz. A modern commercial bipolar process would offer devices with an f T of 25-35 GHz. Page 19-28
Lecture Summary Current flow in bipolar structure experiences external and internal base resistance mechanisms are very different Low frequency small signal equivalent circuit contains r bb and r bx (internal/external base resistances), g π (dependence of back injection current on V BE), g m ( transconductance) and r o the output resistance (Early effect) High frequency small signal equivalent circuit contains the same elements plus C depbc, C depbe (depletion capacitances) and C π (diffusion capacitance) The transit frequency f T incorporates many of the important high frequency parameters, and so is a useful figure of merit of high frequency performance Page 19-29