3 Analysis of Bipolar Junction Transistor Assumptions q

3. Analysis of Bipolar Junction Transistor Assumptions : q This analysis is only for the active region of the transistor. q The emitter, base and collector regions are each uniformly doped. q The emitter is more heavily doped than the base (always valid). q The width of the neutral part of the base (i. e. , excluding the widths of the space charge regions) is much less than the minority carrier diffusion length (valid for the vast majority of cases). q The transistor is biased in the forward active mode, i. e. , the emitterbase junction is forward biased, the collector-base junction is reverse biased. q The entire bias is applied to the respective p-n junction. The voltage drops in the neutral regions of the transistor are negligibly small. L S Tan - 2011 EE 2004 1

3. 1 BJT current components IEn Emitter(n) IEp IE ICn Base(p) Collector(n) IBa ICp IBb + VBC - - VBE + IC IB IEn is the emitter current component due to injection of electrons from the emitter to the base. IEp is the emitter current component due to injection of holes from the base to the emitter. IBa is the hole current from the base which feeds IEp. IBb is the hole current from the base which supplies the holes for recombination in the base. ICn is the collector current component due to the part of the injected emitter electrons (IEn) which do no recombine in the base and manage to reach the collector. ICp is the hole component of the leakage current due to the reverse biased collector base junction; it is usually negligibly small compared to ICn. L S Tan - 2011 EE 2004 2

3. 2 Excess Minority Carrier Concentrations in the BJT x. E x. B p Base δp. E(x) • Short base, x. B << Ln. n Collector Excess Carrier Concentration EMITTER Excess Carrier Concentration n+ Emitter BASE • Assume short emitter, x. E << Lp, in this example, though it is also possible for emitter to be “long”. COLLECTOR δn. B(x) x -n. B 0 L S Tan - 2011 EE 2004 -p. C 0 • The collector is assumed long. This is, however, immaterial to the operation discussed here. 3

x. B Base Colle n ctor p EMITTER BASE δn. B(0)=n. B 0[exp (e. VBE/k. T)-1] δn. B(x. B)= - n. B 0 δp. E(x. E)=0 x. E Rev Bias – Carrier Extraction COL LEC TOR δp. C(0) = - p. C 0 δp. E(0) =p. E 0[exp (e. VBE/k. T)-1] Excess Carrier C onc entration Fwd Bias – Carrier Injection Ex cess Carrier Concentration n+ Emitter x x. B Excess minority carrier concentrations at SCR edges of an npn BJT in forward active mode. L S Tan - 2011 EE 2004 4

Boundary conditions of excess minority carriers at the edges of the space charge regions (SCRs) : Edge of E-B SCR : From theory of a forward-biased p-n junction, Emitter contact : At the ohmic contact, there are no δp. E(0) excess carriers as they have all recombined at the contact. Ohmic metal contact δp. E(x. E) Excess minority carrier concentration Emitter : Base Emitter x” = 0 x” = x. E x” L S Tan - 2011 EE 2004 5

Base : Edge of E-B SCR : From theory of a edge of C-B SCR : From theory of a reverse-biased p-n junction, Note : The total electron (minority carrier) concentration at the edge of the C-B SCR is zero due to the reverse bias of that junction(see theory of p-n junction). The excess carrier concentration is therefore negative. L S Tan - 2011 EE 2004 Excess minority carrier concentration forward-biased p-n junction, δn. B(0) Emitter x=0 Base δn. B(x. B) Collector x = x. B x 6

Edge of B-C SCR : From theory of a reverse-biased p-n junction, Collector ohmic contact : Excess minority carrier concentration Collector : δp. C(x. C) Ohmic metal contact δp. C(0) Base Collector x’ = 0 x’ L S Tan - 2011 EE 2004 x’ = x. C 7

In the neutral region of the base : • Since the width of the neutral base, x. B, is much less than the minority carrier (electron) diffusion length, Ln, practically no excess minority carriers recombine within the neutral base. • The minority carrier (electron) diffusion current through the neutral base is constant. • The distribution of the excess minority carriers in the base is practically a straight line. Excess minority carrier concentration δn. B(x) • The excess minority carriers (electrons in this case) injected from the emitter into the base diffuse through the neutral region of the base towards the collector. Emitter Base x=0 L S Tan - 2011 EE 2004 Collector x = x. B 8

• In a real bjt, since there is some recombination, the minority carrier (electron) distribution in the base is a slightly exponential curve. • However, since the amount of recombination in the base is very small, because x. B << Ln, the curve is very nearly like a straight line. • For our analysis, we shall use the straight line approximation. • IEn is due to the diffusion of the electron through the base, therefore Excess minority carrier concentration δn. B(x) Calculation of IEn n. B 0[exp (e. VBE/k. T)-1] approximate IEn actual Emitter Base x=0 The negative sign of IEn indicates that it is in the – x direction. L S Tan - 2011 EE 2004 -n. B 0 x Collector x = x. B Note: A is the cross-sectional area of the BJT. 9

Calculation of IEp • In this example, we assume a short emitter, i. e. , x. E << Lp. • IEp is due to the diffusion of the holes through the neutral part of the emitter. p. E 0[exp(e. VBE/k. T-1] IEp because in forward active operation. Note : 1. IEp as expressed above is in the + x” direction, which is opposite to the + x direction. 2. If the emitter is long, x. E >> Lp, then x. E in the above expressions should be replaced by Lp. L S Tan - 2011 EE 2004 Emitter x’’ = x. E x” Excess minority carrier concentration δp. E(x) • The minority carrier (hole) distribution in the emitter is then also a straight line. Base x’’ = 0 x 10

• The base current consists holes and is made up of two components. • IBa is the component that supplies holes for injection into the emitter. Therefore, IBa = IEp • IBb is the component that supplies holes for recombination with the excess electrons in the base. • Base recombination current IEp = excess base minority charge minority carrier lifetime Excess minority carrier concentration δn. B(x) Calculation of IBa and IBb nb 0[exp (e. VBE/k. T)-1] δn. B(x) Emitter Base x=0 = IBa IBb -n. B 0 Collector x = x. B where A is the cross-sectional area of the BJT. L S Tan - 2011 EE 2004 11

• The integral represents the • Since x. B << Ln, the recombination in the base is very small, and the excess minority carrier distribution (dashed line), δn. B(x), can be approximated by a straight line (solid line). • Also, we can make the approximation δn. B(x. B) = 0. • The integral is then given by the area under the straight line approximation of δn. B(x), i. e. , i. Ep Excess minority carrier concentration δn. B(x) area under the curve of δn. B(x) in the base. nb 0[exp (e. VBE/k. T)-1] δn. B(x) Emitter Base x=0 because L S Tan - 2011 EE 2004 i. Ba i. Bb -n. B 0 Collector x = x. B in forward active operation. 12

Calculation of ICn IEn Emitter(n) IEp IE ICn Base(p) Collector(n) IBa ICp IBb + VBC - - VBE + IC IB • ICp, the hole leakage current that flows through the reverse-biased collector-base junction, is very small and can be neglected under forward active condition. • ICn, is made up of the flow of those electrons that diffuse across the base and did not recombine with holes within the base. Therefore, L S Tan - 2011 EE 2004 13

Collector current, because the leakage current ICp is very small. because the recombination current IBb << IEn. Therefore, where The collector current, IC , that flows through the collector-base junction is controlled by the voltage across the emitter-base junction, VBE. ----- TRANSISTOR ACTION Note that we equate the magnitudes of IC, ICn and IEn without considering the sign because they are all defined in the same (- x ) direction. See previous slide. L S Tan - 2011 EE 2004 14

Example 1 Objective: To calculate the collector current given the minority carrier injection in the base. Consider an npn transistor with a uniform cross-sectional area of 0. 1 mm 2 which has an excess electron concentration of 8. 5 x 1015 cm-3 maintained at the emitter-base junction. The collector-base junction is reverse biased. If the width of the neutral base region is 5 μm and the electron mobility is 1500 cm 2 V-1 s-1, estimate the collector current. The doping in the base, NAB = 2. 25 x 1017 cm-3. The minority carrier lifetime in the base, τn = 0. 1 X 10 -6 s. The intrinsic carrier concentration, ni = 1. 5 x 1010 cm-3. L S Tan - 2011 EE 2004 15

δn. B(x) Excess minority carrier concentration Solution to Example 1 8. 5 x 1015 cm-3 approximate i. En actual Emitter Base x=0 L S Tan - 2011 EE 2004 i. C -n. B 0 x Collector x = x. B 16