6.3. STATIC CHARACTERISTICS: CURRENT - VOLTAGE RELATION 255
Mode of operation EBJ bias CBJ bias
Forward active Forward (VBE> 0 ) Reverse(VCB> 0 )
Cutoff Reverse (VBE< 0 )Reverse(VCB> 0 )
Saturation Forward (VBE> 0 )Forward(VCB< 0 )
Reverse active Reverse (VBE< 0 )Forward(VCB< 0 )Table 6.1: Operation modes of thenpnbipolar transistor. Depending upon the particular appli-
cation, the transistor may operate in one or several modes.
Later we will study the effect of the two widths being different. Using thep-ndiode theory,
we have the following relations for theexcess carrier densities in the various regions are(see our
discussion on carrier decay in chapter 3 and chapter 4):
δpe(xe=0) = excess hole density at the emitter side of the EBJ
= peo[exp (eVBE/kBT)−1] (6.3.1)
δnb(xb=0) = excess electron density on the base side of the EBJ
= nbo[exp (eVBE/kBT)−1] (6.3.2)
δnb(xb=Wbn)=excess electron density at the base side of the CBJ
(collector-base junction)
= nbo[exp (−eVCB/kBT)−1] (6.3.3)
δpc(xc=0) = excess hole density at the collector side of the CBJ
= pco[exp (−eVCB/kBT)−1] (6.3.4)As shown in figure 6.7 in these expressions the subscriptspeo,nbo,andpcorepresent the
minority carrier equilibrium densities in the emitter, base, and collector, respectively. The total
minority carrier concentrationspein the emitter,nbin the base, andpcin the collector are shown
schematically in figure 6.7b. Assuming 100% ionization of the dopants, the majority carrier
densities areneo =Nde,pbo =Nab,andnco =Ndcfor the emitter, base, and collector. We
will assume that the emitter and collector regions are longer than the hole diffusion lengthsLp,
so that the hole densities decrease exponentially away from base regions.
To find the current flow we have to calculate the spatial variation of carrier densities. In
the base region, the excess electron density is given at the edges of the neutral base region by
equation 6.3.2 and equation 6.3.3 To obtain the electron density in the base we must solve the
continuity equation using these two boundary conditions, as discussed in section 3.9. The excess
minority carrier density in the base region is given by
δnb(xb)=nbo
sinh(
Wbn
Lb)
{
sinh(
Wbn−xb
Lb)[
exp(
eVBE
kBT)
− 1
]
+sinh(
xb
Lb)[
exp(
−
eVCB
kBT