694
24 Transistors
j tot = j n + j p ,
(24.1c)
where p B is the equilibrium minority carrier density in the base. From the discussion of the pndiode, we know that at the boundary of the depletion layer the minority carrier density is increased
by exp(eV /kT ) (cf. (21.102a,b)). At the boundaries of the emitter–base diode (for geometry see
Fig. 24.5a)
δ p(0) = p(0) − p B = p B
exp (βV EB ) − 1
(24.2a)
δn(−x E ) = n(−x E ) − n E = n E
exp (βV EB ) − 1
,
(24.2b)
where n E and p B are the equilibrium minority-carrier densities in the emitter and base, respectively.
Accordingly, at the boundaries of the base–collector diode we have
δ p(w) = p(w) − p B = p B
exp (βV CB ) − 1
(24.3a)
δn(x C ) = n(x C ) − n C = n C
exp (βV CB ) − 1
.
(24.3b)
These are the boundary conditions for the diffusion equations in the p-doped layers and in the neutral
region of the n-doped base. For the p-layers (with infinitely long contacts), the solution is (similar to
(21.128)) for x < −x E and x > −x C , respectively
n(x) = n E + δn(−x E ) exp
x + x E
L E
(24.4a)
n(x) = n C + δn(x C ) exp
−
x − x C
L C
.
(24.4b)
L E and L C are the minority carrier (electron) diffusion lengths in the emitter and collector, respectively.
The solution for the hole density in the neutral region in the base (0 < x < w) is
p(x) = p B +
δ p(w) − δ p(0) exp (−w/L B )
2 sinh (w/L B )
exp
x
L B
(24.5)
−
δ p(w) − δ p(0) exp (w/L B )
2 sinh (w/L B )
exp
−
x
L B
.
We shall denote the excess hole density at x = 0 and x = w as δ p E = δ p(0) and δ p C = δ p(w),
respectively. Typical (‘normal’) operation condition in the common base circuit is that δ p C = 0
(Fig. 24.8a). In the ‘inverted’ configuration, the role of emitter and collector are reversed and δ p E = 0.
We can write (24.5) also as
p(x) = p B + δ p E
sinh[(w − x)/L B ]
sinh [w/L B ]
+ δ p C
sinh [/L B )]
sinh [w/L B ]
.
(24.6)
If the base is thick, i.e. w → ∞, or at least large compared to the diffusion length (w/L B 1), the
carrier concentration is given by
p(x) = p B + δ p(0) exp
−
x
L B
(24.7)
and does not depend on the collector. In this case there is no transistor effect. A ‘coupling’ between
emitter and collector currents that are given by the derivative ∂ p/∂x at 0 and w, respectively, is only
present for a sufficiently thin base.
24 Transistors
j tot = j n + j p ,
(24.1c)
where p B is the equilibrium minority carrier density in the base. From the discussion of the pndiode, we know that at the boundary of the depletion layer the minority carrier density is increased
by exp(eV /kT ) (cf. (21.102a,b)). At the boundaries of the emitter–base diode (for geometry see
Fig. 24.5a)
δ p(0) = p(0) − p B = p B
exp (βV EB ) − 1
(24.2a)
δn(−x E ) = n(−x E ) − n E = n E
exp (βV EB ) − 1
,
(24.2b)
where n E and p B are the equilibrium minority-carrier densities in the emitter and base, respectively.
Accordingly, at the boundaries of the base–collector diode we have
δ p(w) = p(w) − p B = p B
exp (βV CB ) − 1
(24.3a)
δn(x C ) = n(x C ) − n C = n C
exp (βV CB ) − 1
.
(24.3b)
These are the boundary conditions for the diffusion equations in the p-doped layers and in the neutral
region of the n-doped base. For the p-layers (with infinitely long contacts), the solution is (similar to
(21.128)) for x < −x E and x > −x C , respectively
n(x) = n E + δn(−x E ) exp
x + x E
L E
(24.4a)
n(x) = n C + δn(x C ) exp
−
x − x C
L C
.
(24.4b)
L E and L C are the minority carrier (electron) diffusion lengths in the emitter and collector, respectively.
The solution for the hole density in the neutral region in the base (0 < x < w) is
p(x) = p B +
δ p(w) − δ p(0) exp (−w/L B )
2 sinh (w/L B )
exp
x
L B
(24.5)
−
δ p(w) − δ p(0) exp (w/L B )
2 sinh (w/L B )
exp
−
x
L B
.
We shall denote the excess hole density at x = 0 and x = w as δ p E = δ p(0) and δ p C = δ p(w),
respectively. Typical (‘normal’) operation condition in the common base circuit is that δ p C = 0
(Fig. 24.8a). In the ‘inverted’ configuration, the role of emitter and collector are reversed and δ p E = 0.
We can write (24.5) also as
p(x) = p B + δ p E
sinh[(w − x)/L B ]
sinh [w/L B ]
+ δ p C
sinh [/L B )]
sinh [w/L B ]
.
(24.6)
If the base is thick, i.e. w → ∞, or at least large compared to the diffusion length (w/L B 1), the
carrier concentration is given by
p(x) = p B + δ p(0) exp
−
x
L B
(24.7)
and does not depend on the collector. In this case there is no transistor effect. A ‘coupling’ between
emitter and collector currents that are given by the derivative ∂ p/∂x at 0 and w, respectively, is only
present for a sufficiently thin base.