24.4 JFET and MESFET
707
this case, the two-dimensional Poisson equation for the potential distribution V can be used by solving
it along the y direction (channel depth) for all x-positions (adiabatic approximation),
∂
2 V
∂ y 2 = −
ρ(y)
s
.
(24.34)
The geometry is shown in the inset of Fig. 24.15b.
The depth h of the depletion layer in the abrupt approximation is given by (cf. (21.111), reverse
voltages are counted as positive here)
h =
2 s
eN D
(V bi + V G + V (x)) .
(24.35)
Here, we have assumed homogeneous doping, i.e. N D does not depend on y (or x). The built-in voltage
(for a p
+ n gate diode) is given by V bi = β
−1 ln(N D /n i ) (21.101a). The voltage V is the applied source–
drain voltage in relation to the source. The depth of the depletion layer at x = 0 (source) and x = L
(drain) is given by
y 1 = h(0) =
2 s
e N D
(V bi + V G )
(24.36a)
y 2 = h(L) =
2 s
e N D
(V bi + V G + V D ) .
(24.36b)
The maximum value of h is a. Therefore, the pinch-off voltage V P , at which V P = V bi + V G + V D is
such that h = a, is given by
V P =
e N D a
2
2 s
.
(24.37)
The (drift) current density along x is given by (cf. (8.54a))
j x = −e N D μ n E x = e N D μ n
∂V
∂x
(24.38)
for the neutral part of the semiconductor. Therefore, the current in the upper half of the channel is
given by
I D = e N D μ n
∂V (x)
∂x
Z [a − h(x)] ,
(24.39)
where Z is the width of the channel (Fig. 24.14a). Although it seems that I D depends on x, it is of
course constant along the channel due to Kirchhoff’s law.
4 Using the triviality
L
0 I D dx = L I D and
∂V
∂x
=
∂V
∂h
∂h
∂x
with
∂V
∂h
= eN D h/ s from (24.35), we find from (24.39)
I D =
e
2
μ n N
2
D Z a
3
6 s L
3
a 2
y
2
2 − y
2
1
−
2
a 3
y
3
1 − y
3
2
.
(24.40)
4 We neglect recombination, in particular since the current is a majority-carrier current.