524
21 Diodes
(a)
(b)
Fig. 21.7 a Potential ϕ and b charge density ρ across the depletion layer of a Schottky n-GaAs diode. Calculation
parameters are s = 12.5 0 , V bi − V ext = 2 V (small reverse bias), N D = 1 × 10 16 cm −3 , T = 300 K. Abrupt
approximation is shown as solid lines, exact (numerical) calculation as dash-dotted line. Dashed line in (a) indicates
depletion layer width w 0 in the abrupt approximation
with the maximum field strength E m = −eN D w 0 // s < 0 at x = 0. From the condition ϕ(w 0 ) = 0 we
obtain w 0 as
w 0 =
2 s
eN D
V bi .
(21.10)
The charge density and the potential in the abrupt approximation are shown in Fig. 21.7 for GaAs
material parameters.
Beyond the abrupt approximation
The thermal distribution of the majority carriers should be treated with a little more care. The dependence of the charge density ρ = e(N
+
D − n) on the potential ϕ (within the Boltzmann approximation)
is (β = e/kT )
ρ(x) = e N D
1 − exp (β ϕ(x))
.
(21.11)
The actual charge density and the potential, obtained from a numerical solution of (21.7), are shown in
Fig. 21.7 in comparison with the abrupt approximation. Clearly, at the Schottky depletion layer width
w 0 , the charge varies continuously and the potential does not drop to zero.
We note that for the depletion layer ϕ ≤ 0 and n ≤ N D . The charge difference ρ (due to the
tail of the thermal distribution of the majority charge carriers in the depletion layer) between the real
distribution (21.11) to the abrupt approximation model with constant charge density (ρ 0 = eN D ) in
the depletion layer is
ρ(x) = ρ(x) − ρ 0 = −e N D exp (β ϕ(x)) .
(21.12)
The integration of ρ over the depletion layer yields that the voltage drop V bi across the depletion
layer needs to be corrected by
=
w 0
0
⎡
⎣
x
0
−
)
s
dx
⎤
⎦ dx =
1
β
1 − exp(−β V bi )
≈ β
−1
.
(21.13)
The approximation is valid for βV bi 1. Therefore, (21.10) is corrected to