21.2 Metal-Semiconductor Contacts
523
(a)
(b)
Fig. 21.6 Schematic band structure of a metal-semiconductor junction that is dominated by surface properties of the
semiconductor. a no contact; due to pinning of the Fermi level at surface states of the semiconductor, a depletion-layer
of width w is already present. b Metal and semiconductor in contact
from the semiconductor into the metal upon contact formation are accommodated in the surface states
and the position of the Fermi level at the semiconductor surface changes only very little. Thus, the
space-charge region is not modified and it is identical to the surface depletion region. The Schottky
barrier height is then given by the band bending at the (bare) semiconductor surface F Bn (Fig. 21.6d)
and does not depend on the metal work function at all (Bardeen model [1695]). For this limiting case
we find for the surface index S = 0.
For actual metal-semiconductor contacts the surface index S takes values between 0 and 1. A
theory involving the semiconductor band structure and midgap (surface) states (MIGS) is needed [695,
1696]. For Si, the experimental result is S = 0.27; the corresponding density of surface states is
D s = 4 × 10
13 cm
−2 eV
−1 .
21.2.2 Space-Charge Region
The width w of the space-charge region is calculated next. First, we make the so-called abrupt approximation. In this approximation (Schottky-Mott model), the charge density ρ in the space-charge region
(0 ≤ x ≤ w) is given by the doping, i.e. ρ = +eN D . Outside the space-charge region the semiconductor is neutral, i.e. ρ = 0 and the electric field is zero, i.e. dϕ/dx = 0. As further boundary conditions
the potential at the interface is ϕ(0) = −V bi < 0. The potential drop in the space-charge region is
determined by the one-dimensional Poisson equation
d
2
ϕ
dx 2 = −
ρ
s
,
(21.7)
where s = r 0 is the static dielectric constant of the semiconductor. Using the ansatz ϕ(x) =
ϕ 0 + ϕ 1 x + ϕ 2 x
2 we find
ϕ(x) = −V bi +
e N D
s
w 0 x −
x
2
2
,
(21.8)
w 0 being the depletion layer width at zero bias.
The spatial dependence E(x) of the electric field strength is
E(x) = −
e N D
s
(w 0 − x) = E m +
e N D
s
x ,
(21.9)
523
(a)
(b)
Fig. 21.6 Schematic band structure of a metal-semiconductor junction that is dominated by surface properties of the
semiconductor. a no contact; due to pinning of the Fermi level at surface states of the semiconductor, a depletion-layer
of width w is already present. b Metal and semiconductor in contact
from the semiconductor into the metal upon contact formation are accommodated in the surface states
and the position of the Fermi level at the semiconductor surface changes only very little. Thus, the
space-charge region is not modified and it is identical to the surface depletion region. The Schottky
barrier height is then given by the band bending at the (bare) semiconductor surface F Bn (Fig. 21.6d)
and does not depend on the metal work function at all (Bardeen model [1695]). For this limiting case
we find for the surface index S = 0.
For actual metal-semiconductor contacts the surface index S takes values between 0 and 1. A
theory involving the semiconductor band structure and midgap (surface) states (MIGS) is needed [695,
1696]. For Si, the experimental result is S = 0.27; the corresponding density of surface states is
D s = 4 × 10
13 cm
−2 eV
−1 .
21.2.2 Space-Charge Region
The width w of the space-charge region is calculated next. First, we make the so-called abrupt approximation. In this approximation (Schottky-Mott model), the charge density ρ in the space-charge region
(0 ≤ x ≤ w) is given by the doping, i.e. ρ = +eN D . Outside the space-charge region the semiconductor is neutral, i.e. ρ = 0 and the electric field is zero, i.e. dϕ/dx = 0. As further boundary conditions
the potential at the interface is ϕ(0) = −V bi < 0. The potential drop in the space-charge region is
determined by the one-dimensional Poisson equation
d
2
ϕ
dx 2 = −
ρ
s
,
(21.7)
where s = r 0 is the static dielectric constant of the semiconductor. Using the ansatz ϕ(x) =
ϕ 0 + ϕ 1 x + ϕ 2 x
2 we find
ϕ(x) = −V bi +
e N D
s
w 0 x −
x
2
2
,
(21.8)
w 0 being the depletion layer width at zero bias.
The spatial dependence E(x) of the electric field strength is
E(x) = −
e N D
s
(w 0 − x) = E m +
e N D
s
x ,
(21.9)