21.4 Bipolar Diodes
555
-w/2
w/2
x
x
(a)
-x p
x n
-x p
x n
x
x
N B
(b)
(c)
(d)
-w/2
w/2
N A -N D
N A -N D
Fig. 21.40 Description of doping distribution with (a, b) abrupt approximation and with (c, d) linearly graded junction.
(a, c) show real impurity concentration, (b, d) idealized doping profile
junctions that are fabricated by diffusion, the abrupt approximation is suitable for alloyed, ion-implanted
and shallow-diffused junctions. For deep-diffused junctions a linearly graded approximation is better
(Fig. 21.40), which is treated in more detail in [574]. If one doping level is much higher that the other,
the junction is termed a one-sided (abrupt) junction. If n p ( p n), the junction is denoted as an
n
+ p-diode (p
+ n-diode).
The thermodynamical equilibrium of a pn diode is considered here only for the electronic system.
The thermodynamic stability of the atomic doping distribution is discussed in [1761]. Typically, thermodynamics works to randomize the chemical concentration gradient; the existence of a pn junction
is due to the extremely low diffusion coefficient of dopants in the semiconductor lattice. Elevated
temperatures can cause the destruction of the pn diode via enhanced dopant diffusion (Sect. 4.2.3).
However, a thermodynamically stable concentration gradient and thus a built-in field can exist in a
multi-component system [1761].
21.4.2 Space-Charge Region
In thermodynamical equilibrium the Fermi level is constant (∇ E F = 0). The built-in (positive) voltage
V bi is given by (see Fig. 21.41c)
e V bi = E g + e V n + e V p ,
(21.100)
where V n is the difference between conduction band and Fermi level on the n-side, −eV n = E C − E F .
V p is the difference between valence band and Fermi level on the p-side, −eV p = E F − E V . For the
nondegenerate semiconductor V n , V p < 0 and (using (7.12), (7.10) and (7.11))
e V bi = kT ln
N C N V
n
2
i
−
kT ln
N C
n n 0
+ kT ln
N V
p p 0
= kT ln
p p 0 n n 0
n
2
i
= kT ln
p p 0
p n 0
= kT ln
n n 0
n p 0
(21.101a)
∼ = kT ln
N A N D
n
2
i
.
(21.101b)
The electron and hole densities on either side of the junction (n p 0 and p p 0 at x = −x p and n n 0 and p n 0
at x = x n ) are related to each other by (from rewriting (21.101a))
555
-w/2
w/2
x
x
(a)
-x p
x n
-x p
x n
x
x
N B
(b)
(c)
(d)
-w/2
w/2
N A -N D
N A -N D
Fig. 21.40 Description of doping distribution with (a, b) abrupt approximation and with (c, d) linearly graded junction.
(a, c) show real impurity concentration, (b, d) idealized doping profile
junctions that are fabricated by diffusion, the abrupt approximation is suitable for alloyed, ion-implanted
and shallow-diffused junctions. For deep-diffused junctions a linearly graded approximation is better
(Fig. 21.40), which is treated in more detail in [574]. If one doping level is much higher that the other,
the junction is termed a one-sided (abrupt) junction. If n p ( p n), the junction is denoted as an
n
+ p-diode (p
+ n-diode).
The thermodynamical equilibrium of a pn diode is considered here only for the electronic system.
The thermodynamic stability of the atomic doping distribution is discussed in [1761]. Typically, thermodynamics works to randomize the chemical concentration gradient; the existence of a pn junction
is due to the extremely low diffusion coefficient of dopants in the semiconductor lattice. Elevated
temperatures can cause the destruction of the pn diode via enhanced dopant diffusion (Sect. 4.2.3).
However, a thermodynamically stable concentration gradient and thus a built-in field can exist in a
multi-component system [1761].
21.4.2 Space-Charge Region
In thermodynamical equilibrium the Fermi level is constant (∇ E F = 0). The built-in (positive) voltage
V bi is given by (see Fig. 21.41c)
e V bi = E g + e V n + e V p ,
(21.100)
where V n is the difference between conduction band and Fermi level on the n-side, −eV n = E C − E F .
V p is the difference between valence band and Fermi level on the p-side, −eV p = E F − E V . For the
nondegenerate semiconductor V n , V p < 0 and (using (7.12), (7.10) and (7.11))
e V bi = kT ln
N C N V
n
2
i
−
kT ln
N C
n n 0
+ kT ln
N V
p p 0
= kT ln
p p 0 n n 0
n
2
i
= kT ln
p p 0
p n 0
= kT ln
n n 0
n p 0
(21.101a)
∼ = kT ln
N A N D
n
2
i
.
(21.101b)
The electron and hole densities on either side of the junction (n p 0 and p p 0 at x = −x p and n n 0 and p n 0
at x = x n ) are related to each other by (from rewriting (21.101a))