560
21 Diodes
21.4.4 Current-Voltage Characteristics
Ideal Current-Voltage Characteristics
Now, the currents in thermodynamical equilibrium (V = 0) and under bias are discussed. A diode
characteristic will be obtained. We work at first with the following assumptions: abrupt junction,
Boltzmann approximation, low injection, i.e. the injected minority carrier density is small compared
to the majority carrier density, and zero generation current in the depletion layer, i.e. the electron and
hole currents are constant throughout the depletion layer. In the presence of a bias, electrons and holes
have quasi-Fermi levels and the carrier densities are given by (cf. (7.55a, b))
n = N C exp
F n − E C
kT
(21.119a)
p = N V exp
−
F p − E V
kT
.
(21.119b)
Using the intrinsic carrier concentration (7.19,7.20), we can write
n = n i exp
F n − E i
kT
= n i exp [β(ψ − φ n )]
(21.120a)
p = n i exp
−
F p − E i
kT
= n i exp
β(φ p − ψ)
,
(21.120b)
where φ and ψ are the potentials related to the (quasi-) Fermi level and the intrinsic Fermi levels,
−eφ n,p = F n,p and −eψ = E i . The potentials φ n and φ p can also be written as
φ n = ψ − β
−1 ln
n
n i
(21.121a)
φ p = ψ + β
−1 ln
p
p i
.
(21.121b)
The product np is given by
np = n
2
i exp
β(φ p − φ n )
.
(21.122)
Of course, at thermodynamical equilibrium (zero bias) φ p = φ n and np = n
2
i . For forward bias
φ p − φ n > 0 (Fig. 21.45a) and np > n
2
i . For reverse bias φ p − φ n < 0 (Fig. 21.45b) and np < n
2
i .
The electron current density (per unit area) is given by (8.60a) that reads here with E = −∇ψ and
n given by (21.120a) as
23
j n = −e μ n
n E + β
−1
∇n
= e n μ n ∇φ n .
(21.123)
Similarly, we obtain for the hole current density (using (8.60b) and (21.120b))
j p = e μ p
p E − β
−1
∇ p
= −e p μ p ∇φ p .
(21.124)
Both these currents through the depletion/space-charge region are constant since no recombination/generation is assumed for now. The gradient of the quasi-Fermi levels in the depletion layer
23 We remind the reader that μ n was defined as a negative number.
21 Diodes
21.4.4 Current-Voltage Characteristics
Ideal Current-Voltage Characteristics
Now, the currents in thermodynamical equilibrium (V = 0) and under bias are discussed. A diode
characteristic will be obtained. We work at first with the following assumptions: abrupt junction,
Boltzmann approximation, low injection, i.e. the injected minority carrier density is small compared
to the majority carrier density, and zero generation current in the depletion layer, i.e. the electron and
hole currents are constant throughout the depletion layer. In the presence of a bias, electrons and holes
have quasi-Fermi levels and the carrier densities are given by (cf. (7.55a, b))
n = N C exp
F n − E C
kT
(21.119a)
p = N V exp
−
F p − E V
kT
.
(21.119b)
Using the intrinsic carrier concentration (7.19,7.20), we can write
n = n i exp
F n − E i
kT
= n i exp [β(ψ − φ n )]
(21.120a)
p = n i exp
−
F p − E i
kT
= n i exp
β(φ p − ψ)
,
(21.120b)
where φ and ψ are the potentials related to the (quasi-) Fermi level and the intrinsic Fermi levels,
−eφ n,p = F n,p and −eψ = E i . The potentials φ n and φ p can also be written as
φ n = ψ − β
−1 ln
n
n i
(21.121a)
φ p = ψ + β
−1 ln
p
p i
.
(21.121b)
The product np is given by
np = n
2
i exp
β(φ p − φ n )
.
(21.122)
Of course, at thermodynamical equilibrium (zero bias) φ p = φ n and np = n
2
i . For forward bias
φ p − φ n > 0 (Fig. 21.45a) and np > n
2
i . For reverse bias φ p − φ n < 0 (Fig. 21.45b) and np < n
2
i .
The electron current density (per unit area) is given by (8.60a) that reads here with E = −∇ψ and
n given by (21.120a) as
23
j n = −e μ n
n E + β
−1
∇n
= e n μ n ∇φ n .
(21.123)
Similarly, we obtain for the hole current density (using (8.60b) and (21.120b))
j p = e μ p
p E − β
−1
∇ p
= −e p μ p ∇φ p .
(21.124)
Both these currents through the depletion/space-charge region are constant since no recombination/generation is assumed for now. The gradient of the quasi-Fermi levels in the depletion layer
23 We remind the reader that μ n was defined as a negative number.