570
21 Diodes
Avalanche Multiplication
Avalanche multiplication due to impact ionization is the most important mechanism for the breakdown
of pn diodes. It limits the maximum reverse voltage for most diodes and also the collector voltage in a
bipolar transistor or the drain voltage in a field-effect transistor. Avalanche multiplication can be used
for the generation of microwave radiation or for photon counting (cf. Sect. 22.3.6).
Impact ionization was discussed in Sect. 8.4.4. The most important parameters are the electron and
hole ionization coefficients α n and α p . For discussion of the diode breakdown, we assume that at x = 0
a hole current I p 0 enters the depletion layer. This current is amplified by the field in the depletion layer
and impact ionization. At the end of the depletion layer (x = w), it is M p I p 0 , i.e. M p = I p (w)/I p (0).
Similarly, an electron current is increased on its way from w to x = 0. The incremental change of the
hole current due to electron-hole pairs generated along a line element dx is
dI p = (I p α p + I n α n ) dx .
(21.160)
The total current in the depletion layer is I = I p + I n and is constant in stationary equilibrium.
Therefore,
dI p
dx
−
α p − α n
I p = α n I .
(21.161)
The solution is
I p (x) = I
1
M p
+
x
0
α n exp
−
x
0
(α p − α n ) dx
dx
exp
−
x
0
(α p − α n ) dx
.
(21.162)
For x = w we find for the multiplication factor
1 −
1
M p
=
w
0
α n exp
⎡
⎣ −
x
0
(α p − α n ) dx
⎤
⎦ dx .
(21.163)
Avalanche breakdown is reached for M p → ∞, i.e. when
w
0
α n exp
⎡
⎣ −
x
0
(α p − α n ) dx
⎤
⎦ dx = 1 .
(21.164)
A corresponding and equivalent equation is obtained when the consideration is started with the electron
current. If α p = α n = α, (21.164) simplifies to
w
0
α dx = 1 .
(21.165)
This means that per transit of one carrier through the depletion layer, on average another carrier is
created such that the process just starts to diverge. The breakdown voltage for various semiconductor
materials is shown in Fig. 21.54a as a function of the doping level. The depletion-layer width w at
breakdown and the maximum electric field E m are depicted in Fig. 21.54b.
The average impact ionization coefficient α has been given as [1773]
21 Diodes
Avalanche Multiplication
Avalanche multiplication due to impact ionization is the most important mechanism for the breakdown
of pn diodes. It limits the maximum reverse voltage for most diodes and also the collector voltage in a
bipolar transistor or the drain voltage in a field-effect transistor. Avalanche multiplication can be used
for the generation of microwave radiation or for photon counting (cf. Sect. 22.3.6).
Impact ionization was discussed in Sect. 8.4.4. The most important parameters are the electron and
hole ionization coefficients α n and α p . For discussion of the diode breakdown, we assume that at x = 0
a hole current I p 0 enters the depletion layer. This current is amplified by the field in the depletion layer
and impact ionization. At the end of the depletion layer (x = w), it is M p I p 0 , i.e. M p = I p (w)/I p (0).
Similarly, an electron current is increased on its way from w to x = 0. The incremental change of the
hole current due to electron-hole pairs generated along a line element dx is
dI p = (I p α p + I n α n ) dx .
(21.160)
The total current in the depletion layer is I = I p + I n and is constant in stationary equilibrium.
Therefore,
dI p
dx
−
α p − α n
I p = α n I .
(21.161)
The solution is
I p (x) = I
1
M p
+
x
0
α n exp
−
x
0
(α p − α n ) dx
dx
exp
−
x
0
(α p − α n ) dx
.
(21.162)
For x = w we find for the multiplication factor
1 −
1
M p
=
w
0
α n exp
⎡
⎣ −
x
0
(α p − α n ) dx
⎤
⎦ dx .
(21.163)
Avalanche breakdown is reached for M p → ∞, i.e. when
w
0
α n exp
⎡
⎣ −
x
0
(α p − α n ) dx
⎤
⎦ dx = 1 .
(21.164)
A corresponding and equivalent equation is obtained when the consideration is started with the electron
current. If α p = α n = α, (21.164) simplifies to
w
0
α dx = 1 .
(21.165)
This means that per transit of one carrier through the depletion layer, on average another carrier is
created such that the process just starts to diverge. The breakdown voltage for various semiconductor
materials is shown in Fig. 21.54a as a function of the doping level. The depletion-layer width w at
breakdown and the maximum electric field E m are depicted in Fig. 21.54b.
The average impact ionization coefficient α has been given as [1773]