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Depletion
This is the case where an electrical field of arbitrary origin repulses the majority carriers and a space charge region
develops.
Starting with the Poisson equation for doped semiconductors and all dopants ionized, we have
d2∆EC
e2 · N
= –
dx2
εε0

1

EC
– exp –
kT



In contrast to the case of quasi-neutrality, we now have +∆EC >> kT and the sign is important!
This allows an simple approximation:
EC
≈ 0
1 – exp –
kT
The Poisson equation for the part of the semiconductor that contains this carrier density reduces to
d2∆EC
e2 · ND
= –
dx2
εε0
We have treated this case already in the more basic considerations. The result was
e · ND
· x2 – 2dSCR · x + dSCR2
U(x) =
2εε0
1
dSCR =
·
e
 2∆EC(x = 0) · εε0 1/2




ND
With ∆EC(x = 0) = ∆E for brevity, we can rewrite the expression for the width of the space charge layer in terms
of the Debye length LDb
LDb =
 εε0 · kT) 1/2


 e2 · ND 
and obtain
dSCR = LDb ·
 2∆E  1/2


 kT 
If we express ∆E in terms of the the voltage U between the ends of the sample by e · U = ∆E, we have the final
result
Semiconductor - Script - Page 1
dSCR = LDb ·
 2 · e · U  1/2


 kT 
Remember that LDb is a purely material related quality and thus a constant for a given semiconductor. The width of
the space charge region can be expressed very simply in terms of LDb, it is always larger by the factor {2eU/kT}1/2
Since kT at room temperature ≈ 1/40 eV, while applied voltages may be up to 1000 V, dSCR may exceed LDn by
several orders of magnitude. This is shown in the illustration below (the numbers are basically correct, but not in
detail).
The breakdown limit indicates that the SCR, being an dielectric insulator, will eventually experience electrical
breakdown if the field strength exceeds an upper limit.
Semiconductor - Script - Page 2