
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 


d^{2}DE_{C}
dx^{2} 
= – 
e^{2} · N_{}
ee_{0} 
æ
ç
è 
1 – 
exp – 
E_{C}
kT 
ö
÷
ø 




In contrast to the case of
quasineutrality, we now
have +DE_{C}
>> kT and the sign is
important! 

This allows an simple approximation: 





The Poisson equation for the part of the semiconductor that
contains this carrier density reduces to 


d^{2}DE_{C}
dx^{2} 
= – 
e^{2} · N_{D}
ee_{0} 



We have
treated this case already in
the more basic considerations. The result was 


U(x) 
= 
e · N_{D}
2ee_{0} 
· x^{2} – 2d_{SCR}
· x + d_{SCR}^{2} 
d_{SCR} 
= 
1
e 
· 
æ
ç
è 
2DE_{C}(x = 0) ·
ee_{0}
N_{D} 
ö
÷
ø 
1/2 




With DE_{C}(x = 0) = DE for brevity, we can rewrite the expression
for the width of the space charge layer in terms of the
Debye length
L_{Db} 


L_{Db} 
= 
æ
ç
è 
ee_{0} ·
kT)
e^{2} · N_{D} 
ö
÷
ø 
1/2 




and obtain 


d_{SCR} 
= 
L_{Db} · 
æ
ç
è 
2DE
kT 
ö
÷
ø 
1/2 




If we express DE in
terms of the the voltage U between the ends of the sample by e
· U = DE, we have the final
result 


d_{SCR} 
= 
L_{Db} · 
æ
ç
è 
2 · e · U
kT 
ö
÷
ø 
1/2 



Remember that L_{Db} 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
L_{Db}, 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, d_{SCR} may exceed
L_{Dn} 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. 





