
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 – 
DE_{C} kT 
ö ÷ ø 




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

This leads to a 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 D E_{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 
e e_{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. 
 

 
