
We start from a (constant)
distribution of positive charges (for ndoped semiconductors) in the
space charge region. 


The corresponding negative charges
are all on the surface; the charge distribution is shown in the first frame of
the illustration. 


Poisson's equation states
that (for the onedimensional case). 


ee_{0} 
d^{2}V(x)
dx^{2} 
= – 
r(x) 
= 
e · N_{D} 




For 0 < x <
d_{SCR} and = 0 everywhere else. (We can also use the
voltage U(x) instead of V(x) if we
think as V(x = ¥) = 0).
That will also be reflected in the choice of boundary conditions made
below. 

The drawing below shows the situation, including
the slight approximation implicit in our choice of r(x). Note that the xdirection
ist to the left in this case. 





The first straightforward
integration yields dU/d(x) which is the electrical field
strength E_{x} = –dU/dx, or 


ee_{0} 
dV(x)
dx 
= – ee_{0} E_{x} 
= 
e · N_{D} · x + const. 




With the boundary condition
E_{x}(x = d_{SCR}) = 0, we obtain (always for the
interval x = 0 and x = d_{SCR}, of
course): 


e ·N_{D} · d_{SCR} +
const 
= 
0 



const 
= 
– e ·N_{D} ·
d_{SCR} 

E_{x} 
= 
1_{ }
ee_{0} 
· 
(e · N_{D}d_{SCR} – e
· N_{D} · x) 




The second integration yields 


ee_{0} ·
U(x) 
= 
e · N_{D} · x^{2}
2_{ }^{ } 
– e · N_{D} ·
d_{SCR} · x + const. 




With the boundary condition
U(d_{SCR}) = 0, we obtain . 


– 
e · N_{D} · d ^{2}_{SCR}
2_{ } 
+ const. 
= 
0 









const. 
= 
e · N_{D} · d
^{2}_{SCR}
2_{ } 




Using the proper expression for the
integration constant gives gives us the complete voltage function or the shape
of the band bending 


ee_{0} ·
U(x) 
= 
e · N_{D} · x^{2}
2^{ }^{ } 
– 
e · N_{D} · d_{SCR} ·
x 
+ 
e · N_{D} · d ^{2}_{SCR}
2_{ }^{ } 




The width of the space charge region
can be obtained by considering the voltage at x = 0, where we
have U(x = 0) = DE_{F}/e.Using this we obtain 


ee_{0}
e _{ } 
· DE_{F} 
= 
e · N_{D} · d ^{2}_{SCR}
2_{ } 



This gives us the final result for the width of
the space charge region 


d_{SCR} 
= 
1
e 
· 
æ
ç
è 
2DE_{F} ·
e e_{0}
N_{D} 
ö
÷
ø 
1/2 



The corresponding curves are shown in the drawing
above. We obtained the same
formula as before, but now we have a better awareness of the approximations
it contains. 


The positive charge distribution was assumed to be
boxshaped and uniform. This is a rather good approximation; the drawing
indicates the precise shape of the charge distribution for comparison. 


The counter charges are described by a dfunction at the surface; these charges only enter
the calculation in the indirect form of a boundary condition. 

