


Lets first look at the basic situation
as we had it
before for large diodes: 




We have an excess of minority carriers at the edge
of the space charge region stemming form the majority carriers injected into
the other part of the junction. 


The difference of the actual concentration
n^{p,n}_{e,h}(U) and the equilibrium
concentration n^{p,n}_{e,h}(U = 0)
was given
by 


D n^{p, n}_{e,
h} 
÷
÷ 
edge
SCR 
= 
n ^{p, n}_{e, h}(U) –
n ^{p, n}_{e, h}(U = 0) 
= 
n^{p,n}_{e,h}(equ) ·

æ
ç
è 
exp 
U
kT 
– 1 
ö
÷
ø 



Neglecting the – 1
for forward conditions, we have the exceedingly simple
general relation that the current flowing is simply the diffusion
current at the edge of the SCR following from the concentration gradient
via Ficks 1st law. Lets look at this a bit closer. 


All that counts is the slope dDn^{min}/dx of excess minority
carrier concentration at the edge of the SCR. It gives directly the
minority carrier current at the edge of the SCR  and that is the only
current we need to consider. 


Since it is the only
component of the current flowing at this point of the junction, (we neglected
the other principal terms for the forward condition), and since the current is
constant throughout the junction, it simply is the current. We don't have to worry about the other
side of the junction or anything else. 


The junction current j thus is 


j 
= – q · D · 
dDn^{min}
dx 
÷
÷ 
edge
SCR 



What about the current deeper in the Si?
The slope is smaller and this must lead to a smaller current, too. Yes  but
now we have a majority carrier current,
too. Whatever we loose due to recombination in the minority carrier current
component, we gain in the majority carrier current component and the total
current stays constant. 

In order to compute it, we need the slope and thus
Dn^{min}(x). 


We always obtain Dn^{min}(x) as the solution of a
diffusion problem, taking into account boundary conditions, e.g. Dn^{min}(x = 0), i.e. at the
edge of the SCR, or the disappearance via recombination. 


One boundary condition is clear: At the edge of the SCR
the excess concentration will be at a fixed value controlled by the applied
potential as described above. 


The second boundary condition is less clear. When we
derived the
relation



Dn(x) = Dn_{0} · exp – 
x
L 




we also
got the
current 


j ^{min}(x = 0) 
= 
q · D
L 
· Dn^{min}(x = 0) 




we implicitly assumed that the size of the diode was infinite
and that minority carriers simply disappear by recombination. 

For a small
diode now, with xdimensions much smaller than
L, we have to reconsider the diffusion problem. 


Assuming that after a distance d_{Con}
<< L we now have an ohmic contact, we must ask what the excess
minority carrier density will be at x =
d_{Con}. 


To make life easy, we now simply include in the definition of a "good"
ohmic
contact that minority carriers reaching it will recombine instantaneously.
While this is pretty much true for real contacts, it is not necessarily
obvious. 


With this assumption we simply have as the important boundary
condition for a small diode . 




This makes the solution to the diffusion problem
very simple. 


Since practically no recombination in the bulk will take place
 all minorities die at the contact  the current everywhere is simply the minority carrier current.
This necessitates that 


dDn^{min}
dx 
= const = – 
Dn^{min}
÷ _{edge SCR}
d_{Con} 




The current then is 


j ^{min} 
= j = – q · D ·

dDn^{min}
dx 
÷
÷ 
edge
SCR 
= 
q · D
d_{Con} 
· Dn^{min} (x =
0) 



This is exactly the same formula as for the large
diode  except that we now have
d_{Con} instead of L as the important
length scale of the device. 


Moreover, minority carriers will now disappear by
recombination at the contact after an average time t_{trans} called transit time given by the time they need for
traveling the distance d_{Con}. Obviously, we have 


d_{Con} 
= 
æ
è 
D · t_{trans} 
ö
ø 
1/2 




in complete analogy to the
relation between
lifetime and diffusion length 

Of course, this is still a rather simple
description of a small diode. We only restricted one dimension, since we still treated a
onedimensional case. 


Real diodes might be small in more than one dimension, and all
kinds of other complications can be imagined. Nevertheless, the device
dimensions and the transit time will in one form or other replace the bulk
diffusion length and lifetime. 


The importance of this can not be overestimated. Device sizes
in integrated circuits are in the subµm region and critically
influence device behavior. 


