
Again, we start from the equation for the net recombination U_{DL} via
deep levels 
 
U_{DL} = 
v · s
^{e} · N_{DL} · (n^{e} · n^{h} – n_{i}^{2}) 
n^{e} + n^{h} + 2n _{i} · 
cosh  E_{DL} – E_{MB}
kT 
 = _{ } 
1/t · (n^{ e} · n^{h} – n_{i}^{2}) 
n^{e} + n^{h} + 2n_{i} ·  cosh
 E_{DL} – E_{MB} kT 





with 1/t = v ·s
^{e} ·N_{DL} as we know by now. 

 

The carrier densities n^{e} and n ^{h}
may be expressed via their QuasiFermi energies as
E_{ F}^{e} and E_{F}^{h}
, respectively. For their product we get 
 
n^{e} · n^{h}  =
 n_{i}^{2} · exp – 
E_{F}^{e} – E_{F}^{h}
kT^{ } 



For the forward direction we have E_{F}^{e} – E_{
F}^{h} < 1 and thus 
 



This leaves us with 
 
U_{DL}  = 
1 t 
· 
n^{e} · n^{h}
n^{e} + n^{h} 



The maximum value for U_{ DL} gives
the upper limit for the net recombination rate and thus the maximum current due to recombination
in the SRC, too. The maximum is defined by 


¶{(n^{e} · n^{h})/( n^{e} + n^{h})}
¶n^{e} 
= 
¶{(n^{e} · n^{h})/(n^{e}
+ n^{h})}
¶n^{h} 
= 0 




which gives us n^{e} = n^{h} for maximum current. With n^{e}
· n^{h} = n_{i}^{ 2} · exp – [(E_{F}^{e} –
E_{F}^{h} )/k T] from above, we have 


n^{e} = n^{h}_{ } 
=  n_{i} · exp – 
E
_{F}^{e} – E_{F}^{h} 2kT_{ }^{ } 



What we need now is an equation for the difference of the QuasiFermi energies. Lets look at
the situation in a banddiagram 
 



Whatever the exact positions of the QuasiFermi energies, their difference E_{ F}^{e}
– E_{F}^{h} is about equal to the difference in the
bulk Fermi energy and thus 
 
E_{F}^{e} – E_{F}^{h}
 » 
e · U 




(The "about equal" contains roughly the same approximation
as the "average barrier height" from the simple derivation!) 

This gives us the final result 


U_{DL} (max) 
» 
1 2t 
· n_{i} · exp – 
e · U 2kT 



Again , this is the net recombination rate at any point in the space
charge region. To obtain the current density, we have to multiply with the width d of the SCR (and
the elementary charge) and obtain for the maximum current from the SCR in forward direction: 
 
j_{F}(SRC)  = 
e · n_{i} · d_{SCR}
2t_{ } 
· exp – 
e · U 2kT 



Considering that we needed the whole formalism of ShockleyReadHall recombination theory,
QuasiFermi energies, some junction theory, and lots of assumptions and approximations to
get the same result as before, this does not appear to be a much better way of getting an idea about the influence of
the SCR on the diode characteristic than the "quick and dirty" way. 
 
But don't deceive yourself! The treatment given here is not only physically sound, but transparent at every
step. If you want to do more precise calculations, you would know  at least in principle  what to do. 
  