Forward Currents from the Space Charge Region

Again, we start from the equation for the net recombination UDL via deep levels
UDL  =  

v · s e · NDL · (ne · nh  –  ni2)

nenh  +  2n i · cosh EDLEMB

1/t · (n e · nh  –  ni2)

nenh  +  2ni · cosh EDLEMB
with 1/t = v ·s e ·NDL as we know by now.
The carrier densities ne and n h may be expressed via their Quasi-Fermi energies as E Fe and EFh , respectively. For their product we get
ne · nh  =  ni2 · exp –   EFeEFh
For the forward direction we have EFeE Fh < 1 and thus
ne, nh  >>  ni
This leaves us with
UDL  =  1
  ·   ne · nh
ne  +  nh
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
{(ne · nh)/( ne + nh)}
  =  {(ne · nh)/(ne + nh)}
 =  0
which gives us ne = nh for maximum current. With ne · nh = ni 2 · exp – [(EFeEFh )/k T] from above, we have
ne =  nh   =  ni · exp –  E Fe  –  EFh
What we need now is an equation for the difference of the Quasi-Fermi energies. Lets look at the situation in a band-diagram
FOrwardly biased p-n-junction
Whatever the exact positions of the Quasi-Fermi energies, their difference E FeEFh is about equal to the difference in the bulk Fermi energy and thus
EFe   –  EFh  »   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
UDL (max)  »  1
  · ni · exp – e · U
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:

jF(SRC)  =  e · ni · dSCR
  · exp – e · U

Considering that we needed the whole formalism of Shockley-Read-Hall recombination theory, Quasi-Fermi 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.

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go to 5.2.1 Total Efficiency of Light Generation

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© H. Föll (Semiconductors - Script)