# 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)
ne + nh + 2n i · | cosh | EDL – EMB kT
--- | --- | ---
| = |
1/t · (ne · nh – ni2)
ne + nh + 2ni · | cosh | EDL – EMB kT
--- | --- | ---
|  | 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 – | EFe – EFh kT
--- | --- | --- | ---
| For the forward direction we have EFe – E Fh < 1 and thus
ne, nh | >> | ni
--- | --- | ---
|  | This leaves us with
UDL | = | 1 t | · | 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 | = | ¶{(ne · nh)/(ne + nh)} ¶nh | = 0
--- | --- | --- | ---
|  | which gives us ne = nh for maximum current. With ne · nh = ni 2 · exp – [(EFe – EFh )/k T] from above, we have
ne = nh | = | ni · exp – | E Fe – EFh 2kT
--- | --- | --- | ---
| What we need now is an equation for the difference of the Quasi-Fermi energies. Lets look at the situation in a band-diagram
|  | Whatever the exact positions of the Quasi-Fermi energies, their difference E Fe – EFh 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 2t | · ni · 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:
jF(SRC) | = | e · ni · dSCR 2t | · exp – | e · U 2kT
--- | --- | --- | --- | ---
| 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.

With frame

5.2.1 Total Efficiency of Light Generation

2.3.5 Junction Reconsidered

© H. Föll (Semiconductors - Script)