Forward Currents from the Space Charge Region

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_i²)

n^e + n^h + 2n _i ·	cosh	E_DL – E_MB kT

1/t · (n^e · n^h – n_i²)

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 Quasi-Fermi energies as E_F^e and E_F^h , respectively. For their product we get

n^e · n^h	=	n_i² · exp –	E_F^e – E_F^h kT

For the forward direction we have E_F^e – E_F^h < 1 and thus

n^e, n^h	>>	n_i

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_i² · 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 Quasi-Fermi energies. Lets look at the situation in a band-diagram

Whatever the exact positions of the Quasi-Fermi 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 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