Diode Ideality
Consider an n+/p diode in forward bias. We draw the band-edge vs position diagram
as follows:
Figure 1: Band-edge vs position diagram for the forward biased diode
Here are a few things to note. The “IMREFs” represent non-equilibrium Fermi levels. In
the bulk semiconductor, far from the space-charge or the minority-flooded space charge
boundaries, the system is “just about” in equilibrium, and so there is only one Fermi level
in these regions. The volume of semiconductor which is not in equilibrium is the space
charge and its immediate surroundings. Here, the Fermi level splits into two “IMREFs.”
The IMREFs don’t droop through the space charge. At the metallurgical junction, xmj ,
the mobile electron and hole concentrations are still equal, but they are elevated over ni
1
due to the bias-induced lowering of the charge injection barrier. Further note: the charge
injection barrier is (before bias is applied) just the built-in voltage, φbi . As the mobile
charge densities depend exponentially on the separation of the IMREF from mid-gap, and
since the electron and hole densities are equal, the electron IMREF is as much above the
mid-gap position as the hole IMREF is below it. These two potential energy separations
are the same (Va /2.) And so, we can write:
q Va
p = ni exp ( )( )
(1)
kT 2
q Va
n = ni exp ( )( )
(2)
kT 2
In our previous work deriving the diode transport equations, we were only concerned
about what happens at the space charge edges. We said diffusion was the “rate limiting”
process in transport and by evaluating the diffusion currents right at the space charge
boundary, we could find the total diode current. This neglected what was happening inside
the space charge. Inside the space charge layer, we have many excess electrons and holes
present due to injection. These mobile carriers can recombine, creating a recombination
current, Irec .
Recombination is governed by the Shockley-Read recombination equation:
U=
1 n2i − np
τ n + p + 2ni
(3)
where τ is the recombination lifetime. The right-hand fraction numerator is the degree to
which the system is locally “out of equilibrium.” In equilibrium, this numerator is zero by
constancy of the pn product (np=n2i . So the formula is just a re-statement of Le Chatelier’s
principle: the system attempts to return to equilibrium at a rate proportional to the degree
it’s out of equilibrium. The proportionality constant is the reciprocal lifetime.
To proceed, we make two assumptions. First, we assume that in the flooded junction
space charge, n and p are much bigger than ni . Next, we assume that recombination
proceeds rapidly only when n is about equal to p. For example, if p is much bigger than n,
the excess electrons may recombine rapidly, but there just aren’t a lot of them to contribute
to the overall current. If these assumptions are true, we can write:
U=
n
2τ
(4)
We could also have written:
p
2τ
It doesn’t matter which you choose, But you have to pick one to proceed.
U=
2
(5)
You can turn U into a Coulombic current by multiplying it by q and the volume over
which recombination is significant. The volume is just the junction area, A times some
“effective” space charge thickness, x0sc , and we have:
qni Ax0sc
q Va
Irec =
exp ( )( )
(6)
2τ
kT 2
Sincew this is a forward current and the exponential term is much bigger than 1, we can
re-write this into a form similar to the Shockley diode equation:
qVa
0
)−1
(7)
Id = Is exp(
2kT
Where:
qni Ax0sc
(8)
2τ
But, in fact, the diode diffusion current and the recombination current are both present
in a single diode, and the actual current expression looks like this:
qVa
0
Id = Is exp(
)−1
(9)
nkT
Is0 =
The term n is referred to as the diode ideality factor. What factors define n? Well,
recombination is enhanced by defects. In fact, defects drive the recombination process.
More defects, mean more space-charge recombination. If there were no defects present,
the total diode current would be a diffusion current and n would be 1. This would be the
“ideal” diode case. More defects drive n up to 2 (the “non-ideal” case). So you see why
we call n the ideality factor.
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