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We will now give some thought to the
second and third question raised before:
How much light is produced by
recombination?. This raises the question for the value of the
quantum efficiency hqu mentioned before and the total or
external efficiency hex in absolute terms (third
question). |
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First we will define the quantum efficiency again (and somewhat more
specifically) and relate it to some other efficiencies. |
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The
quantum efficiency
is defined as
hqu = {Number of photons generated in
the recombination zone}/{Number of recombining carrier pairs in the
recombination zone}. |
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It could be
expressed as
hqu =
|
1
1+ trad / tnon-rad
|
|
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In the high
injection approximation the number of carriers is about equal to the number
of carriers injected (across a junction) into the recombination zone. |
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That part of the total recombination occurring
via a radiative channel determines the quantum efficiency. However, the surplus
carriers in the recombination zone have one more "channel", not
considered so far, for disappearing from the recombination zone: They simply
move out! |
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In other words: parts of the injected carriers
will simply flow across the recombination zone and leave it at "the other
end". |
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This effect can be
described by the current
efficiency hcu; it is defined as |
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hcu =
{Number of recombining carrier pairs in the recombination zone}/{Number of
carrier pairs injected into the recombination zone}. |
 |
We now define the
optical efficiency
hopt as |
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hopt = {
Number of photons in the exterior}/{Number of photons generated in the
recombination zone}. |
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The optical efficiency takes care of the (sad)
fact that in most devices a large part of the photons generated become
reabsorbed or are otherwise lost and never leave the device. |
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The total or
external efficiency hex now becomes simply |
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 |
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If we want to optimize the external
efficiency, we must work on all three factor - none of them is
negligeable. |
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We already "know" how to optimize the
quantum efficiency hqu by
looking at the equation above. We must look for the best combination of
materials producing radiation at the desired wavelength and then dope it in
such a way as to maximize the radiative channel(s) by minimizing the
corresponding lifetimes. While this is not easy to do in practice, it is clear
in principle. |
 |
We do not yet know how to attack the
two other problems: Maximized current efficiency and maximized optical
efficiency. And theses problems are far from being solved in a final, or just
semi-final way - intense world-wide research efforts center on new solutions to
these problems. |
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While there are no general solutions to these
problems and only some useful equations, a few general points can still be
made. We will do this in the remainder of this subchapter for the current
efficiency and in a separate subchapter for the
optical efficiency |
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The question to ask is: Why is the
current efficiency not close to 1 in any case? |
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After all, if we consider a simple p-n-junction
biased in forward direction in a direct semiconductor (e.g. GaAs), we inject
electrons into the p-part and holes into the n-part, where they will become
minority carriers. Some of the injected carriers will recombine in the space
charge region, all others eventually in the bulk region. |
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While the quantum efficiency may be different in
the different regions because the strength of the recombination channels
depends on the carrier density which is not constant across the junction, we
still could assign some kind of mean quantum efficiency to the diode so that
hcu =1. |
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 |
However, we defined the efficiencies relative to
a "recombination zone", i.e we are not interested in radiation
produced elsewhere for various reasons (to be discussed later). If we take the
recombination zone to be identical to the SCR, only that part of the injected
carriers that recombines in the SCR will contribute. This is exactly that part
of the forward current that we had to introduce to account for real
I-V-characteristics of p-n- junctions - cf. the
simple and
advanced version in the relevant
modules. |
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That part of the current that injects the
carriers that recombine in the SCR was given by
jrec
(SCR) = |
e ni
d
2 t
|
exp( |
e U
2KT
|
) |
d was the width of the SCR. |
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The current efficiency in this case would be
given by
hcu = |
jrec
jdiode
|
= |
jrec
jnon-rec +
jrec
|
|
or |
|
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hcu = |
1
1+ (jnon-rec /
jrec)
|
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With
jnon-rec (assuming that the electron and hole
contributions and parameters are equal) given by the "simple" diode
equation as |
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 |
jnon-rec = |
2 e L
ni2
t NDop [exp (eU
/ KT) - 1]
|
|
|
|
 |
This gives for hcu (neglecting the -1 after the
exponential). |
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 |
hcu = |
1
1 + 4(L / d) t
ni NDop exp (eU / KT)
|
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|
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hcu
thus decreases exponentially with the applied voltage and it would not make
sense to include this effect in some averaged hqu. |
 |
Why are we looking at radiation only
form some confined part of the device and not at the total volume, which
demands a finer look at the efficiencies? There are practical reasons,
e.g.: |
|
 |
If we consider a semiconductor Laser, only the
radiation inside the "resonator" counts - everything outside of this
specific recombination volume is of little interest. |
|
 |
If we look at a light emitting diode - a
LED - made from GaP doped with N (in
addition to the normal doping) to produce the isolectronic impurities
needed to bind the excitons responsible for the
radiative recombination channel, it only radiates from the p-side because
only electrons become primarily bound to the isoelectronic impurity and than
attract a hole. In other words, only the electron part of the injected current
will contribute to radiation. |
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We must confine light production to
areas close to the surface as shown in the next
subchapter. |
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Looking ahead we will learn that many
optoelectronic devices are extremely complicated heterostructures for several
reasons, including a precise definition of the recombination volume. |
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Nevertheless, current efficiencies,
while helpful in thinking about devices and radiative processes, are relatively
hard to describe in detail. |
|
 |
If we accept that we want light only from a
defined recombination volume, we now can look at a simple way to understand the
current efficiency. If we look at the disappearance of the injected carriers,
we have the simple rate equation |
|
 |
dn / dt = (1 / q)
div( j ) - R
total. |
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 |
The first term describes the appearance or
disappearance of carriers because they flow in or out of the volume as a
current; the second term the recombination processes. |
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 |
A high current efficiency then simply means to
make divj as negative as possible for electrons and as positive as
possible for holes (remember
that q = -e for electrons and +e for holes). In other words, maximize
the flow of carriers into the volume, and minimize the flow out of it. |
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|