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The optical efficiency hopt is easy to understand by looking a
the mechanisms that prevent photons from leaving the device. We have two basic
mechanisms: |
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The emitted photon is absorbed before
arriving at a (internal) surface of the device. |
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The emitted photon makes it to the
(internal) surface of the device but is reflected back into the interior and
than absorbed. |
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We thus have to worry about
absorption of light in semiconductors in general and about reflections at
surfaces. |
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The first topic is a science in
itself. In due time, a chapter might be added dealing with this issue in more
detail, here we only note a few of the major points: |
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In direct analogy to the various
modes of radiative recombination, we have the reverse process also: A photon
creates an electron hole pair occupying some levels; including, e.g. exciton
levels. |
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All the conservation laws must be
obeyed; phonons or other third particles may have to assist the process. |
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The dominating absorption process
usually is the direct band-band process; i.e. straight up in a (reduced) band
diagram from an (occupied) position in the valence band to an (unoccupied)
position in the conduction band. For indirect semiconductors this requires a
larger energy than the band gap! |
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The band-band
absorption process is also called the
fundamental (absorption) process, it is described phenomenologically by
Beer's law: |
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The intensity I of the
light at a depth z in the semiconductor, I ( z
), is given by |
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I ( z ) =
I0·exp - ( a·z )
with I0 = intensity at z = 0
and a = a
(hn) =
absorption coefficient of the
material. |
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It is clear that a must be a strong function of the energy hn of the photons. |
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For hn < Eg(direct), no
electron hole pairs can be created, the material is transparent and
a is small. |
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For hn >= Eg(direct), absorption
should be strong. All mechanisms other than the fundamental absorption may add
complications (e.g. "sub band gap absorption" through excitons), but
usually are not very pronounced. |
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The absorption coefficients of major
semiconductors indeed follow this predictions as can be seen in the following
diagram |
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There are many more points to the
absorption of light in semiconductors but we will not pursue the issue further
at this point. |
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If we now look at a
LED, we notice that light with wavelengths corresponding to the absorption edge
thus will be absorbed within a few µm of the material -
and that automatically applies to the light emitted by
radiative recombination. |
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If we look at a naive cross section
of a light emitting diode (on the left), we see that only light from the edges
of the p-n-junction has a change to make it to the surface of the device.
Obviously, this is not a good solution for a large optical efficiency. |
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If we make a junction more like in an
integrated Si circuit (above right), the situation is somewhat improved, but it
might be difficult to drive high currents in the central region of the device,
far away from the contacts. |
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We might be better
off in choosing a n-type material with a larger bandgap than the p-type
material and see to it that light is generated in the p-type material. Its
photon energy then would be too small for absorption the large bandgap material
and it could escape without absorption. In other words: We utilize a heterojunction |
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This sequence demonstrates several
important points about the realization of LEDs: |
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A large optical efficiency is not
easy to achieve. Generally, much of the light produced will never leave the
device. |
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The typical structures from Si integrated circuit
technology may or may not be useful for optoelectronic applications. In
general, we have to develop new approaches. |
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We always should try
to produce the light close to the (possibly internal) surface of the active
material - in other words we need a defined recombination zone that is not in
the bulk of the active material |
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Heterostructures - meaning the combination
of different semiconductors - come up quickly in optoelectronics (while
virtually unknown in Si technology). |
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Next, lets assume that the photons
make it to the surface of the device - the question now is if they are
reflected back into the interior or if the can escape to the outside
world. |
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This is a question that can be
answered by basic optics. The relevant quantities are shown in the next
picture |
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For the light beams coming from the
interior of the semiconductor to the interface (air in the picture; more
generally a medium with a refractive index n2), Snellius
law is valid: |
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n1·sin
Q1 = n2·sin Q2
with n1 = index of refraction of the
semiconductor; n1 = index of refraction of the outside
(= 1 if it is air) |
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Since relevant semiconductors have
rather large refractive indexes (simply given by the square
root of the dielectric constant), refraction is quite severe. |
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As soon as Q2 reaches 90o; light will be
reflected back into the semiconductor, this happen for all angles
Q1 larger than Qcrit, the critical angle for total reflection
which os obviously given by |
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Qcrit = arcsin (n2
/ n1). |
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For typical refractive indices of 3,5
(or dielectric constants er = 12,25), we have Qcrit = 17o This is a severe
limitation of hopt: Assuming that
radiation is produced isotropically, a cone of 17o contains only
about 2% of the radiation! |
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But the situation is even worse
because photons within the critical angle may also become reflected - the
probability is <1, but not zero. |
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For the transmissivity T,
the fraction of light that does not get reflected, the following relation
holds: |
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T = 1-( |
n1 cosQ1-
n2 cosQ2
n1 cosQ1+n2 cosQ2
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)2 = 1-R |
with R = reflectivity |
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This can be simplified to an expression for the
total fraction of light leaving the semiconductor
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Ttotal» |
4 n1
n2
(n1 +n2)2
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For n1 » 3,5 and n1 = 1 (aair)
we have Ttotal = 0,69, so only about 2/3 of the
radiation contained within the critical angle leaves the semiconductor. |
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The total optical
efficiency of LEDs with isotropic generation of radiation thus is in the 1%
region - something we must worry about! |
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The simplest solution is to "grade" the
refractive index - lower it in steps. This is most easily achieved with a
"drop" of epoxy or some other polymer; how it works becomes clear
from the drawing: |
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Two light rays at the edge of some aperture have
been traced; the relevant angles are shown as pink triangles for the red light
beam. The critical angle for total reflection at both interfaces is now
considerably larger. Not that the angle in the lower index medium is always
larger and that this leads to a certain, not necessarily isotropic radiation
characteristics of the system. |
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The polymer layer, in other words, acts as an
optical system, and by giving it specific shapes we can influence the radiation
characteristics to some extent. |
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In total, we see that getting the
light out of the device (and having it more or less focussed or otherwise
influenced in its directional characteristics), is a major part of
optoelectronic technology. |
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In fact, a totally new field of research with
some bearing to these problems has recently be opened by the (first
theoretical, and then experimental "discovery" of so-called
photonic crystals) - activate the link
for some details |