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Luminescence is the word for light emission
after some energy was deposited in the material. |
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Photoluminescence describes light emission
stimulated by exposing the material with light - necessarily with higher energy
than the energy of the luminescence light. Sometimes this is also called
fluorescence if the emission happens less
than about 1 µs after the excitation and phosphorescence if it takes long - up to
hours and days. |
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Cathodoluminescence describes excitation
by energy-rich electrons, chemoluminescence provides the necessary
energy by chemical reactions. |
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Here we are interested in
electroluminescence, in particular in
injection luminescence. |
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Injection luminescence
occurs if surplus carriers are injected
into a semiconductor which then recombine via a radiating channel. |
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This implies non-equilibrium, i.e.
ne·nh > ni2 and net
recombination rates U given by the basic equation from the
Shockley-Read-Hall theory for
direct semiconductors:
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U = R -
Gtherm = v s
(ne nh -ni2) = r
ni2[ exp( |
EFe - EFh
KT
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)-1] |
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Some, but not necessarily all of the
recombination events described by U produce light, and these recombination channels are of particular interest
for optoelectronics. |
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Since optoelectronic
devices usually are made to produce plenty of light, the deviation of the
carrier concentrations from equilibrium must be large to obtain large values of
U. |
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If we write the
concentrations, as before, as
ne,h = ne,h(equ) + Dne,h, we now may use the simplest possible approximation called
high injection approximation:
Dne,h >>
nmin(equ); i.e. the minority carrier concentration is far above
equilibrium. |
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The surplus carriers
contained in Dne,h are always
injected into the volume under
consideration (called recombination
zone), usually by forward currents across a junction. They always must
come in equal numbers, i.e. in pairs to maintain charge neutrality; otherwise
large electrical fields would be generated that would restore neutrality. We
thus have |
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Dne = Dnh
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The recombination volume usually is
the space charge region of a junction or an other volume designed to have low
carriere concentrations in equilibrium. Since the equilibrium concentration of
both carrier types in the SCR are very low in the SCR, we may easily reach the
high injection case. For a bulk piece of a (doped) semiconductor this is much
more difficult - you would have to illuminate with extremely high intensity to
increase the majority carrier density by some factor. |
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The surplus concentration
of carriers decays with a characteristic lifetime t which is given by the individual life times of all
recombination channels open to the carriers. Since R >> Gtherm
for the high injection case, we have
in analogy to the approximation
made for (small) deviations from equilibrium: |
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U = R - Gtherm
=ca. R = n/t. |
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We call this approximation (where we
negelct G) "high-injection"
approximation or the high injection case because the high density of surplus
carriers is usually provided by injecting them over a forwardly biased junction
into the region of interest. |
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Note that while the rate equations
are formally the same for high or low injection (or everything in between),
t is not a constant but may depend on the degree of
injection as we will see. |
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Now we have to look at all the
possibilities for recombination - called recombination channels - that are open
for carriers as possible ways back to equilibrium. Recombination channels
generating light we will call radiative
channels. |
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The
band-band recombination
channel (with which we started above, using the full Shockley-Read-Hall
equations) can now be extremely simplified: |
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Rb-b = v·s(n2), or, considering that v·s may no longer be totally correct as the proportionality
factor, |
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Rb-b =
Bb-b(n2) - and the index "b-b"
denotes band-band recombination. The proportionality constant B is
occasionally called a recombination
coefficient. |
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If we use the same
approximations for the recombination channel via deep levels, we obtain a
rather simple relation, too, for the recombination rate
Rdl |
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Rdl =
Bdln
with Bdl = recombination coefficient for this case. |
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Recombination via
band-band transitions and via deep levels was all we considered so far. What
kind of other recombination channels are available, especially for direct
semiconductors and the high injection case? |
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There are several, some very special
and specific and only relevant for certain materials and/or doping. In this
subchapter we will look at the most important ones. |
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Before we do that,
however, we will give some thought to the equilibrium case. |
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In thermal equilibrium, we still have
generation and recombination described by the equilibrium rates Gtherm
and Rtherm and Utherm =
Gtherm - Rtherm = 0. |
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Now a tough question comes up: If recombination occurs via
band-band recombination and results in the emission of a photon, does this mean
that our piece of semiconductor, just lying there, would emit photons and thus
glow in the dark? |
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Obviously that can not be. Energy
would transported out of the semiconductor which means it would become cooler
just lying there, a clear violation of the "second law". On the other
hand, a single recombination event "does not know" if it belongs to
equilibrium or non-equilibrium, so radiation must be produced, even in
equilibrium. We seem to have a paradox. |
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The apparent paradox
becomes solved as soon as we consider that any piece of a material
"glows" in the dark (or in the bright) because it emits and absorbs
radiation leading to an equilibrium distribution of radiation intensity versus
wave length - the famous "black
body" radiation of Max
Planck fame. |
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Recombination events in equilibrium
do produce light - but the photons mostly will become reabsorbed and, in
general, will not leave the material. The small amount that does escape to the
environment must be exactly balanced by electromagnetic radiation absorbed from
the environment. |
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This topic will be
considered in more detail in an advanced
module |
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So far we considered only band-band
recombination and recombination via deep levels. There are, however, more
recombination channels, some of which are particular to compound
semiconductors. |
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But first we look at universal
mechanisms occurring in all semiconductors. They are: |
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Auger recombination. In this case the energy of
the recombination event is transferred to another electron in the conduction
band (or hole on the valence band), which then looses its surplus energy by
"thermalization", i.e. by transferring it to the phonons of the
lattice. This means that no light is produced. |
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Donor - Acceptor recombination or recombination
via "shallow levels". This
includes transitions from a donor level to an acceptor level or to the valence
band, and transitions form the conduction band to an acceptor level. |
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Mixed forms: From a donor level via a deep
level to the conduction band, etc. |
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Now
for material specific recombination channels. The most important one with
direct technical uses is recombination via "localized excitons". |
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Excitons are something like
hydrogen atoms - except that a hole and not a proton forms the nucleus. They
are thus electron - hole pairs bound by electrostatic interaction. They can
form in any semiconductor, are mobile and do not live very long at room
temperature because their binding energy is very small. They decompose
("get ionized") into a free electron and a free hole. They will not
normally disappear by recombination because they can not have the same wave
vector and thus need a third partner. |
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If you wonder why they do not simply recombine,
think about it. They can not possibly have the same wave vector (how would they
"circle" each other then?) and thus need a third partner. |
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On occasion, however, they might become trapped
at certain lattice defects and then recombine, emitting light. GaP, though an
indirect semiconductor, can be made to emit light by enforcing this
mechanism. |
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We will cone back to excitons, more about them
can be found in an advanced module. |
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The picture below
illustrates these points. |
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The picture is far too simple and we
will have to consider some of the processes in more detail later; especially
recombination via excitons. Here we look at Auger recombination and donor -
acceptor recombination. |
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Even without going into details, it is rather
clear that (radiating) donor - acceptor recombination in all 4 variants is not
all that different from direct (and radiating) band-band recombination.
Especially for relatively high doping concentrations, when the individual
energy levels from the doping atoms overlap forming a small band in the band
gap, we might simply add the dopant states to the states in the conduction or
valence band, respectively. |
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We then can treat donor-acceptor recombination as
subsets of the band-band recombination, possibly adjusting the recombination coefficient Bdl somewhat. |
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This leaves us with
Auger recombination. This is an
important recombination process that can not be avoided and that always reduces the quantum yield of radiation
production. |
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It has not been covered in the treatment of
Shockley-Read-Hall recombination
before, and we will not attempt a formal treatment here. It is, however, simple
to understand in the context of the high-injection approximation used for
optoelectronics. |
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Since you need three carriers at the same time at the same place
(the e- and h+ that recombine plus a third carrier to
remove the energy), the Auger recombination rate, RA,
must be proportional to the third power of the carrier density n,
RA = BA(n3). |
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This means that for large carrier concentrations
n (always way above equilibrium) Auger recombination sooner or later will be
the dominating process, limiting the yield of radiating transitions. |
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All recombination processes will
occur independently and the total recombination rate will be determined by the
combination of all channels. |
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The situation is totally analogous to the flow of
current through several resistors switched in parallel. The individual
recombination rates Ri add up (like the currents) and for the total
recombination rate we have |
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Rtotal = SRi = Sn/ti = nS1/ti. |
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The total recombination time ttotal is thus defined by |
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1
ttotal
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= |
1
tb-b
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+ |
1
tdl
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+ |
1
tA
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+ |
1
texciton
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+..... |
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Since we are only interested in radiative and
non-radiative channels, we may write this as
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1
ttotal
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= |
1
trad
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+ |
1
tnon-rad
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and
Rtotal =
Rrad + Rnon-rad = |
n
trad
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+ |
n
tnon-rad
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The quantum
efficiency hqu introduced
before now can be calculated. It is given by the fraction of
Rrad relative to Rtotal, or |
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hqu = |
Rrad
Rtotal
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= |
1/trad
1/trad +tnon-rad
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, or
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hqu = |
1
1+ trad /tnon-rad
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That is easy enough, but now need
some numbers for the recombination coefficients in order to get some feeling
for what is going on in different semiconductors. |
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It should be clear that the Bi
defined above are related to quantities like the thermal velocity, the capture
cross sections, the density of deep (and shallow) levels, and so on - they
depend to some extent on the particular circumstances of the semiconductor
considered. e.g. doping, cleanliness, defect density, etc. |
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It should also be clear the Bi
are not absolute constants for a given materials but only useful as long as
the approximations used are holding. in other words, there are no universal
numbers for a certain semiconductor. We only can give typical numbers. |
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With this disclaimers in mind, we use the
following values (if two numbers are included, they come from different
sources). Yellow denotes the indirect semiconductors and the GaP value is for
the very unikely direct recombination without excitons. |
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(T = 300K) |
Si |
Ge |
GaAs |
InP |
GaP |
B |
t
[µs] |
B |
t
[µs] |
B |
t
[µs] |
B |
t
[µs] |
B |
t
[µs] |
Bdl
[s-1] |
1·105 |
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1·108 |
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Bb-b
[cm3s-1] |
1·10-14
1,8·10-15 |
5500 |
5,3·10-14 |
200 |
3·10-10
7,2·10-10 |
0,015 |
1,26·10-9 |
0,008 |
5,4·105 |
2000 |
BA
[cm6s-1] |
2·10-32 |
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1·10-27 |
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Now we can construct a recombination rate -
surplus carrier concentration diagram as follows: |
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We can see a few interesting
points: |
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The recombination rate in Si is generally much
smaller than in GaAs - a direct effect of the much larger lifetimes. |
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Direct recombination in Si is not strictly
forbidden - it is just very unlikely. At a typical carrier concentration of
1018 cm-3 we have about 1022 photons generated
per s and cm-3 compared to about 5·1026 in
GaAs. |
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Rb-b in GaAs is comparable to
the recombination rates of the Auger and deep level channels at concentrations
of about (3 - 4) ·1017 cm-3; while in Si
Rb-b is always much smaller than the other recombination
rates. |
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While for large carrier concentrations the Auger
recombination process always dominates, it may still be useful to increase n:
While the quantum efficiency goes down, the amount of light produced still
increases linearly with n. |
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For very large carrier concentrations (say
1019 cm-3 and beyond as occasionally encountered in power
circuits), even Si may produce some visible light. |
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The GaAs curves now provide a first
answer to our second question about the quantum
efficiency. |
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For n = 1616 cm-3,
we have about 3·1022 radiating recombination events per s and
cm-3 out of a total of about 1024per s and
cm-3 which gives a quantum efficiency of 3%. |
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At the high concentration end, around n =
1616 cm-3, the situation is similar, the quantum
efficiency is in the few percent range. |
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The highest quantum efficiency is around 30 % for
concentrations around n = 5·1017 cm-3. |
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Of course, given the values of the
recombination coefficients, we could calculate the quantum efficiency
precisely, but that would not be very helpful because real devices are more
sophisticated than the simple forwardly biased junction implicitly assumed in
this consideration. |
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This means that we now must look more closely at
the important compound semiconductors, especially on how they are doped and
what typical differences to Si occur. |
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We will, however, first do a little exercise for
injection across a straight p-n-junction in order to get acquainted with some
real numbers for carrier densities produceable by injection. |