High Injection Approximations for Recombination Rates
Luminescence is the word for light emission after some energy was deposited in the material.  
Photoluminescence describes light emission stimulated by exposing the material to light – by necessity with a higher energy than the energy of the luminescence light. Photoluminescence is also called fluorescence if the emission happens less than about 1 µs after the excitation, and phosphorescence if it takes long times – up to hours and days – for the emission.  
Cathodoluminescence describes excitation by energyrich electrons , chemoluminescence provides the necessary energy by chemical reactions.  
Here we are interested in electroluminescence , in particular in injection luminescence.  
Injection luminescence occurs if surplus carriers are injected into a semiconductor which then recombine via a radiative channel.  
This implies nonequilibrium, i.e. n_{e} · n_{h} > n_{i}^{2} and net recombination rates U given by the basic equation from the ShockleyReadHall theory for direct semiconductors:  
 
Some, but not necessarily all of the recombination events described by U produce light, and these radiative recombination channels are of particular interest for optoelectronics.  
Since optoelectronic devices usually are made to produce plenty of light, the deviation of the carrier densities from equilibrium must be large to obtain large values of U.  
If we write the densities, as before, as n_{e,h} = n_{e,h}(equ) + Dn_{e,h}, we now may use the simplest possible approximation called high injection approximation:  
 
i.e. the minority carrier density is far above equilibrium.  
That is different from the approximation made
before, where we assumed that 

The surplus carriers contained in Dn_{e,h} are always injected into the volume under consideration (called recombination zone or recombination volume), 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  
 
The recombination volume usually is the space charge region of a junction or an other volume designed to have low carrier densities in equilibrium. Since the equilibrium density of both carrier types in the SCR is automatically very low, 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 minority carrier density by some factor.  
The surplus density 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  
 
We call this approximation (where we neglect G) "highinjection " 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.  
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).  
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.  
The bandband recombination channel (with which we started above, using the full ShockleyReadHall equations) can now be extremely simplified:  
 
or, considering that v · s may no longer be totally correct as the proportionality factor,  
 
and the index "bb " denotes bandband recombination. The proportionality constant B is occasionally called a recombination coefficient.  
If we use the same approximations for the recombination channel via deep levels, we obtain a rather simple relation, too, for the recombination rate R_{dl}  
 
With B_{dl} = recombination coefficient for this case.  
Recombination via bandband 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?  
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.  
Before we do that, however, we will give some thought to the equilibrium case.  
In thermal equilibrium, we still have generation and recombination
described by the equilibrium rates G_{therm
} and R_{therm} and  
Now a tough question comes up: If recombination occurs via bandband 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?  
Obviously that can not be. Energy would be 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 nonequilibrium, so radiation must be produced, even in equilibrium. We seem to have a paradox .  
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.  
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.  
This topic will be considered in more detail in an advanced module.  
Additional Recombination Channels  
So far we considered only bandband recombination and recombination via deep levels. There are, however, more recombination channels, some of which are particular to compound semiconductors.  
But first we look at universal mechanisms occurring in all semiconductors. They are:  
Auger recombination . In this case the energy of the recombination event is transferred to another electron in the conduction 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.  
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.  
Mixed forms: From a donor level via a deep level to the valence band, etc.  
Now for material specific recombination channels. The most important one with direct technical uses is recombination via "localized excitons".  
Excitons are something like hydrogen atoms (or, even closer in similarity, positronium = atom consisting of electron and positron) – except that a hole and not a proton is the partner of the electron. 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.  
If you wonder why they do not simply recombine, think about it. They can't possibly have the same wave vector (how would they "circle" each other then?) and thus need a third partner for the recombination process to occur.  
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.  
We will come back to excitons later; more about them can be found in an advanced module.  
The picture below illustrates these points.  

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.  
Even without going into details, it is rather clear that (radiative) donor–acceptor recombination in all 3 variants (note that two of the four arrows shown in the drawing above refer to equivalent processes) is not all that different from direct (and radiative) band–band recombination. Especially for relatively high doping densities, 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.  
We then can treat donoracceptor recombination as subsets of the bandband recombination, possibly adjusting the recombination coefficient B_{bb} somewhat.  
This leaves us with Auger recombination. This is an important recombination process that cannot be avoided and that always reduces the quantum yield of radiation production.  
It has not been covered in the treatment of ShockleyReadHall recombination before, and we will not attempt a formal treatment here. It is, however, simple to understand in the context of the highinjection approximation used for optoelectronics.  
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, R_{A}, is proportional to the third power of the carrier density n:  
 
This means that for large carrier densities n (always way above equilibrium), and therefore large doping, Auger recombination sooner or later will be the dominating process, limiting the yield of radiative transitions.  
Total Recombination and Quantum Yield  
All recombination processes will occur independently and the total recombination rate will be determined by the combination of all channels.  
The situation is totally analogous to the flow of current through several resistors switched in parallel. The individual recombination rates R_{i} add up (like the currents) and for the total recombination rate we have  
 
The total recombination time t_{total} is thus defined by  
 
Since we are only interested in radiative and nonradiative channels, we may write this as  
 
The quantum efficiency h_{qu} introduced before now can be calculated. It is given by the fraction of R_{rad} relative to R_{total}, or  
 
Obviously, the result is  
 
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.  
It should be clear that the B_{i} 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.  
It should also be clear the B_{i} 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.  
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 unlikely direct recombination without excitons.  
 
Now we can construct a recombination rate vs. surplus carrier density diagram as follows:  
 
We can see a few interesting points:  
The recombination rate in Si is generally much smaller than in GaAs – a direct effect of the much larger lifetimes.  
Direct recombination in Si is not strictly forbidden – rather, it is just unlikely. At a typical carrier density of 10^{18} cm^{–3} we have about 10^{22} photons generated in Si per s and cm^{3} compared to about 3 · 10^{26} in GaAs.  
R_{bb} in GaAs is similar to the recombination rates of the Auger and deep level channels at densities around 4 · 10^{17} cm^{–3}, whereas in Si, for most densities R_{bb} is much smaller than the other recombination rates.  
Although for large carrier densities 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 with n.  
For very large carrier densities (say 10^{19} cm^{–3} and beyond as occasionally encountered in power circuits), even Si may produce some visible light.  
The GaAs curves now provide a first answer to our second question about the quantum efficiency.  
For n = 10^{16} cm^{–3}, we have about 4 · 10^{22} radiative recombination events per s and cm^{3} out of a total of about 10^{24} per s and cm^{3} , which gives a quantum efficiency of 4 %.  
At the high density end, around n = 10^{19} cm^{–3}, the situation is similar, the quantum efficiency is in the few percent range.  
The highest quantum efficiency is around 30 % for densities around n = 5 · 10^{17} cm^{–3}.  
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.  
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.  
We will, however, first do a little exercise for injection across a straight pn junction in order to get acquainted with some real numbers for carrier densities produceable by injection.  

© H. Föll (Semiconductors  Script)