LASERs and Stimulated Emission of Radiation
In principle, anything that emits electromagnetic radiation can be turned into a "LASER", but what is a Laser?  
The word "Laser" was (and of course still is) an acronym, it stands for "Light Amplification by Stimulated Emission of Radiation" By now, however, it is generally perceived as a standard word in any language meaning something that is more than the acronym suggests (and we will no longer write it with capital letters)!  
A Laser in the direct meaning of the acronym is a black box that emits (= outputs) more light of the same frequency than what you shine ( = input) on it  that is the amplifier part. But the "stimulated emission" part, besides being the reason for amplification, has a second indirect meaning, too: The light emitted is exactly in phase (or coherent to) the light in the input. Unfortunately, Lasers in this broad sense do not really exist. Real Lasers only amplify light with a very specific frequency  its like electronic amplifiers for one frequency only.  
A Laser in the general meaning of the acronym thus produces intense monochromatic electromagnetic radiation in the wavelength region of light (including infrared and a little ultra violet; there are no sharp definitions) that it is coherent to the (monochromatic) input. If you "input" light containing all kinds of frequencies, only one frequency becomes amplified.  
A Laser in the specific meaning of everyday usage of the word, however, is more special. It is a device that produces a coherent beam of monochromatic light in one direction only and, at least for semiconductor Lasers, without some input light (but with a "battery" or power source hooked up to it). It is akin to an electronic oscillator that works by internally feeding back parts of the output of an amplifier to the input for a certain frequency.  
Before the advent of hardware Lasers in the sixties, there were already "Masers"  just take the "M" for "microwave" and you know what it is.  
And even before that, there was the basic insight or idea behind Masers and Lasers  and, as ever so often  it was A. Einstein who described the "Stimulated Emission part in 1917/1924. More to the history of Lasers can be found in an advanced module.  
Obviously, for understanding Lasers, we have to consider stimulated emission first, and then we must look at some feedback mechanism.  
Understanding stimulated emission is relatively easy; all we have to do is to introduce one more process for the interaction between light and electrons and holes. So far we considered two basic processes, to which now a third one must be added:  
1. Fundamental absorption, i.e. the interaction of a photon with an electron in the valence band resulting in a electron(C)  hole(V) pair.  
2. Spontaneous emission of a photon by the (spontaneous and direct) recombination of an electronhole pair.  
3. Stimulated emission, as the third and new process, is simply the interaction of a photon with an electron in the conduction band. It forces recombination and thus the emission of a second photon.  
All three processes are schematically shown in the band diagram below.  


Looking at this picture, you should wonder why one obvious process is missing? How about an electron in the conduction band simply absorbing a photon?  
In other words: An electron in the conduction band absorbs a photon, moves up the amount h · n in the conduction band, and comes back to the band edge by tranferring its surplus energy to phonons.  
This process does take place, but is not very strong if we do not have many electrons in the conduction band. More important: It is not necessary for "lasing", but rather detrimental  we will cover it later.  
Stimulated emission, however, is not just the reverse of absorption. Photons usually interact with electrons in the conduction band by transferring their energy to the electron, which moves the electron to some higher energy level in the band (or to the next band, or, if the photons are very energetic (meaning Xrays), even out of the crystal). In other words: The photons are absorbed.  
Stimulated emission is a resonant process; it only works if the photons have exactly the right energy, corresponding to the energy that is released if the electron makes a transition to some allowed lower level. This also means that the two photons are exactly in phase with each other. For semiconductors, this is pretty much the energy of the band gap, because all conduction band electrons are sitting at the conduction band edge (with some small DE, of course), and the only available lower energy level are the free positions occupied by holes at the valence band edge.  
Stimulated emission thus may be seen as a competing process to the fundamental bandband absorption process described before. But while all photons with an energy hn > E_{g} may cause fundamental absorption because there are many unoccupied levels above E_{g}, only photons with hn = E_{g} (give or take some small DE) may cause stimulated emission.  
Einstein showed that under "normal" conditions (meaning conditions not too far from thermal equilibrium), fundamental absorption by far exceeds stimulated emission. Of course, Einstein did not show that for semiconductors, but for systems with well defined energy levels  atoms, molecules, whatever.  
However, for the special case that a sufficiently large number of electrons occupies an excited energy state  this is called inversion,  stimulated emission may dominate the electronphoton interaction processes. Then two photons of identical energy and being exactly in phase come out of the system for one photon going into the system.  
The kind of inversion we are discussing here should not be mixed up with the inversion that turns ntype Si into ptype or vice versa that we encounterd before. Same word, but different phenomena!  
These two photons may cause more stimulated emission  yielding 4, 8, 16, ... photons, i.e. an avalanche of photons will be produced until the excited electron states are sufficiently depopulated.  
In other words: One photon hn impinging on a material that is in a state of inversion (with the right energy difference hn between the excited state and the ground state) may, by stimulated emission, cause a lot of photons to come out of the material. Moreover, these photons are all in phase, i.e. we have now a strong and coherent beam of light  amplification of light occurred!  
We are now stuck with two basic questions:  
1. What exactly do we mean with "inversion", particularly with respect to semiconductors?  
2. How do we induce a state of "inversion" in semiconductors?  
Let's look at these questions separately  
Obtaining Inversion in Semiconductors  
If you shine 10 input photons on a crystal, 6 of which disappear by fundamental absorption, leaving 4 for stimulated emission, you now have 8 output photons. In the next round you have 2 · (8 · 0,4) = 6,4 and pretty soon you have none.  
Now, if you reverse the fractions, you will get 12 photons in the first round, 2 · (12 · 0,6) = 14,4 the next round  you get the idea.  
In other words, the coherent amplification of the input light only occurs for a specific condition (the light eventually produced by recombination of the electron hole pairs generated by fundamental absorption is not coherent to the input and does not count!):  
There must be more stimulated emission processes than fundamental absorption processes if we shine light with E = hn = E_{g} on a direct semiconductor  this condition defines "inversion" in the sense that we are going to use it.  
We only look at direct semiconductors, because radiant recombination is always unlikely in indirect semiconductors, and while stimulated emission is generally possible, it also needs to be assisted by phonons and thus is unlikely, too.  
We will find a rather simple relation for the dominance of stimulated emission, but it is not all that easy to derive. Here we will take a "shortcut", leaving a more detailed derivation to an advanced module.  
Lets first consider some basic situations for inversion in full generality. For the most simple system, we might have two energy levels E_{1} and E_{2} for atoms (take any atom), the lower one (E_{1}) mostly occupied by electrons, the upper one (E_{2}) relatively empty. Inversion then means that the number of electrons on the upper level, n_{2}, is larger or at least equal to n_{1}. 


In equilibrium, however, we would simply have  


With DE = E_{1} – E_{2}, and D_{1,2} = the maximum number of electrons allowed on E_{1,2} (the "density of states").  
In words: In equilibirum we have far more electrons at E_{2} than at E_{1}  
For inversion to occur, we must be very far from equilibrium if DE is on the order of 1 eV as needed for visible light. In fact, the systems would have a negative temperature for such a distribution (this is something you should figure out by yourself).  
Stimulated emission would quickly depopulate the E_{2} levels, while fundamental absorption would kick some electrons back. Nevertheless, after some (short) time we would be back to equilibrium.  
To keep stimulated emission going, we must move electrons from E_{1} to E_{2} by some outside energy source. Doing this with some other light source providing photons of the only usable energy DE would defeat the purpose of the game; after all that is the light we want to generate. In semiconductors we could inject electrons from some other part of the device, but a twolevel system is not a semiconductor, so that is not possible.  
In short: Two level systems are no good for practical uses of stimulated emission  
What we need is an easy way to move a lot of electrons to the energy E_{2}. This can be achieved in a three level system as shown below (and this was the way it was done with the first ruby Laser).  
The essential trick is to have a whole system of levels  ideally a band  above E_{2}, from which the electrons can descend efficiently to our single level E_{1}  but not easily back to E_{2} where they came from. Schematically, this looks like the figure on the right. 


The advantage is obvious. We now can take light with a whole range of energies  always larger than DE  to "pump" electrons up to E_{2} via the reservoir provided by the third level(s).  
The only disadvantage is that we have to take the electrons from E_{1}. And no matter how hard we pump (this is the word used for this process), the probability that a quantum of the energy we pour into the system by pumping will find an electron to act upon, will always be proportional to the number (or density) of electrons available for kicking up to E_{2}. In the three level system this is still at most D_{1}. If we sustain the inversion, it is at most 0,5 · D_{1}, because by definition we have at least onehalf of the available electrons on E_{2}.  
It is clear what we have to do:
Provide a fourth level (even better: A band
of levels) below
E_{1}, where you have a lot of electrons that can be
kicked up to E_{2} via the third 

We simply introduce a system of energy states below E_{1} in the picture from above. We now have a large reservoir to pump from, and a large reservoir to pump to. 


All we have to do is to make sure that pumping is a oneway road, i.e. that there are no (or very few) transitions from the levels 3 to levels 4.  
This is not so easy to achieve with atoms or molecules, but, as you should have perceived by now, this is exactly the situation that we have in many direct band gap semiconductors. All we have to do to see this, is to redraw the 4  level diagram at the right as a band diagram. To include additional information, we do this in kspace.  
We have the following general situation for producing inversion in semiconductors:  


Electrons may be pumped up from anywhere in the valence band to anywhere in the conduction band  always provided the transition goes vertically upwards in the reduced band diagram.  
The electrons in the conduction band as well as the holes in the valence band will quickly move to the extrema of the bands  corresponding to the levels E_{2} and E_{1} in the general four level system.  
"Quickly" means within a time scale defined by the dielectric relaxation time. This time scale is so small indeed that it introduces some uncertainties in the energies via the uncertainty relation which is considered in the advanced module but need not bother us here.  
We have now everything needed for a "quick and dirty" derivation for the inversion condition in the sense introduced on top.  
The Inversion Condition
The condition for inversion was that there where at least as many stimulated emission processes as fundamental absorption processes. The recombination rate by stimulated emission we now denote R_{se}, and the electronhole pair generation rate by fundamental absorption is R_{fa} We thus demand:  


In general, fundamental absorption and stimulated emission can happen in a whole range of frequencies for semiconductors. While we expect that the electrons that are being stimulated to emit a photon will occupy levels right at the conduction band edge, stimulated emission is not forbidden for electrons with a higher energy somewhere in the conduction band. While these electrons are in the (fast) process of relaxing to E_{C}, they still might be "hit" by a photon of the right energy at the right time and place  it is just more unlikely that at E_{C}.  
We thus must expect both rates, R_{se} and R_{fa}, to be proportional to:  
1. The spectral intensity of the radiation in the interesting frequency interval.  
The differential frequency interval considered extends from n to n + Dn; the spectral intensity in this interval we name u(n)Dn or, expressing the frequency n in terms of energy via E_{phot} = hn, u(E)DE.  
The value of u(E) times DE essentially gives the number of photons in this frequency interval.  
2. The density of states available for the processes.  
The probability that a photon with a certain frequency n and therefore energy E_{phot} = hn will be absorbed by an electron at some position E ^{v} in the valence band, will be proportional to the density of states in the valence band, D_{V}(E ^{v}) and to the density of states exactly E_{phot} above this position in the conduction band, i.e. D_{V}(E ^{v} + hn).  
Contrariwise, the probability that emission takes place stimulated by a photon with energy hn, is proportional to the density of states in the conduction band and to the density of states at E ^{c} – hn below in the valence band.  
This is a crucial part of the consideration  and a rather strange one, too. That both densities of states must be taken into account  where the particle is coming from and where it is going to  is a quantum mechanical construct (known as Fermis golden rule) that has no classical counterpart.  
3. The probability that the states are actually occupied (or unoccupied).  
The density of states just tells us how many electrons (or holes) might be there. The important thing is to know how many actually are there  and this is given by the probability that the states are actually occupied (necessary for absorption) or unoccupied (necessary for the transition of the electrons to this state).  
In other words, the Fermi Dirac distribution comes in. In the familiar nomenclature we write it as f(E, E_{F}^{e}, T) or f(E, E_{F}^{h}, T) with E_{F}^{e,h} = Quasi Fermi energy for electrons or holes, respectively.  
The crucial point is that we take the quasi Fermi energies, because we are by definition treating strong nonequilibrium between the bands, but (approximately) equilibrium in the bands.  
We also, for ease of writing define a direct Fermi distribution for the holes as outlined before and distinguish the different distributions by the proper index:  


That is all. However, the density of states are complicated functions of E ^{v} and E ^{c}, and the spectral density of the radiation we do not know  it is something that should come out of the calculations.  
But we are doing shortcuts here, and we do know that the radiation density will have a maximum around hn = E_{g} = E_{C} – E_{V}. So lets simply assume that the necessary integrations over u(E) · D(E)DE will be expressible as N_{eff} · u(n) · Dn with N_{eff} = effective density of states. Moreover, we assume identical N_{eff} in the valence and conduction band.  
The rates R_{se} for stimulated emission and R_{fa} for fundamental absorption than can be written as  


The A_{fa} and the A_{se} are the proportionality coefficients and we always use f_{h in V} if we discuss carriers in the valence band and f_{e in C} if we discuss the conduction band.  
Enter Albert Einstein. He showed in 1917 that the following extremely simple relation always holds for fundamental reasons:  


We will just accept that (if you don't, turn to the advanced module for a derivation) and now form the ratio R_{se}/R_{fa}. Most everything then just drops out and we are left with  


With some shuffling of the terms (see the exercise below) we obtain  


with E_{C}^{c} and E_{V}^{v} denoting some energy level in the conduction or valence band, respectively.  
This is a rather simple, but also rather important equation. It says that we have more stimulated emission between E ^{c}_{C} and E ^{v}_{V} than fundamental absorption between E ^{v}_{V} and E ^{c}_{C}, if the difference in the quasi Fermi energies is larger than the difference between the considered energy levels.  
The smallest possible difference between some energy levels in the valence band and some energy levels in the conduction band that are connected by a direct transition is E_{g} for direct semiconductors.  
Since we are also most interested in the stimulated emission from E_{C} to E_{V} , we have as the first Laser condition:  


We call this "Laser condition", because "lasing" requires inversion, i.e. at least as many electrons at the conduction band edge as we have electrons (not holes!) at the valence band edge. 
It is clear that this involves heavy nonequilibrium conditions.  
We need to inject a lot of electrons into the conduction band and a lot of holes ( = taking electrons out) into the valence band.  
And we have to keep the injection rates at least as large as the stimulated emission rate, i.e. we have to supply electrons (and holes) just as fast as stimulated emission takes them away if we want to keep the rate of radiation constant.  
Now we know what is needed to obtain light amplification in principle. But how much amplification do we get from a piece of semiconductor kept in inversion? This will be the topic in the next subchapter.  


© H. Föll (Semiconductor  Script)