6. Principles of the Semiconductor LASER

6.1 LASER conditions

6.1.1 Interaction of Light and Electrons and Inversion

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 electron-hole 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.
Photon-electrOn-interactions
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 X-rays), 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 band-band absorption process described before. But while all photons with an energy hn > Eg may cause fundamental absorption because there are many unoccupied levels above Eg, only photons with hn = Eg (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 electron-photon 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 n-type Si into p-type 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 = Eg 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 "short-cut", 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 E1 and E2 for atoms (take any atom), the lower one (E1) mostly occupied by electrons, the upper one (E2) relatively empty. Inversion then means that the number of electrons on the upper level, n2, is larger or at least equal to n1.  
TwO level system
 Two level system in inversion condition
In equilibrium, however, we would simply have
 
n1
n2
 =  D1
D2
 · exp – DE
kT
   
With DE = E1E2, and D1,2 = the maximum number of electrons allowed on E1,2 (the "density of states").
In words: In equilibirum we have far more electrons at E2 than at E1
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 E2 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 E1 to E2 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 two-level 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 E2. 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 E2, from which the electrons can descend efficiently to our single level E1 - but not easily back to E2 where they came from. Schematically, this looks like the figure on the right.  
Three level Laser
The advantage is obvious. We now can take light with a whole range of energies - always larger than DE - to "pump" electrons up to E2 via the reservoir provided by the third level(s).
The only disadvantage is that we have to take the electrons from E1. 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 E2. In the three level system this is still at most D1. If we sustain the inversion, it is at most 0,5 · D1, because by definition we have at least one-half of the available electrons on E2.
 
It is clear what we have to do: Provide a fourth level (even better: A band of levels) below E1, where you have a lot of electrons that can be kicked up to E2 via the third level(s). It is clear that we are talking semiconductors now, but lets first see the basic system:
 
We simply introduce a system of energy states below E1 in the picture from above. We now have a large reservoir to pump from, and a large reservoir to pump to.  
FOur level Laser
All we have to do is to make sure that pumping is a one-way 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 k-space.
 
We have the following general situation for producing inversion in semiconductors:
 
SemicOnductor asfOur level Laser structure
 
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 E2 and E1 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 Rse, and the electron-hole pair generation rate by fundamental absorption is Rfa We thus demand:
Rse   ³ Rfa
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 EC, 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 EC.
We thus must expect both rates, Rse and Rfa, 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 Ephot = 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 Ephot = 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, DV(E v) and to the density of states exactly Ephot above this position in the conduction band, i.e. DV(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, EFe, T) or f(E, EFh, T) with EFe,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 non-equilibrium 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:
fe or h(E, EFe,h, T)     =  probability that some level at energy E
is occupied by an electron or hole
       
1  –  fe or h(E, EFe,h, T)    =    probability that some level at energy E
is not occupied by an electron or hole
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 = Eg = ECEV. So lets simply assume that the necessary integrations over u(E) · D(E)DE will be expressible as Neff · u(n) · Dn with Neff = effective density of states. Moreover, we assume identical Neff in the valence and conduction band.
The rates Rse for stimulated emission and Rfa for fundamental absorption than can be written as
  Rfa  =  Afa · Neff2 · u(n) · Dn · æ
è
1  –  fh in V (E v, EFh, T) ö
ø
 ·  æ
è
1  –  fe in C (E c, EFe, T) ö
ø

Rse  =  Ase · Neff2 · u(n) · Dn · æ
è
fe in C (E C, EFe, T) ö
ø
 ·  æ
è
fh in V (E V, EFh, T) ö
ø
The Afa and the Ase are the proportionality coefficients and we always use fh in V if we discuss carriers in the valence band and fe 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:
Afa  =  Ase
We will just accept that (if you don't, turn to the advanced module for a derivation) and now form the ratio Rse/Rfa. Most everything then just drops out and we are left with
Rse
Rfa
  =  [1  –  fh in V (E v, EFh, T)] · [1  –  fe in C (E c, EFe, T)]
[fe in C (E c, EFe, T)] · [fh in V (E v, EFh, T)]
With some shuffling of the terms (see the exercise below) we obtain
Rse
Rfa
  =  EFe  –  EFh
ECc  –  EVv
with ECc and EVv 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 cC and E vV than fundamental absorption between E vV and E cC, 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 Eg for direct semiconductors.
Since we are also most interested in the stimulated emission from EC to EV , we have as the first Laser condition:
EFeEFh  ³  Eg  ³    hn
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 non-equilibrium 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 sub-chapter.
Exercise 6.1-1
Do the Math for the 1st Laser condition
 

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