As in the "short-cut" derivation of the inversion condition, we look at the number of stimulated emission and fundamental absorption processes per time and volume unit (i.e., the rate density). We use the same symbols, however, with some additional indices if necessary. We have: | |||||||||||||||||||||||||||||||||||||

rate for R_{fa}(E_{1}, E_{2}) =fundamental
absorption of a photon by an electron sitting in the valence band in the energy interval defined by
and E_{1} that is then moved "up" into
the conduction band to the energy interval E_{1} + DE_{1} and under the condition that the wave vector of the electron (in a reduced
band diagram) does not change.E_{2}, E_{2} + DE
_{2} | |||||||||||||||||||||||||||||||||||||

rate for R_{ se}(E_{1}, E_{2}) =stimulated emission
of a photon by a photon from an electron sitting in the conduction band in the energy interval defined by
and E_{2} that is then moved "down" into the valence band to the energy interval E_{2} + DE
_{2} and under the condition that the wave vector
of the electron (in a reduced band diagram) does not change.E_{1},
E_{1} + DE_{1} |
|||||||||||||||||||||||||||||||||||||

We now define the rates relative to an energy interval
because we will see that the photon does not need to have exactly the energy h ·
n = to induce transitions – this follows from E_{2} – E_{1}
Heisenberg's
uncertainty relations. | |||||||||||||||||||||||||||||||||||||

As before, the rates must be proportional to the number of electrons available for the interaction with
a photon and the number of empty states available for the electrons to occupy after
the interaction | |||||||||||||||||||||||||||||||||||||

At the respective energies, the density in some small energy interval D
(ideally being zero) around some energy value E or E
_{2} in the conduction or valence band, is always
given by the density of states at the chosen energy times probability of occupations times small energy interval, i.e.
E_{1} | |||||||||||||||||||||||||||||||||||||

| |||||||||||||||||||||||||||||||||||||

The density of states are functions of D_{C, V}, of
course.E_{2, 1} | |||||||||||||||||||||||||||||||||||||

Since we use the quasi Fermi energies, we have to use two
different Fermi distributions for the two quasi Fermi energies; we have f( and E, E_{F}^{ e},
T)f()E, E_{ F}^{h} , T | |||||||||||||||||||||||||||||||||||||

However, for a full appreciation of the symmetries in the problem, it may be advantageous to define alltogether
four Fermi distributions, two of which we then will really use as already pointed out when quasi Fermi energies were introduced. |
|||||||||||||||||||||||||||||||||||||

Besides f, giving the _{e}(E, E_{F}^{e}, T)direct probability of finding electrons in the conduction band, we simply define the direct probability for finding holes in the valence band as f. Watch out for the indizes _{
h}(E, E_{F}^{h}, T) = 1 – f_{e}(E , E_{F}^{
h}, T) and ^{h} in this!_{e} | |||||||||||||||||||||||||||||||||||||

Next we simply agree to use only
f if we discuss what is going on in the valence band, and _{h}only
f for the conduction band._{e} | |||||||||||||||||||||||||||||||||||||

Using this convention, the equations from above, augmented by two more equations to account for all densities, we have | |||||||||||||||||||||||||||||||||||||

| |||||||||||||||||||||||||||||||||||||

This should be clear, simply recall that f is the _{h}
(E_{1}, E_{F}^{h}, T)direct probability of finding holes in ; we therefore must take E_{1}DE_{1}
(1 – f for the probability of finding _{h}(E_{1} , E_{
F}^{h}, T)electrons
if we only use f in the valence band. _{h}(E_{1},
E_{F}^{h}, T) | |||||||||||||||||||||||||||||||||||||

This allows us to formulate the electron and hole density part of the rates we are trying to calculate.
R | |||||||||||||||||||||||||||||||||||||

These rates are also proportional to the number of photons with the energy hn
»
, i.e., to E_{2} – E_{1}. We use u(n )"»" instead of "=", because one of the goals of this module is to see
how far off the photon energy could be from the band gap energy while still giving some effects. | |||||||||||||||||||||||||||||||||||||

Using the proportionality constants and A_{fa}
, we now can formulate the same
equations as we had before, but in a more precise and general way (omitting the temperature to save space):A_{se} | |||||||||||||||||||||||||||||||||||||

| |||||||||||||||||||||||||||||||||||||

The terms in the large brackets are (from left to right)- The proportionality coefficients or Einstein coefficients
- The density of unoccupied states (= density of holes) available for the electrons at the end point of the transition.
- The density of electrons present for interactions with the photons at the energy interval considered, i.e. the occupied states.
- The density of photons available for the processes considered
| |||||||||||||||||||||||||||||||||||||

This kind of relation for a transition probability of a particle from one state to another state is an
example of an universal principle known as "Fermi's
golden rule". While it is very easy to grasp the way it is presented here,
it is actually a bit strange: What the electrons do depends on the density of unoccupied energy levels
somewhere and sometime else. How do the electrons "know" if there
is some free space "down there"? More to that in another advanced module. |
|||||||||||||||||||||||||||||||||||||

Einstein showed that A_{se}
= A_{fa} = A =Einstein coefficient
for fundamental absorption under all circumstances. We will derive this relation in another
modul – it is not so easy (Einstein was a great physicist indeed). |

We will now calculate , the R^{net}_{se}net
rate of stimulated emission, which is | ||||||||||||||||||

| ||||||||||||||||||

Using the formula from above and gong back to the regular Fermi distribution, we obtain | ||||||||||||||||||

| ||||||||||||||||||

We now must look more closely at the D
terms and consider their physical significance. E_{1, 2} | ||||||||||||||||||

The key point is that we are looking at electron (or hole) states that are very short lived.
Consider the electron that was knocked into the state with the energy in the conduction band
(or the hole state left at E_{2}). They will disappear very quickly (within a time given by the dielectric relaxation time of the system,
which is around E_{ 1}10 for typical optoelectronic semiconductors).^{–12} s | ||||||||||||||||||

In other words, the electron in the conduction band will very quickly thermalize by interactions
with phonons and settle at energies close to .E_{ C} | ||||||||||||||||||

Heisenberg's uncertainty principle for energies states that | ||||||||||||||||||

| ||||||||||||||||||

This means that for electrons occupying some state for some small
average (= uncertain individual) time E_{2}Dt, we have an uncertainty in the precise value
of the energy. It is not given precisely, but only within some uncertainty D. E = h/Dt | ||||||||||||||||||

More precisely, things can happen even if the energy available is not precisely what is needed
- all the it is required that it happens so fast (small Dt) that the difference in energy
is covered by the resulting uncertainty. In other words, there is a certain likelihood that transitions may occur for not
exactly matching energies, too |
| |||||||||||||||||

In yet other words: A photon with the precise energy of h · n may
create electron–hole pairs with energies of h · n ± D
with some probability that depends on ED and EDt
. | ||||||||||||||||||

This means that there is a certain probability distribution L( for obtaining a transition even if the photon does not have the exactly right energy E_{ 2}, E_{1}, t_{C}, t
_{V})hn
= , but deviates by some E_{2} – E_{1}D.
E | ||||||||||||||||||

We cannot calculate that distribution at this point, but we can be pretty sure that it must be something as shown in the picture on the right. | ||||||||||||||||||

In total, we have the possibility that photons somewhat off the proper energy still may induce a transition.
The probability, however, will go down rapidly as the deviation D is increased or E
t increases. | ||||||||||||||||||

The reverse then is also true: Photons with only one well defined energy
hn will cause transitions not only between energy states , but also between states with an energy E_{2} –
E_{1} = hn( and the probability for some E_{2}
– E_{1}) ± DED
is given by the EL curve. | ||||||||||||||||||

The total
transition rates of electrons in response to photons with some density )
at the frequency u(nn is thus obtained by integrating over all possible transitions weighted
by the probability function L(hn). If we include this into the formula for the net emission
rate we obtain | ||||||||||||||||||

| ||||||||||||||||||

and we have to integrate over the valence and conduction band. | ||||||||||||||||||

Now we reached the end of our tether. Because we neither know L(, nor could we easily do the integration
if we would know it (it must be, after all, some bell-shaped kind of probability distribution).E_{2}, E_{1}
, t_{C}, t_{V}) | ||||||||||||||||||

We thus introduce the first approximation into our so far rather rigid derivations: We assume
a d-function for L; in other words, we neglect the energy uncertainties and allow
only transitions with identical energies to occur. This does not mean that the energy
levels are now fixed, only that the energy level you reach from some level is fixed. | ||||||||||||||||||

One integration now becomes trivial. We integrate over
the conduction band, rewrite as E_{2}
(we can do that now because we now have a sharp level in the conduction band), and obtainE_{1} + hn | ||||||||||||||||||

| ||||||||||||||||||

Far simpler, but his equation still contains an integration over the density of states, which is not so easy to do even in the simplest approximations. | |||||||||||||

In other words: We still allow transitions from different
energy states to the corresponding states hn larger in energy (and always at the
same wave vector ). So we seem to be stuck again. k | |||||||||||||

Well, not completely so, because the last equation also contains the inversion
condition in a slightly veiled form. Let's see how this can be verified: | |||||||||||||

For inversion we defined that , and that implies that R_{se} ³
R_{fa}
must be zero or R^{net} = R_{se} – R_{fa}
positive. | |||||||||||||

Since , A, and u(n)
are D_{C, V}always positive, only applies if R^{ net}
³ 0 | |||||||||||||

| |||||||||||||

This is a condition for any combination of two energy levels
from the valence and conduction band with energy difference hn. | |||||||||||||

Let's see what that means by looking at a special combination of energies:
, E_{1} = E_{V}, E_{2} = E_{C}hn
= . Then,E_{g} f( is the probability that we find electrons at the conduction
band edge, and E_{1} + hn, E_{F}^{e})
= f(E_{2}, E_{F}^{e})1 – f( is the probability that the valence band edge states areE_{1}, E_{F}^{ h}) = 1 – f(E_{V},
E_{F}^{h})
not occupied by holes and therefore occupied by electrons. | |||||||||||||

The inversion condition as expressed in the equation above then states that stimulated emission only surpasses fundamental absorption if it is more likely to find an electron at the conduction
band edge than at the valence band edge. | |||||||||||||

This statement applies to any pair of energies. Inversion occurs as soon as it is more likely to find an
electron at some energy in the conduction band, than at this energy minus hn in the valence
band. | |||||||||||||

Spelling out the Fermi distribution functions, | |||||||||||||

| |||||||||||||

and so on, we find once more as the inversion condition | |||||||||||||

| |||||||||||||

What did we gain by this procedure? Quite a lot, because we now know where we cut corners. We know how we can calculate rates for stimulated emission and fundamental absorption with sufficient precision – all we have to do is some numerical work. | |||||||||||||

But actually, what we really want to do is to make a laser diode and not to calculate emission rates. Making
an operating Laser diode entails that we have some specific volume where we need to
have inversion conditions. Then we send some photons into this volume to start stimulated emission, and, since this would
deplete the electron population in the conduction band, we also inject electrons into the conduction band (and, of course,
holes in the valence band), with the aim of achieving some steady state in the output of light produced by stimulated emission! |
|||||||||||||

In other words, we must look at the dynamics of inversion, at how all
quantities involved change with time. And this can be done by working with the formulas
derived here – but in a different module. | |||||||||||||

6.1.1 Interaction of Light and Electrons; Inversion

Fermi's Golden Rule and Puzzles of Quantum Theory

© H. Föll (Semiconductors - Script)