Page 383 - Electrical Properties of Materials

Electro-absorption in quantum well structures 365

with an energy somewhat less than the gap energy may also be absorbed be-
cause creating a bound electron–hole pair (i.e. an exciton) needs less energy
than creating a free electron–hole pair. The difference between the two is the
binding energy. How can we determine this binding energy? We may simply
argue that the binding energy between an electron and a hole is the same type
as that between an electron and a proton, and we may then use the hydrogen
atom formulae into which we need to substitute the actual values of dielectric
constant and effective mass. In bulk GaAs the binding energy may be calcu- This is the same calculation as
lated to be 4.2 meV, and the orbit of the exciton in its lowest state (called the 1s we did in Section 8.3 for silicon
state in analogy with the hydrogen atom) is about 30 nm. for finding the ionization energy
Does this mean that we see a sharply defined energy level below the gap? needed to promote an electron
No, because we have to take into account that the lifetime of an exciton is from a donor level into the con-
not more than a few hundred femtoseconds and, therefore, the line will be duction band. The physics for the
considerably broadened. The outcome of all this is that, in bulk materials at energy levels of excitons is fairly
room temperature, the excitonic resonance can hardly be noticed. similar.

13.11.2 Excitons in quantum wells

Let us now consider the effect of excitons upon the absorption spectrum in
a quantum well material. If the width of the well is larger than the exciton
orbit then, obviously, the excitons will hardly be affected. But if the width
of the well is less than the diameter of the lowest orbit, then the exciton has
no other option but to get squashed in the direction perpendicular to the layers.
What will happen in the other directions? A good indication can be obtained by
solving the problem of the two-dimensional hydrogen atom. It turns out that the
diameter will reduce by a factor of 4. So we may come to the conclusion that the
size of the exciton will be reduced in all three dimensions, and consequently
its binding energy will be increased. Calculations yield a binding energy of
10 meV, indicating that the exciton absorption effect is much stronger. We may
expect to see exciton peaks associated with each absorption step.
Experiments show that our expectations are correct. For a GaAs quantum
well of 10 nm width, the measured absorption spectrum is shown in Fig. 13.26.
Both the peaks and the steps may be clearly distinguished. Do not worry about
the double peak. It is simply due to the fact that there are both heavy and light
holes present, which has not so far been mentioned, because they do not affect
the basic argument.

13.11.3 Electro-absorption
This means optical absorption in the presence of an electric field. Will the
excitons know that an electric field is present? They certainly will. An electric
field will try to move the two particles in opposite directions. Since there is a
tendency anyway for the two particles to part company, they can more easily
do so in the presence of an electric field. Consequently, the exciton is more
likely to be ionized (this is called field ionization), its lifetime will be shorter,
and the absorption line will be broader. All this is true for the bulk material.
But will the argument change in any way for a quantum well material? If the
electric field is applied in the direction of the layers, then the same argument
will still roughly hold: the chances of being able to ionize the exciton with