Semiconductor Optoelectronics

Prof. M. R. Shenoy
Department of Physics
Indian Institute of Technology, Delhi

Lecture - 24
Electro-absorption Modulator

We start with the first device: Electro-absorption modulators or Quantum Well modulators.
Before I start, I will briefly discuss the answer for the quiz.

(Refer Slide Time: 00:42)

This was the quiz. A particular semiconductor laser amplifier with indium gallium arsenide
phosphide active medium of Eg equal to 1.24 eV has a peak gain coefficient of 75 cm-1
inverse and amplification bandwidth of 100 nm. Making use of the above data, draw
qualitatively the variation of gain coefficient with wavelength in the amplification band.
(Refer Slide Time: 01:33)

So, the answer is here. The fact that Eg was given 1.24 e V should have been a hint to you that

λ=1 μm , because λ=1.24/ E g γp

it will give . The peak gain coefficient is given. We

know that if frequency is increasing in this direction then wavelength will be decreasing in

λg
this direction. h ν starts from Eg, so will end there, and you can find the starting

λ corresponding to E – E from the amplification bandwidth that is given.

fc fv

(Refer Slide Time: 02:48)

Quantum Well Modulators work on the principle of Quantum Confined Stark Effect. We
know what is Stark effect. Stark effect refers to shift in the absorption spectrum. This was
originally observed in the absorption spectrum of hydrogen by Stark, which was shift in the
absorption lines of hydrogen atom in the presence of an applied electric field. We are looking
at Quantum Confined Stark effect, so Quantum here is referring to the Quantum Well.

QCSE, Quantum confined Stark effect is made use of to realize Quantum Well modulators or
electro absorption modulators. So let us see first absorption in semiconductors. We know the
absorption spectrum in semiconductors as well as in Quantum Well structures, as we have
seen this in the last classes.

(Refer Slide Time: 05:17)

Let me quickly recall what we have seen. In a bulk semiconductor the attenuation varies
something like this, as shown, starting from Eg. If we see at a lower temperature, two things
we observe, one is the absorption edge is shifted to higher energies. Absorption edge is the
energy or wavelength at which the absorption suddenly changes rapidly. The second thing
that you observe is these peaks are more pronounced. Actually in a bulk semiconductor at
room temperature we do not see any excitonic peaks. However, if a measurement is made at
low temperature, then you can see them.

Several people measured this in the early 1960s. At room temperature the absorption
coefficient does not show any peaks but if you lower the temperature then there are excitonic
resonance peaks, and the peak becomes sharper if you go to lower temperature. Excitons, if
you recall, correspond to electrostatically bound electron hole pair close to band edge where
kinetic energy is minimum. The binding energy of these excitons at room temperature is of
the order of 4 to 5 meV, Some wide band gap semiconductors like gallium nitride has a much
higher excitonic binding energy about 25 meV also.

But most of the semiconductors have in this range, 2 to 10 meV. As I discussed earlier that at
room temperature the average kinetic energy, thermal energy is of the order of 25 meV. So
due to the large thermal energy in the lattice these excitons are immediately ionized and
dissociated, so they are no more held together. Excitons do not exist at room temperature, or
they form and immediately dissociate. So you do not see any resonance peaks. If you lower
the temperature then the thermal energy kT reduces and therefore the binding energy of the
excitons becomes comparable and then you start seeing resonance peaks in the absorption.
These are for bulk, this is not Quantum Well, this is bulk gallium arsenide. In fact this is very
important proof for the existence of excitons, as very clearly experimentally you can see
existence of excitons in semiconductors. The band edge is shifting with temperature because
band gap Eg is temperature dependent for semiconductors. Generally, Eg is higher at lower
temperature.

(Refer Slide Time: 13:49)

In the case of Quantum Well structures you already have a small peak even at room
temperature. This is because of quantum confinement, which leads to higher binding energy
of excitons. The excitons now have generally about 3 to 4 times that of bulk because of
quantum confinement. What is this quantum confinement? Let me take two states here, E q1
and Eq2 in the conduction band, and similarly Eq1’ and Eq2’ in the valence band.

In the case of valence band if you recall I had mentioned that there are light holes and heavy
holes. They can be degenerate or they need not be degenerate. If they are not degenerate at k
equal to 0, which means this is light hole band which is the case normally in Quantum Well
structures. I am showing by dashed line the light hole bands. q equal to 1 determines the k z
value, but kx and ky are almost continuously varying and therefore we have formation of
energy sub bands. So for every level correspondingly there is energy sub bands in Quantum
Wells. Corresponding to heavy hole and light hole there are two sub bands here. Both
transitions are allowed as they correspond to q = 0.

q = 2 to q = 1 is not allowed. Quantum mechanically you can find out what are the allowed
transitions. Please remember allowed transition does not mean a transition will not take place.
But the probability is very low; for allowed transition the probability is high. So forbidden
transitions can still take place, but with low probability. So, corresponding to this excitons
formed from the heavy hole to this, we have the first peak. The second small peak that you
see is from the light hole to the upper band. That is why you see two peaks, because of light
hole and heavy hole. It is also possible to have only one peak, by using strained layer
Quantum Wells, so it is possible to merge these heavy hole and light hole levels, making them
degenerate. Therefore, in future discussions I will not bring these two peaks into the
discussion separately. The important point to note is there is the absorption peak because of
excitonic resonances. So you have an excitonic peak and you can see excitons at room
temperature in Quantum Well structures, because of quantum confinement. What is this
quantum confinement?
(Refer Slide Time: 21:01)

So let me draw again the same structure, because now I want to show wave functions. In the
case of a Quantum Well structure, I consider one energy level Eq1 and Eq1’. What I am plotting
is the wave function associated with the electron here and the wave function associated with
the hole. I can draw it upward or downward, it does not matter, it simply indicates that there

is a phase change π . Even in the case of bulk if you have an electron here and a hole here,

it is like a packet and therefore there is an associated wave function, but that wave function is
not confined by any potential barrier. So a slight energy supplied dissociates the exciton.

But in a Quantum Well structure the wave function is confined by this potential energy
barrier, and this is called quantum confinement; there is no confinement in the bulk case, it is
only because of the Coulomb attractive potentials that they hold on together with a weak
binding energy. Due to Quantum confinement in this case now, the binding energy is higher,
typically 3 or 4 times, as I said before. For example, if I take gallium arsenide which has a
binding energy of about 4 meV in bulk, then in Quantum Well structure it will have a binding
energy of 15 to 20 meV. The binding energy is comparable to the thermal energy at room
temperature and hence they do not dissociate so quickly, and you can see the peaks at room
temperature.

Now we will see what Quantum confined Stark effect is. It refers to the shift in absorption
peaks or absorption lines, or the absorption edge in Quantum Well structures. Let me
illustrate it and then it will become clear. We saw the shift because of change in temperature.
Now, we will see a similar shift will be there because of an applied electric field, and that is
called Quantum confined Stark effect.

(Refer Slide Time: 25:28)

Say I take gallium arsenide, and this is the absorption spectrum at T = 300 K, room
temperature, and no electric field is applied. What happens if you apply an electric field? A
piece of semiconductor (bulk) has an energy band diagram like this, you have Ev, Ec and Eg. It
is not a p-n junction, just a piece of semiconductor. So what will happen to the band diagram
if we apply an electric field? This end is at a positive potential which means electron potential
energy is lower, and therefore the band bends, but there is no change in the energy band gap
Eg. For a bulk semiconductor, absorption starts at E g because a photon of that energy can take
an electron at the edge of the valence band to the edge of the conduction band. If I increase
the electric field, the band will tilt more. An electron is at a particular value of x, x = 0. But
due to the tilting of the band, there is an allowed state at x = L also (say) too. So there is an
allowed state in the conduction band at that same energy as that of the valence band, but at a
different position. Therefore, in principle the electron can exist here or here (x = 0 or L), so
the electron can tunnel into the allowed state in the conduction band because it is at the same
energy. This is quantum mechanical tunneling.

For those who are not familiar, this is elementary quantum mechanics. If there is an allowed
state inside a well, and there is another identical well but they are separated by a barrier,
tunneling can occur. The two wave functions in the two wells can interact and energy can be
transferred from one to the other. So the electron’s evanescent wave tail from the forbidden
region comes into the allowed state where it starts oscillating and so the electron can tunnel
from here to this. The tunneling coefficient or the probability that the electron can tunnel
depends on the height and width of the barrier. Classically it cannot go, but quantum
mechanically it can go to the other side, i.e., tunnel through the barrier. I come back to our
discussion, this much elementary concept is sufficient, we cannot go into too much of
quantum mechanics here.

If I take a small band gap, and make the tilt steeper, which means I have applied a stronger
electric field, then this electron will now have a much better probability to tunnel because the
barrier width is small.

In the case of a semiconductor we can also have photon assisted tunneling. What is this

photon assisted tunneling? There is a photon which is incident, which has energy hν less

than Eg. It is able to lift the electron only up to some energy level in the forbidden region, and
then then electron tunnels into an allowed state in the conduction band, because the distance
is very small, so the tunneling probability is very high. This is called photon assisted
tunneling. What would happen to the absorption spectrum of this bulk? This will now shift in
the presence of applied electric field, because photons of energy less than E g can also be
absorbed due to photon assisted tunneling. This famous experiment was done by Franz and
Keldysh and is known as the Franz-Keldysh effect.
(Refer Slide Time: 38:01)

So this was discovered in 1958. A German scientist Franz and a Russian scientist Keldysh
discovered this effect, that in the presence of an applied electric field the absorption edge
shifts to lower energy or higher wavelength. It is called red shift of the absorption edge in
semiconductors. It is explained by the phenomena of photon assisted tunneling. Why did I
discuss this?

See, red shifting of the absorption edge in bulk semiconductors is called Franz-Keldysh
effect. If you apply an electric field to Quantum Well structure you will see a similar red shift
of the absorption edge. So red shifting of the absorption edge in a Quantum Well structure is
called QCSE, quantum confinement stark effect. I have shown with E = 0 and E
50 kV/cm. In bulk you need a little higher electric field, about 100 kV/cm.
These are not a very big electric field, because the unit is per centimeter, and the dimension
of the quantum well is very small.

Why the name QCSE? It is Quantum Confined because of this step like variation, with an
excitonic peak because of quantum confinement, and it is Stark Effect because the shift is
because of an applied electric field. Basically it is Franz-Keldysh effect but in a Quantum
Well structure.
(Refer Slide Time: 41:32)

When I apply the electric field, the electron wave function becomes asymmetric. Why do you
think electron wave function becomes asymmetric? Because I have applied a positive voltage
here. So the electron is getting attracted towards the positive voltage, it is trying to go to that
side. If I had taken a simple bulk semiconductor the electron would immediately rush to that
end and dissociate. Here it is trying to go, but the barrier is stopping it from going. So the
wave functions have separated out, the peaks do not lie at the same position now. What is
important is that there’s a change in energy, i.e., difference in the effective band gap, which is
not smaller. So there is red shift of the band edge after applying electric field. And because of
quantum confinement, the electron fields are still held together and therefore if exciton is
formed then the exciton is still there.

Where is the modulator? This plot is in terms of energy. If I were to plot this in terms of
wavelength, would it look? The shift would be towards right, to higher wavelength, as it
corresponds to a lower energy. So, where is the Quantum Well modulator? Suppose there are

λ1 λ2
two wavelengths here, and which are the band edges. The operating wavelength

λop
corresponds to the excitonic peak after the field has been applied. This is the Quantum

λop
Well structure and I am putting a beam of wavelength . When there is no electric field,

what is the attenuation coefficient? Almost nil, as you can see form the absorption spectrum.
So it is transparent to the operating wavelength. When I apply the electric field, this curve has
shifted, and so the attenuation coefficient is very high, and nothing will come out. So by

applying electric field the α has increased by orders of magnitude, because now the

second curve is applicable, not the first curve, because I’ve applied electric field.

So what is the point? If I see the transmission T versus time, if I apply an electric field pulse
like this, then the optical transmission is low when the field is high, and vice-versa. The
response in an electro-absorption modulator is very small, of the order of 10 to 20
picoseconds. So you can really have electro absorption modulators working at 40 GhZ, 50
Ghz, and so on. So this is the principle of how the Quantum Confined Stark effect can be
used to realize electro absorption modulator. We will stop here and discuss the device
configurations later. So we will discuss about the device in the next class.