These are some common band structures.
Most III-V compounds have a direct bandgap (exception: GaP, AlP).
You will remember that the top of the valence band and the bottom
of the conduction band are found at the same place (k=0).
Figure 1. Sketch of the band structures of GaAs
and CdTe. Both are direct-gap semiconductors.
Source??
Figure 2. Sketch of the band
structures of Si and GE at room temperature.
Both are indirect-gap semiconductors.
Source??
Si, Ge and GaP have an indirect bandgap; the top of the valence
e band and the conduction band are not aligned.
Let's take a moment to try and develop a good intuition for the
material: Imagine that you are walking through the crystal. You
see different rows of atoms, depending on the direction in which
you are walking. Although your energy remains constant, you will
find, however, that your velocity is not fixed; this depends on
the other forces that are acting on you (from the lattice).
Plotted in this figures is the energy of the valance electrons
in different directions, giving rise to different shapes. The vertical
axis is energy; on the horizontal, x is momentum. The lines represent
different orbitals.
In principle, you can calculate what will happen to the free electron.
Using the GaAs figure as an example (above), the energy of the lowest-energy
electron is found at the red dot:
If the velocity (momentum) of the electron remains the same, the energy
will change because it is interacting with the lattice. For the electron
with zero momentum, the total energy moves to the right. Taking the case
of Si with bonding electrons, the extra lines in the CB are for different
orbitals of the electrons (due to an SP coupling, etc.). There are techniques
available that allow you to separate the energies. In a solid, the electrons
can be successively pulled out and their energy calculated, thus permitting
us to calculate the distribution of energy in the valence e band (This
technique is known as XPS.).
Figure 3.
Kasap, 1999,
The band structure diagrams that we are presently looking at represent
with more detail what we previously saw lumped together in Figure
3. When the material changes, the lines will also change because
the nature of the bonding is different.
When
GaAs is alloyed with AlAs, which orbitals will change? Check
your answer.
Absorption
By using a band gap diagram, we may determine the absorption coefficient
a as a function of light,n
.
We know that glass is transparent to visible light; GaAs is transparent
only for low frequency light (its band gap is smaller). At the low
end of n , absorption is negligible,
but becomes significant as we reach the visible region. In the previous
lectures on insulators, our discussion was limited to infrared and
Raman. With semiconductors, our focus shifts to visible light; we
want our source and detection to take place in the visible region.
Figure 4.
Effect of Light on n-type
semiconductors
There is a change of nature in the bandgap due to composition. Examples
are: AlAs, an indirect band gap material, and GaAs, a direct bandgap.
Click
on button to see animation.
Figure 5. The illumination of an n-type semiconductor gives
rise to an excess concentration of electrons and holes. After illumination,
the material returns to a state of equilibrium, wherein electrons
and holes recombine.
Figure 6.
Kasap(1999).Optoelectronics.
In
the graph to the left, we have a light source being switched
on at t = 0 and off at t = t_{off }. What happens
to the excess minority carrier concentration p_{n}
(t) at these two instances? Check
your answer.
Relation between the density
of states and absorption coefficient
A is the minimum amount of energy of the photon that will be absorbed.
hn indicates a higher energy; in this
band diagram the absorption is increasing as we move to the right.
If
the energy of the electron is increased to two times that of the
band gap, what happens? Check
your answer.
If the number of electrons at various energies is the same, the number
of possible conduction states remains constant and the spacing between
the number of energy levels, i.e., is the same, it makes no difference
where the electron goes. The challenge arises when the density of states
is different. The absorption coefficient depends on the two density of
states -- in the VB and in the CB. The joint optical density of states
is a combination of the information from the valence e band and from the
conduction band.
This is represented qualitatively in the diagram to the right (Fig
7 b), which shows a uniform density of states for the CB (without
peaks or troughs), typical of amorphous semiconductors like silicon.
Let's look at the animation: In the diagramto the left
and at the bottom, A represents the threshold energy; from
0 to 2 the absorption coefficient is small because the density of
states is small. At point B, we see a lot of absorption;
then because the energy becomes too high and the density of states
in the valence band decreases, the absorption is less, point C.
We will not calculate the density of states in this course because
it would require an entire course. Let us just say that computer
approximations are not trustworthy; to engineer a good product for
industrial purposes, most developers would also want supporting
experimental data on absorption.
Click
on start to see animation.
Figure 7.
Kasap(1999).Optoelectronics
An Overview of Sources of
Absorption
1. Direct interband transition: From excitation of e from
VB to CB with negligible change in the momentum of e. We have already
discuss this at length in the previous figures.
2. Indirect interband transition: From excitation of e from
VB ® CB but with a change in momentum
with help from phonons. These transitions have lower probability
than the direct transitions.
3. Impurity to band and impurity-impurity transitions: Generally
Non-radiative transition for e from VB ®
AB (i.e. h going from AB ® VB);
The e excitation from DB ® CB is
generally not observed as kT is enough for this transition. Most
radiative transitions are AB « DB.4.
Excitonic transition: Absorption of sub bandgap light by
excitons in direct bandgap materials. Non-radiative transition of
e from CB ® EB followed by radiative
transition from EB ® VB.
5. Intraband transition: Unlikely within VB due to selection
rule requiring Dl = ±
1. Free e in CB may gain KE by photon absorption, followed by thermalization
with the lattice.
6. Phonon transitions. Transfer of energy between phonons
and photons as observed in Raman, and Brillouin scattering.
7. Defect transitions: Similar to dopant transitions, except
that the states may be deeper in the band. Defects are present unintentionally
and difficult to control than usual dopants.
We will be treating each of these topics separately in the rest of this
lecture.
Absorption in Direct Bandgap
Material
In Figure 8, we see the minimum of the bands in different directions
(These representations include only small sections of the bands).
When working with indirect bandgap material, the electrons that
respond the most easily and with a minimum of energy will be those
at the top of the valence e band. Depending on the density of states,
these electrons may not be the largest in number, but these are
the ones that will determine the threshold energy for absorption.
In indirect bandgap materials, for the electrons in the VB to reach
the bottom of x in the CB, there must be a change in momentum k
as well in energy. This is the difference with direct bandgap materials.
Figure 8. Optical absorption transitions that
promote a valence electron to a conduction-band state. In direct-gap
materials, the k-values of the initial and final states are the
same, so the electron goes directly to the central G
point valley. In indirect gap material, the kvalues of the initial
and final states for the lowest energy transition must be different,
so the valence electron reaches the X point valley. In this case,
a phonon must take part in the excitation, through either absorption
(+hu _{p}) or (-hu
_{p})
Source??
There is always conservation of energy and conservation of momentum in
the absorption process. The photon kicks the electron out of the bonds
and takes it to a higher state in the CB. The momentum comes from the
phonons, which also have energy and move the electron to the X.
The term phonon absorption means that the energy of the photon will be
less by that amount; in other words, the sum of the phonon and photon
energy must be equal to the energy of the transition. There can also be
the emission of the phonon, meaning that the photon of slightly higher
energy will be absorbed if a phonon is emitted out. Absorption may be
described in the following terms:
The absorption coefficient depends on the optical transition probability,
which is the matrix element that can be calculated for the initial and
final wave function multiplied by the occupancy of the two states involved
in the transition.
N_{i} and N_{f} are drawn in these figures as simple
parabolas and can be treated as free electrons and holes, with some modification
to the effective masses. The resulting derivation is:
n_{ch}^{2} is the carrier density
2m * is the
reduced mass of the e-h pair
x _{G}
is the bandgap
.
The
above is an expression for direct bandgap material with k=0. What
can be changed in this expression so as to play with the absorption
coefficient? Check
your answer.
Absorption in Indirect Band
Gap Material
The math for the indirect band gap materials has been worked out along
the same lines as the direct except that the phonon has been introduced
into the transmission. The transmission probability will now depend on
the density of states of the phonons (how many phonons, their energy and
their momentum). Therefore, in this case, we need to look at the dispersion
diagram for optical phonons.
for phonon absorption
The phonon emission and absorption have to do with
the energy of the phonon. The momentum must always be of the right
kind. In indirect bandgap material, a
is related to the square of energy as opposed to the square root
in direct bandgap materials.
for phonon emission
Figure 9. Typical absorption
behavior for an indirect-gap semiconductor.
Elliott, 398
In this diagram, we see three components of the absorption:
+x _{p} or -x
_{p} , x _{p} being
the energy of the phonon. Keep in mind that we are plotting the square
root of the absorption coefficient. x
_{G} -x _{p }indicates
that the phonon is absorbed; x _{G}
+x _{p }that the phonon is
emitted. We see the interplay of the phonon absorption emission; then,
as the energy increases, the direct transmission takes over.
At
low energy, only phonon absorption can mediate the electron
excitation. What happens to the absorption probability as
the temperature is lowered
As
the temperature increases, is Ge likely to absorb more or
fewer photons?
Figure 10. The absorption edge
of crystalline Ge. In the vicinity of indirect transitions,
measured at various temperatures as indicated.
MacFarlane et al., (1957). Phys.
Rev. 108, 1377
Excitons
An exciton is a bound e-h pair, which has lower energy than a free e
and h. In ionic crystals it is an electron in an excited state with a
stronger binding. The binding energy can vary from 1 meV to 1 eV.
Figure 11.
Brown, F. and Schmidt, A.
Which
component after InSb has the smallest exciton binding energy?
If the exiton electron were completely free, it would
migrate entirely into the CB and have no interaction with the hole.
By staying in the area around the hole, it lowers its energy. Moreover,
it could recombine with the hole completely. According to this latter
scenario, in the ultimate equilibrium state the e-h would disappear.
Figure 12a. An exiton is a bound-electron-hole
pair, usually free to move together through the crystal. In
some respects, it is similar to an atom of positronium, formed
from a positron and an electron. The exiton shown is a Mott-Wannier
exciton: it is weaklybound, with an average electron-hole
distance that is large in comparison with a lattice constant.
In the case of an anti-impurity, the electron was bound to an ionized
donor and then liberated. For exitons, the electron is taken from
the VB into the CB without involving an impurity. The energetics
of this electron can be worked out using the hydrogenic model.
The same concept can be applied to alkalide halides (Fig. 12b).
Exitons have a lifetime; but if you are constantly applying a light
to the material, you will continuously create a dynamic equilibrium
where some e-h's recombine and others are created.
What is the effect of an exiton
on absorption?
Possible Answers:
1. You will reduce the probability of transition A and the absorption
curve will uniformly decrease.
2. You will increase the absorption; the curve will increase.
3. For the real answer, see the following section.
Figure 12b. A
tightly bound or Frenkel exciton shown localized on one
atom in an alkali halide crystal. An ideal Frenkel exciton
will travel as a wave throughout the crystal, but the electron
is always close to the hole.
Kittel.
Effect of Excitons on Light
Absorption and Luminescence
Figure 13. The effect of an exciton
level on the optical absorption of a semiconductor for photons
of energy near the gap E_{g} in gallium arsenide at
21 K.
M..D. Sturge.
The presence of exciton energy levels in this example using gallium
arsenide introduces additional absorption of light near the band gap.
For the same reason there is an additional peak in the recombination
luminescence at less than the Eg energy. In fact, states have been
created in the bandgap because the e-h pair is now bound; binding
means lowering of energy. In terms of absorption, it will take more
energy to move the photon into the CB, so the absorption in that area
might decrease. However, there will be absorption at sub-bandgap light.
Energy Levels of Excitons
Additional states are created by the exciton just below
the bottom of the C.B (Fig 14). This implies that not all material
is transparent at sub bandgap levels.
Figure 14. Exciton levels in relation
to the conduction band edge.
Kittel
As previously mentioned, excitons can be treated like
hydrogen with the electron orbiting the hole. Using that equation,
the energy is quantized. In semiconductors because the dielectric
constant is large, Excitons near the direct gap in an indirect bandgap
crystal are unstable and decay into free e and h (Fig 15).
Figure 15. Energy levels of an exiton created ina direct
process.
Kittel
Associated excitons
Figure 16. Table of Wannier exciton binding
energies and Bohr radii.
Source?
Which
compound has the smallest effective electron mass?
How
does the binding energy scale with the exciton Bohr radius?
Figure 17. Schematic of the simpler exciton-associated
defects.
Source?
Which
excitons are likely to have a smaller binding energy? Check
your answer.
Excitons can move, but do not carry any current. They can be modified,
however, when they are trapped near a real dopant impurity. They then
have their own energy.
Intrinsic and Urbach band
tails
The tail of the band is important for applications. The two types of
tails are referred to as intrinsic and Urbach. For the former, absorption
does not change with temperature; according to the latter model, absorption
does change with temperature.
Impurities, strain and defects locally distort the e-lattice interactions,
resulting in a change in the band structure. This occurs near the edges
where the states represent the e and h of the lowest energy and creates
a tail in the absorption band. In the case of intrinsic tailing of the
band edges, which does not depend on T as long as the structure does not
change, examples are:
In antimaterial, when the e is excited from the impurity, the ionized
donor remains behind. The ionized donor then attracts the e in the conduction
band and repels the h in valence band (the phosphorous atom in silicon).
Non-uniform compressive stresses decrease the unit cell size, resulting
in an increase of the coulomb interactions and of the Eg.
Defects destroy lattice symmetry and thus locally modify the bandgap.
Figure 18. Optical absorption
edges of amorphous semiconductors.
Elliott, 1990
Why
is Si preferred over As_{2}Te_{3} as a detector
material? Check
your answer.
Lattice vibrations cause temporary changes in the lattice parameter, hence
in the local band structure. Calculations that were originally made for
a fixed lattice should be modified, taking into consideration the temperature.
Tails that are exponentially T dependent follow the Urbach relation:
d(ln a)
/ d(hn) = 1/kT
Band tail, say for C.B., can be observed from optical absorption of a
heavily doped p-type material for which E_{F} is in the VB and
hence the tail of the VB is unimportant for the transition.
Materials Issues for Photodetectors
The diffusion lengths of electrons(L_{e}) and holes (L_{h}),
which should be as high as possible for maximum minority-carrier collection
from both sides of the p-n junction. This means that the minority 0
carrier lifetime should be as long as possible since L_{e,h is }
where
is the carrier diffusivity.
Surface recombination of photogenerated carriers, which reduces the
collection efficiency.
The leakage current, which should be as low as possible. Therefore,
defects in the junction space-charge region or diode periphery should
be minimized. Avalanche photodiodes should be microplasma-free. This
means that local field enhancement, owing to inclusions or precipitates
in the space-charge region, must be avoided and surface fields must
be minimized.
Degree of crystallinity. Although useful solar cells can be produced
from polycrystalline material, the grain boundaries can degrade the
junction current voltage properties and may increase the lateral resistance
of the device.
Figure 19.
Which
photodetector material is most suitable for extremely low intensity
light signals?
Which
compound has the broadest spectral range?
UP
1.What
region of light is of interest to us when working with semiconductors?
2.Which
best describes indirect bandgap materials?
3.Which
Absorption source is best described as a: •generally non-radiative transition for e from VB
® AB
(i.e. h going from AB ® VB);
•the e excitation from DB ®
CB is generally not observed as kT is enough for this transition;
• most radiative transitions are AB «
DB.
4.Check
off ALL the sources of absorption in semiconductors.