Material Science 435: Photonic Materials
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Lecture 26: Properties of Semiconductors - Materials for LED's, Detectors: The Absorption of Light by Semiconductors

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Lectures in Series:

Lecture 1: Optical Characteristics

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Band Structure of Common Semiconductors

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 = toff . What happens to the excess minority carrier concentration pn (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 diagram to 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.

Ni and Nf 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:

  • nch2 is the carrier density
  • 2m * is the reduced mass of the e-h pair
  • x G is the bandgap

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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
The absorption probability increases
The absorption probability decreases
The absorption probability remains the same.
As the temperature increases, is Ge likely to absorb more or fewer photons?
More
Fewer


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?
LiF
GaAs
GaP

 

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 weakly bound, 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 Eg 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?
GaAs
Si
CdS
How does the binding energy scale with the exciton Bohr radius?
The binding energy increases as the radius becomes smaller.
The binding energy decreases as the radius becomes smaller.

 


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 As2Te3 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 EF is in the VB and hence the tail of the VB is unimportant for the transition.

Materials Issues for Photodetectors

  1. The diffusion lengths of electrons(Le) and holes (Lh), 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 Le,h is where is the carrier diffusivity.
  2. Surface recombination of photogenerated carriers, which reduces the collection efficiency.
  3. 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.
  4. 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?
Ge
InGaAs
GaAlAsP
Which compound has the broadest spectral range?
InGaAs
Si
Ge

 

UP
1.What region of light is of interest to us when working with semiconductors?
Infrared
Ultraviolet
Visible
2.Which best describes indirect bandgap materials?
The k-values of the initial and final states are the same
The k-values of the initial and final states for the lowest energy transition are different
The k-values of the entitle and final states for the lowest energy transition are different so the valence electron reaches the X point valley. A phonon must take part in the excitation, through either absorption
(+hu p) or (-hu p).
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.
Excitonic transition
Intraband transition
Impurity to band and impurity-impurity transitions
4.Check off ALL the sources of absorption in semiconductors.
Direct interband transitions.
Indirect interband transition
Contraband transitions
Impurity to band and impurity-impurity transitions
Excitonic transition
Intraband transition
Phonon transitions.
Defect transitions
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