Carrier recombination is an effect in which electrons and holes (carriers) interract with each other in a way in which both particles are eliminated. The energy given off in this process is related to the difference between the energy of the initial and final state of the electron that is moved during this process. Recombination can be stimulated by temperature changes, exposure to light or electric fields. Radiative recombination occurs when a photon is emitted in the process. Non-radiative recombination occurs when a phonon (quanta of lattice vibrations) is given off rather than a photon. A special case known as “Auger recombination” causes kinetic energy to be transferred to another electron.

Band to band recombination occurs when an electron moves from one band to another. In thermal equilibrium, the carrier generation rate is equal to the recombination rate. This type of recombination is dependent on carrier density. In a direct bandgap material, this will radiate a photon.

An atom of a different type of defect in the material can form “traps” which can contain one electron when the particle falls into it. Essentially, trap assisted recombination is a two step transitional process as opposed to the one step band to band transition. This is sometimes known as R-G center recombination. A two step recombination is known as “Shockley Read Hall” recombination. This is typically indirect recombinaton, which emits lattice vibrations rather than light.

The final type is Auger Recombination caused by collisions. These collisions between carriers transfer motional energy to another particle. One of the main reasons why this is distinct from the other two types is that this transfer of energy also causes a change in the recombination rate. Like the previous type, this tends to be non radiative.

A distinction should be made for band-to-band recombination between stimulated and spontaneous emission. Spontaneous emission is not started by a photon, but rather due to temperature or some other means (sometimes called luminescence). As stated in a previous post, stimulated emission is what emits coherent light in lasers, however spontaneous emission is responsible for most light emission in general.

Rayleigh scattering is an effect of the scattering of light or electromagnetic radiation by particles much smaller in size than the wavelength. For example, when sunlight emits photons which enter the earth’s atmosphere, scattering occurs. The average wavelength for sunlight is around 500nm, which is in the visible light spectrum. However, it is known that the sunlight also emits Infrared waves and of course, ultraviolet radition. Interestingly enough, Rayleigh scattering influences the color of the sky due to diffuse sky radiation.

The reason why a huge wavelength (compare 400 nm with nitrogen and oxygen molecules which are only hundreds of picometers) can scatter on a small particle is because of electromagnetic interractions. When the nitrogen/oxygen molecules vibrate at a certain frequency, the photons interract and vibrate at the same frequency. The molecule essential absorbs and reradiates the energy, scattering it. Because the horizontal direction is the primary direction of vibration, the air scatters the sunlight. The polarization is dependent on the direction of the incoming sunlight. The intensity is proportional to the inverse of the wavelength to the fourth power. The shorter the wavelength, the more scattering. This can explain why the sky is blue because blue is more likely scattered by Raleigh scattering due to higher frequency (smaller wavelength). It is not dark blue because other wavelengths are also scattered, but much less so.

Rayleigh Scattering is quite important in optical fibers. Because the silica glass have microscopic differences in the refractive index within the material, Rayleigh scattering occurs which leads to losses. The following coefficient determines the scattering.

The equation shows that the scattering coefficient is proportional to isothermal compressibility (β), photoelastic coeffecient, the refractive index  as well as fictive Temperatue and is inversely proportional to the wavelength.

Rayleigh scattering accounts for 96% of attenuation in optical fibers. In a perfectly pure fiber, this would not occur. The scattering centers are typically atoms or molecules, so in comparison to the wavelength they are quite small. The Rayleigh scattering sets the lower limit for propagation loss. In low loss fibers, the attenuation is close to the Rayleigh scattering level, such as in Silica Fibers optimized for long distance propagation.

As previously concluded, solids can be characterized based on energy band diagrams. A conductor has a valence and conduction bands that are very close or overlap. In addition a conductor will have a completely filled valence band and an almost full conduction band. The “forbidden region of the conductor is very small and little energy is required for an electron to move from conduction to valence band. In the presence of an external field, it is very easy for electrons to move from the valence band to the conduction band.

For semiconductors, at absolute zero the valence band is also completely full and the bandgap is typically about 1eV to 3eV, however even a bandgap of .1eV could be considered a semiconductor. Therefore, a semiconductor at 0K is an insulator. Semiconductors are very temperature sensitive. The subsequent figure illustrates the temperature dependence. The resistivity is very high at absolute zero, making the semiconductor behave like an insulator. However at higher temperatures the semiconductor can become quite conductive. At room temperature (300k), the semiconductor behaves more like a conductor.

With band diagrams, not much information is given therefore it is necessary to also analyze an E-K (Energy momentum) diagram. E is the energy require for an electron to traverse the bandgap. For example in Silicon with a bandgap of 1.1eV, it would take an energy level of 1.1eV for an electron to move from conduction to valence band. Energy is given as E = kT where T is a given temperature.

For intrinsic semiconductors like Silicon, the structure is crystalline and periodic. The wavefunction (which describes probability of finding an electron) should therefore be of periodic nature (sinusoidal). From the Schrodinger equation, it can be found that the Energy is periodic with k as well. For the diagrams, E is plotted against k.

The borders of the first Brillouin zone are from -π/a to π/a. These are cells of the crystalline lattice. Since the wavefunction is periodic, we only care about one of the zones. The above figure can be considered the “reduced zone” figure. Sometimes the x axis is given as the moment or wavenumber, since these only differ by a factor of Planck’s constant. From this diagram: the bandgap energy is shown, the effective mass of electrons and holes are shown as well as the density of states. The effective mass is shown by the curvature of the bands. For example, a heavy hole band could be found by observing the diagram that is less curved. From the above diagram, it is also noticeable that the material is direction bandgap (such as GaAs). The basic energy gap diagram compares to the E-k diagram in that the maximums and minimums correspond. However, the original band gap diagram does not give any other characteristics. It is for this reason the E-k diagram is so useful.