Author Archives: jalves61

Photodetectors and Dark Current

A photodetector simply is a device that converts light energy to an electrical current. These devices are very much similar to lasers, although they are designed to operate in reverse bias. “Dark current” is a term that originates from this reverse bias condition. When you reverse bias any diode, there is some leakage current which is appropriately named reverse bias leakage current. For photsensitive devices, it is called dark current because there is no light absorption involved. The main cause of this current is random generation of electrons and holes in the depletion region. Ideally, this dark current is minimal (<< 1).

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The basic structure of the photodiode is the “PIN” structure, similar to a semiconductor laser diode. An intrinsic (undoped) region occurs between the P-doped and N-doped region.  Although PIN diodes are poor rectifiers, they are much better suited for high speed, high frequency applications due to the high level injection process. The wide intrinsic region provides a lowered capacitance at high frequencies. For photodetectors, the process is photon energy being absorbed into the depletion region, causing an electron hole pair to be created when the electron moves to a higher energy level (from valence to conduction band). This is what causes an electrical current to be created from light.

Photodetectors are “photoconductive”. That is, conductivity changes with applied light. Like amplifiers and other devices, photodetectors have “Figures of Merit” which signify characteristics of the device. These will be briefly examined

Quantum Efficiency

Quantum efficiency refers to the number of carriers generated per photon. It is normally denoted by η. It can also be stated as carrier flux/incident photon flux. Sometimes anti-reflection coatings are applied to photodetectors to increase QE.

Responsivity

Responsivity is closely related to the QE (quantum efficiency). The units are amperes/watt. It can also be known as “input-out gain” of any photosensitive or detective device. For amplifiers this is known as “gain”. Responsivity can be increased by maximizing the quantum efficiency.

Response Time

This is the time required for the photodiode to increase its output from 10% to 90% of final output level.

Noise Equivalent power

This value corresponds to units of Watts/sqrt(Hz). It is another measure of sensitivity of the device in terms of power that gives a signal to noise ratio of one hertz per output bandwidth, Small NEP is due to increased sensitivity of the device.

Carrier Recombination

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.

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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

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.

emagspec

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.

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.

The Electronic Oscillator

The semiconductor laser is a device that can be compared to an electronic oscillator. An oscillator can be thought of as a resonator (a circuit that resonates or produces a strong output at a specific frequency) with gain. Resonators naturally decay over time by some factor, so adding in gain (so long as the gain is greater than or equal to the loss) can allow the resonator to become an oscillator that does not decay or dampen.

The stimulation of the oscillations of an oscillator is caused by electronic noise. A block diagram can demonstrate an oscillator in an abstract, easier to understand way.

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The oscillator is built using an amplifier (transistor that is biased into active/saturation region) or op amp with positive and negative feedback. Noise in the circuit begins the oscillation, and this output is fed back into the input and is filtered along the way. This becomes an oscillation at a single frequency.

Oscillators can be built from RC circuits, LC circuits or can be crystal oscillators. RC circuit oscillators tend to be lower frequency oscillators in the audio range. The LC oscillator is often compared to the laser in terms of functionality. The negative reactance of the capacitor and positive inductive reactance cancel at a specific frequency, leaving the circuit with only resistance and a strong current is achieved. LC oscillators are much more important for RF/microwave purposes. A crystal oscillator produces its frequency through mechanical vibrations and has a much higher Q factor than the other resonator types, which provides greater temperature and frequency stability.

Two very important oscillator types for RF/microwave/mmWave circuits are dielectric resonators and SAW (surface acoustic wave) resonators. Dielectric resonators are mainly used as mmWave oscillators to drive antennas. They are generally made of a “puck” of ceramic which oscillates at a certain frequency dependent on its dimensions. Waves are confined inside the material due to an abrupt change in the permittivity. When the waves inside interfere and produce a standing wave, this increase of amplitude creates the resonance effect. SAW resonators are often used in cell phones and have distinct advantages over the LC oscillator or other types due to cost and size.

In a semiconductor laser (laser diode), the source of oscillations is the noise generated by spontaneous emission. Spontaneous emission is the result of recombination of electron and hole pairs within the material which produces photons. This spontaneous emission is how lasers begin their operation, and this is continued by stimulated emission. Stimulated emission is electron hole recombination due to photon energy which also produces a photon. The light emitted by this type of emission is coherent, a characteristic of a laser.

Pseudomorphic HEMT

The Pseudomorphic HEMT makes up the majority of High Electron Mobility Transistors, so it is important to discuss this typology. The pHEMT differentiates itself in many ways including its increased mobility and distinct Quantum well shape. The basic idea is to create a lattice mismatch in the heterostructure.

A standard HEMT is a field effect transistor formed through a heterostructure rather than PN junctions. This means that the HEMT is made up of compound semiconductors instead of traditional silicon FETs (MOSFET). The heterojunction is formed when two different materials with different band gaps between valence and conduction bands are combined to form a heterojunction. GaAs (with a band gap of 1.42eV) and AlGaAs (with a band gap of 1.42 to 2.16eV) is a common combination. One advantage that this typology has is that the lattice constant is almost independent of the material composition (fractions of each element represented in the material). An important distinction between the MESFET and the HEMT is that for the HEMT, a triangular potential well is formed which reduces Coloumb Scattering effects. Also, the MESFET modulates the thickness of the inversion layer while keeping the density of charge carriers constant. With the HEMT, the opposite is true. Ideally, the two compound semiconductors grown together have the same or almost similar lattice constants to mitigate the effects of discontinuities. The lattice constant refers to the spacing between the atoms of the material.

However, the pseudomorphic HEMT purposely violates this rule by using an extremely thin layer of one material which stretches over the other. For example, InGaAs can be combined with AlGaAs to form a pseudomorphic HEMT. A huge advantage of the pseudomorphic typology is that there is much greater flexibility when choosing materials. This provides double the maximum density of the 2D electron gas (2DEG). As previously mentioned, the field mobility also increases. The image below illustrates the band diagram of this pHEMT. As shown, the discontinuity between the bandgaps of InGaAs and AlGaAs is greater than between AlGaAs and GaAs. This is what leads to the higher carrier density as well as increased output conductance. This provides the device with higher gain and high current for more power when compared to traditional HEMT.

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The 2DEG is confined in the InGaAs channel, shown below. Pulse doping is generally utilized in place of uniform doping to reduce the effects of parasitic current. To increase the discontinuity Ec, higher Indium concentrations can be used which requires that the layer be thinner. The Indium content tends to be around 15-25% to increase the density of the 2DEG.

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Basic Energy Band Theory

Band theory is essential in the study of solid state physics. The basic idea tends to center around two bands: the conduction and valence band (for reasons discussed later on). Between the two bands is a forbidden energy level (Energy gap) which depends on the resistivity or conductance of the material. In order to fully understand solid state devices such as transistors or solar cells, this must be discussed.

For a single atom, electrons occupy discrete energy levels called bands. When two atoms join together to form a diatomic element (such as Hydrogen), their orbitals overlap. The Pauli Exclusion Principle states that no two electrons can have the same quantum numbers. Now keep in mind that there are four types of quantum numbers. This means that when these two atoms combine the atomic orbitals must split to compensate so that no two electrons have the same energy. However for a macroscopic piece of a solid, the number of atoms is quite high (on the power of 10^22) and therefore the number of energy levels is also high. For this reason, adjacent energy levels are almost continuous, forming an energy band. The main bands under consideration are the valence (outermost band involved in chemical bonding) and conduction because the inner electron bands are so narrow. Band gaps or “forbidden zones” are leftover energy levels that are not covered by a band.

In order to apply band theory to a solid, the medium must be homogeneous or evenly distributed. The size of material must be considerable as well, which is not unreasonable considering the number of atoms in an appreciable piece of a solid. The assumption also must include that electrons do not interract with phonons or photons.

The “density of states” is a function that describes the number of states per unit volume, per unit energy. It is represented by a Probability Density function.

A Fermi-Dirac distribution function demonstrates the probability of a state of energy being filled with an electron. The probability is given below.

fuck

The μ is generally expressed as EF which is the Fermi energy level or total chemical potential. kT is the familiar thermal energy which is the product of the Boltzmann constant and the temperature. From this equation it is clear that absolute zero temperature, the exponential term increases to infinity, causing the entire term to trend to zero. This leads to the conclusion that semiconductors behave as insulators at 0K.

The density of electrons can be calculated by multiplying this value with the density of states function and integrating over all energy.

Band-gap engineering is the process of changing a material’s band gap. This is usually done to semiconductors by changing the composition of alloys in the material.

Object Oriented Programming and C#: Dictionaries/Hash Tables

A “dictionary” in C# is a ADT (Abstract data type) that maps “keys” to “values”. Normally with an array, the values within this collection of data are accessed using indexing. For the dictionary, instead of indexes there are keys. Another name for a dictionary is a “hash table”, although the distinction can be made in the sense that the hash table is a non-generic type and the dictionary is of generic type. The namespace required for dictionaries is the “System.Collections.Generics” namespace.

The dictionary is initialized much like a list (dynamic array), however the dictionary take two parameters (“TKey”,”TValue”). The first is the data type of the key and the second the data type of the value. Similarly to dynamic arrays, values can be added to the dictionary using a “Add(key,value)” command. Similarly, a value can be deleted using the “Delete(key)” command. However it is important to note that keys do not have to be integers, unlike an index. They can be of any data type imaginable. However, a dictionary cannot contain duplicate keys.

The functionality of a dictionary in C# is similar to a physical dictionary. A dictionary contains words and their definitions and analogously, a programming dictionary maps a key (word) to a value (definition).

The following program illustrates adding values to a dictionary. The key is of type integer and the value of type string. The values “one”, “two” and “three” are added with corresponding integer keys.

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Much like with arrays, a “foreach” statement can be used to iterate over all the values of a dictionary.

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It is important to note for a hash table, the relationship between the key and its value as that this must be one to one. When different keys have the same hash value, a “collision” occurs. In order to resolve the collision, a link list must be created in order to chain elements to a single location.

An important concept with hash table: speed of processing does not depend on size. For arrays, in order to find a specific value a linear search must be performed. This takes a long time to complete if the array is very long. With a hash table, size does not matter because the hashing function is a constant time. The “ContainsKey()” method can be used to find a specific key without the need for a linear search.

When would you use a dictionary/hash table over a list? Dictionaries can be helpful in instances where indexes have special meaning. A particular use of a dictionary could be to count the words in a text using the “String.Split()” method and adding each word to the dictionary. In this instance, the “foreach” statement could easily be used to iterate over every value and find the number of words. In short, the dictionary maps meaningful keys to values whereas the list simply maps indexes to values.