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TransistorsThe two main types of transistors are the bipolar junction transistor (BJT) and the field-effect transistor (FET). Bipolar Junction TransistorsBJTs can have two different polarities, NPN and PNP. An NPN BJT is one where a positively-doped (P-type) semiconductor is sandwiched between two negatively-doped (N-type) semiconductors. A PNP BJT is, obviously, one where an N-type semiconductor is sandwiched between two P-types. Both types of BJTs have an exponential dependence between the input voltage and the current output. For the record, I should state that a semiconductor is basically a material with conductance between that of an insulator and a conductor. Silicon and germanium are the two most well-known semiconductors. Also, doping just means the addition of impurities into a semiconducting material in order for it to either: increase its electron acceptance (P-type) or increase its electron conductance (N-type).

Some specific types of BJTs:

HBT - heterojunction bipolar transistor - These types of transistors are very similar to BJTs except that the two P-type semiconductors in the PNP polarity, or the two N-type semiconductors in the NPN polarity, are doped differently relative to each other. The reason for doing this, simply stated, is to make it more difficult for a transistor to operate in the reverse direction from which is was intended.

Grown-junction transistor - This was the first type of BJT and is self-explanatory. The PN or NP junctions, depending on whether it's of NPN or PNP polarity, respectively, are grown onto a single, solid crystal of semiconductor material. Grown, in this case, means slowly attached, chemically.

Alloy-junction transistor - Similar to a grown-junction transistor except the semiconducting material onto which the PN or NP junctions are grown is specifically germanium.

MAT - micro-alloy transistor - An improved, speedier version of the alloy-junction transistor. The materials of the PN or NP junctions of a MAT are metal-semiconductor, as opposed to semiconductor-semiconductor.

MADT - micro-alloy diffused transistor - An improved, speedier version of the MAT. The dopant material of a MADT is diffused (thinly spread) accross the entire germanium crystal prior to PN or NP growth, as opposed to a MAT where the doping material is only on the metallic side of the PN or NP junction.

PADT - post-alloy diffused transistor - An improved, speedier version of the MADT. A thin, diffused dopant layer of germanium is grown onto the germanium crystal, as opposed to the entire germanium crystal being diffused, which allows the germanium crystal to be as thick as necessary for mechanical strength purposes. The PN or NP junctions are then grown onto this thin layer.

Schottky transistor - These are alloy-junction transistors with a Schottky barrier between the metal-semiconductor junction. All metal-semiconductor junctions act sort of like capacitors with a voltage between the junctions. Often, you'd like to minimize this voltage in order to minimize the saturation (the amount of the germanium crystal) needed for the transistor to work. Minimizing the saturation effectively speeds up the transistor's performance, which is great for things like switches. Schottky barriers use various materials to do exactly this.

Surface-barrier transistor - These are just like Schottky transistors except that both junctions are metal-semiconductor as opposed to only one.

Drift-field transistor - The doping agent of these transistors is engineered to produce a specific electric field. This effectually reduces the electrons' transit time between the junctions of the transistor, thereby making it work faster.

Avalanche transistor - These transistors can operate in the breakdown voltage region of a transistor's junctions. The breakdown voltage is simply the minimum voltage in which an insulator starts acting like a conductor. Thus, these transistors allow for higher currents to be applied to them than their normal counterparts.

Darlington transistor - These are simply two BJTs connected together to further increase the gain of the current output.

IGBT - insulated-gate bipolar transistor - These transistors combine the use of BJTs as switches with an isolated-gate FET (see below) as the input. IGBTs provide much more efficient and faster switching than regular BJTs and are thus some of the most common transistors found in modern appliances.

Photo transistor - These transistors convert electromagnetic radiation in the form of visible light, UV-rays, or X-rays into current or voltage. As opposed to the normal PN junctions found in many transistors, photo transistors use PIN junctions. PIN junctions are similar to PN junctions except that they have an additional intrinsic semiconductor between the P-type and N-type semiconducting regions. This intrinsic semiconductor is a very lightly doped semiconductor which exists, at least for the purposes of photo transistors, to supply a region within the junction where a photon (a particle of electromagnetic radiation with a specific energy) can ionize (knock an electron out of via the photoelectric effect) an atom of this semiconducting material. Because of the electric field caused from the surrounding P-type and N-type semiconducting regions, this ionization causes the photoelectron to move toward one end of the junction, thereby producing what's known as a photocurrent, which is then amplified in the same manner as all other BJTs. I promise that the rest of my answer won't get more complicated than this.

Field-Effect TransistorsFETs use electric fields to control only one-type of charge carrier, as opposed to BJTs which control both types. Now's as good a time as any to introduce the concept of electron holes. Intuitively, electrons carry negative charge and are thus referred to as negative charge carriers. Well, the absence of an electron where one used to be is called an electron hole. These holes act exactly as electrons do in transistors except that they carry positive charge, in the form of missing negative charge, and are thus called positive charge carriers. FETs are designed to control either positive or negative charge carriers, in the form of holes or electrons, but not both. The flow of positive or negative charge carriers occurs through what's called the channel of an FET. FET channels are created within the bulk material of the FET, which is usually silicon. If you find this idea more complicated than what I wrote about photo transistors, that's only because you haven't looked up the physics behind the photoelectric effect yet.

Some specific types of FETs:

CNTFET - carbon nanotube field-effect transistor - These FETs use carbon nanotubes instead of silicon as their channel material. Carbon nanotubes are needed as FETs continue to get smaller in size. They help reduce effects, such as quantum tunneling and overheating, which are beginning to become real problems in small, silicon-based FETs.

JFET - junction gate field-effect transistor - This FET supplies a voltage accross the charge-carrying channel that can pinch it shut, effectively stopping the current through the channel.

MESFET - metal semiconductor field-effect transistor - Similar to, but faster than, JFETs, MESFETs use a Schottky barrier (see above) instead of a PN junction.

HEMT - high electron mobility transistor - The FET version of an HBT (see above). Faster than a MESFET, the charge-carrying channel is between two different materials instead of within a single, doped region. Also known as a heterostructure FET (HFET) or a modulation-doped FET (MODFET).

MOSFET - metal-oxide-semiconductor field-effect transistor - This is the most basic, and most common, type of FET, analogous to the standard BJT (see above). Instead of pinching its charge-carrying channel shut as in a JFET, a MOSFET has an insulator attached to its input electrode which can be turned on or off depending on whether a voltage is supplied accross it. The channel can be N-type (nMOS) or P-type (pMOS), as explained above under the "bipolar junction transistors" heading.

ITFET - inverted-T field-effect transistor - This is simply any type of FET that extends vertically out from the horizontal plane in a T-shape, hence the name.

MuGFET - multiple gate field-effect transistor - A MOSFET where more than one input shares the bulk material of the FET. The idea is to use the same FET, thus the same sized object, for multiple things. This concept came about due to the ever shrinking sizes of transistors.

MIGFET - multiple independent gate field-effect transistor - A MuGFET where the multiple inputs are independently controlled.

Flexfet - A MIGFET with two inputs, one on a JFET and the other on a MOSFET. The JFET and MOSFET are then "stacked" on top of each other. Due to its design, the JFET and MOSFET are coupled to each other; i.e. the channel through one effects the channel through the other and vice versa.

FinFET - A MuGFET where the charge-carrying channel is wrapped around a piece of silicon, called a fin. The reason for doing this is similar to that of a PADT (see above); i.e. mechanical strength.

FREDFET - fast-recovery (or reverse) epitaxial diode field-effect transistor - A cute name for a transistor which is basically designed to quickly turn off when no more voltage is being supplied to it.

TFT - thin-film transistor - An FET where the semiconducting material is placed via thin films over the bulk of the device. This is opposed to the bulk of the device being the semiconductor itself, as in most FETs. The bulk material used in TFTs is often glass. The reason being so that the transistors can work behind a clear display in applications like liquid crystal display (LCD) monitors.

OFET - organic field-effect transistor - An FET with an organic polymer semiconductor as its channel. These are like TFTs except the bulk of the device is plastic, allowing for very cool, flexible LCD monitors.

FGMOS - floating gate MOSFET - A MOSFET with a "floating gate" input; i.e. an electrically isolated input that can store charge, like a capacitor, to be used later. These are the transistors behind flash drives.

ISFET - ion-sensitive field-effect transistor - An FET that changes its current depending on the ion concentration of a solution. The solution itself is used as the input electrode in an ISFET.

EOSFET - electrolyte-oxide-semiconductor field-effect transistor - A MOSFET with the metal replaced by an electrolyte solution. EOSFETs are used to in neurochips to detect brain activity.

DNAFET - Deoxyribonucleic acid (DNA) field-effect transistor - A MOSFET with its input electrode being a layer of immobilized, single-stranded DNA. The current through the MOSFET is modulated by the varying charge distributions that occur when complimentary DNA strands hybridize to the layer of single-stranded DNA on the input electrode. DNAFETs are used, not surprisingly, in DNA sequencing.

My sources all stem from the link below which is also a great place to learn more about transistors.

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Q: What are the various types of transistors?
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