Electron hole

In physics, chemistry, and electronic engineering, an electron hole (often simply called a hole) is the lack of an electron at a position where one could exist in an atom or atomic lattice. Since in a normal atom or crystal lattice the negative charge of the electrons is balanced by the positive charge of the atomic nuclei, the absence of an electron leaves a net positive charge at the hole's location. Holes are not actually particles, but rather quasiparticles; they are different from the positron, which is the antiparticle of the electron. (See also Dirac sea.)

Holes in a metal[1] or semiconductor crystal lattice can move through the lattice as electrons can, and act similarly to positively-charged particles. They play an important role in the operation of semiconductor devices such as transistors, diodes and integrated circuits. If an electron is excited into a higher state it leaves a hole in its old state. This meaning is used in Auger electron spectroscopy (and other x-ray techniques), in computational chemistry, and to explain the low electron-electron scattering-rate in crystals (metals, semiconductors).

In crystals, electronic band structure calculations lead to an effective mass for the electrons, which is typically negative at the top of a band. The negative mass is an unintuitive concept,[2] and in these situations a more familiar picture is found by considering a positive charge with a positive mass.

Electron-hole
When an electron leaves a helium atom, it leaves an electron hole in its place. This causes the helium atom to become positively charged.

Solid-state physics

In solid-state physics, an electron hole (usually referred to simply as a hole) is the absence of an electron from a full valence band. A hole is essentially a way to conceptualize the interactions of the electrons within a nearly full valence band of a crystal lattice, which is missing a small fraction of its electrons. In some ways, the behavior of a hole within a semiconductor crystal lattice is comparable to that of the bubble in a full bottle of water.[3]

Simplified analogy: Empty seat in an auditorium

15-puzzle-02
A children's puzzle which illustrates the mobility of holes in an atomic lattice. The tiles are analogous to electrons, while the missing tile (lower right corner) is analogous to a hole. Just as the position of the missing tile can be moved to different locations by moving the tiles, a hole in a crystal lattice can move to different positions in the lattice by the motion of the surrounding electrons.

Hole conduction in a valence band can be explained by the following analogy. Imagine a row of people seated in an auditorium, where there are no spare chairs. Someone in the middle of the row wants to leave, so he jumps over the back of the seat into another row, and walks out. The empty row is analogous to the conduction band, and the person walking out is analogous to a conduction electron.

Now imagine someone else comes along and wants to sit down. The empty row has a poor view; so he does not want to sit there. Instead, a person in the crowded row moves into the empty seat the first person left behind. The empty seat moves one spot closer to the edge and the person waiting to sit down. The next person follows, and the next, et cetera. One could say that the empty seat moves towards the edge of the row. Once the empty seat reaches the edge, the new person can sit down.

In the process everyone in the row has moved along. If those people were negatively charged (like electrons), this movement would constitute conduction. If the seats themselves were positively charged, then only the vacant seat would be positive. This is a very simple model of how hole conduction works.

Instead of analyzing the movement of an empty state in the valence band as the movement of many separate electrons, a single equivalent imaginary particle called a "hole" is considered. In an applied electric field, the electrons move in one direction, corresponding to the hole moving in the other. If a hole associates itself with a neutral atom, that atom loses an electron and becomes positive. Therefore, the hole is taken to have positive charge of +e, precisely the opposite of the electron charge.

In reality, due to the uncertainty principle of quantum mechanics, combined with the energy levels available in the crystal, the hole is not localizable to a single position as described in the previous example. Rather, the positive charge which represents the hole spans an area in the crystal lattice covering many hundreds of unit cells. This is equivalent to being unable to tell which broken bond corresponds to the "missing" electron. Conduction band electrons are similarly delocalized.

Detailed picture: A hole is the absence of a negative-mass electron

BandDiagram-Semiconductors-E
A semiconductor electronic band structure (right) includes the dispersion relation of each band, i.e. the energy of an electron E as a function of the electron's wavevector k. The "unfilled band" is the semiconductor's conduction band; it curves upward indicating positive effective mass. The "filled band" is the semiconductor's valence band; it curves downward indicating negative effective mass.

The analogy above is quite simplified, and cannot explain why holes create an opposite effect to electrons in the Hall effect and Seebeck effect. A more precise and detailed explanation follows.[4]

A dispersion relation is the relationship between wavevector (k-vector) and energy in a band, part of the electronic band structure. In quantum mechanics, the electrons are waves, and energy is the wave frequency. A localized electron is a wavepacket, and the motion of an electron is given by the formula for the group velocity of a wave. An electric field affects an electron by gradually shifting all the wavevectors in the wavepacket, and the electron accelerates when its wave group velocity changes. Therefore, again, the way an electron responds to forces is entirely determined by its dispersion relation. An electron floating in space has the dispersion relation E=ℏ2k2/(2m), where m is the (real) electron mass and ℏ is reduced Planck constant. Near the bottom of the conduction band of a semiconductor, the dispersion relation is instead E=ℏ2k2/(2m*) (m* is the effective mass), so a conduction-band electron responds to forces as if it had the mass m*.

The dispersion relation near the top of the valence band is E=ℏ2k2/(2m*) with negative effective mass. So electrons near the top of the valence band behave like they have negative mass. When a force pulls the electrons to the right, these electrons actually move left. This is solely due to the shape of the valence band, and is unrelated to whether the band is full or empty. If you could somehow empty out the valence band and just put one electron near the valence band maximum (an unstable situation), this electron would move the "wrong way" in response to forces.

  • Positively-charged holes as a shortcut for calculating the total current of an almost-full band.[4]

A perfectly full band always has zero current. One way to think about this fact is that the electron states near the top of the band have negative effective mass, and those near the bottom of the band have positive effective mass, so the net motion is exactly zero. If an otherwise-almost-full valence band has a state without an electron in it, we say that this state is occupied by a hole. There is a mathematical shortcut for calculating the current due to every electron in the whole valence band: Start with zero current (the total if the band were full), and subtract the current due to the electrons that would be in each hole state if it wasn't a hole. Since subtracting the current caused by a negative charge in motion is the same as adding the current caused by a positive charge moving on the same path, the mathematical shortcut is to pretend that each hole state is carrying a positive charge, while ignoring every other electron state in the valence band.

  • A hole near the top of the valence band moves the same way as an electron near the top of the valence band would move[4] (which is in the opposite direction compared to conduction-band electrons experiencing the same force.)

This fact follows from the discussion and definition above. This is an example where the auditorium analogy above is misleading. When a person moves left in a full auditorium, an empty seat moves right. But in this section we are imagining how electrons move through k-space, not real space, and the effect of a force is to move all the electrons through k-space in the same direction at the same time. In this context, a better analogy is a bubble underwater in a river: The bubble moves the same direction as the water, not opposite.

Since force = mass × acceleration, a negative-effective-mass electron near the top of the valence band would move the opposite direction as a positive-effective-mass electron near the bottom of the conduction band, in response to a given electric or magnetic force. Therefore, a hole moves this way as well.

  • Conclusion: Hole is a positive-charge, positive-mass quasiparticle.

From the above, a hole (1) carries a positive charge, and (2) responds to electric and magnetic fields as if it had a positive charge and positive mass. (The latter is because a particle with positive charge and positive mass responds to electric and magnetic fields in the same way as a particle with negative charge and negative mass.) That explains why holes can be treated in all situations as ordinary positively charged quasiparticles.

Role in semiconductor technology

In some semiconductors, such as silicon, the hole's effective mass is dependent on direction (anisotropic), however a value averaged over all directions can be used for some macroscopic calculations.

In most semiconductors, the effective mass of a hole is much larger than that of an electron. This results in lower mobility for holes under the influence of an electric field and this may slow down the speed of the electronic device made of that semiconductor. This is one major reason for adopting electrons as the primary charge carriers, whenever possible in semiconductor devices, rather than holes. Also, why NMOS logic is faster than PMOS logic.

However, in many semiconductor devices, both electrons and holes play an essential role. Examples include p–n diodes, bipolar transistors, and CMOS logic.

Holes in quantum chemistry

An alternate meaning for the term electron hole is used in computational chemistry. In coupled cluster methods, the ground (or lowest energy) state of a molecule is interpreted as the "vacuum state"—conceptually, in this state there are no electrons. In this scheme, the absence of an electron from a normally filled state is called a "hole" and is treated as a particle, and the presence of an electron in a normally empty state is simply called an "electron". This terminology is almost identical to that used in solid-state physics.

See also

References

  1. ^ Ashcroft and Mermin (1976). Solid State Physics (1st ed.). Holt, Reinhart, and Winston. pp. 299–302. ISBN 978-0030839931.
  2. ^ For these negative mass electrons, momentum is opposite to velocity, so forces acting on these electrons cause their velocity to change in the 'wrong' direction. As these electrons gain energy (moving towards the top of the band), they slow down.
  3. ^ Weller, Paul F. (1967). "An analogy for elementary band theory concepts in solids". J. Chem. Educ. 44 (7): 391. Bibcode:1967JChEd..44..391W. doi:10.1021/ed044p391.
  4. ^ a b c d e Kittel, Introduction to Solid State Physics, 8th edition, pp. 194–196.
Andreev reflection

Andreev reflection (AR), named after the Russian physicist Alexander F. Andreev, is a type of particle scattering which

occurs at interfaces between a superconductor (S) and a normal state material (N). It is a charge-transfer process by which normal current in N is converted to supercurrent in S. Each Andreev reflection transfers a charge 2e across the interface, avoiding the forbidden single-particle transmission within the superconducting energy gap.

Auger effect

The Auger effect is a physical phenomenon in which the filling of an inner-shell vacancy of an atom is accompanied by the emission of an electron from the same atom. When a core electron is removed, leaving a vacancy, an electron from a higher energy level may fall into the vacancy, resulting in a release of energy. Although most often this energy is released in the form of an emitted photon, the energy can also be transferred to another electron, which is ejected from the atom; this second ejected electron is called an Auger electron. The effect was first discovered by Lise Meitner in 1922; Pierre Victor Auger independently discovered the effect shortly after and is credited with the discovery in most of the scientific community.Upon ejection, the kinetic energy of the Auger electron corresponds to the difference between the energy of the initial electronic transition into the vacancy and the ionization energy for the electron shell from which the Auger electron was ejected. These energy levels depend on the type of atom and the chemical environment in which the atom was located.

Auger electron spectroscopy involves the emission of Auger electrons by bombarding a sample with either X-rays or energetic electrons and measures the intensity of Auger electrons that result as a function of the Auger electron energy. The resulting spectra can be used to determine the identity of the emitting atoms and some information about their environment.

Auger recombination is a similar Auger effect which occurs in semiconductors. An electron and electron hole (electron-hole pair) can recombine giving up their energy to an electron in the conduction band, increasing its energy. The reverse effect is known as impact ionization.

The Auger effect can impact biological molecules such as DNA. Following the K-shell ionization of the component atoms of DNA, Auger electrons are ejected leading to damage of its sugar-phosphate backbone.

Avalanche breakdown

Avalanche breakdown is a phenomenon that can occur in both insulating and semiconducting materials. It is a form of electric current multiplication that can allow very large currents within materials which are otherwise good insulators. It is a type of electron avalanche. The avalanche process occurs when carriers in the transition region are accelerated by the electric field to energies sufficient to create mobile or free electron-hole pairs via collisions with bound electrons.

Avalanche photodiode

An avalanche photodiode (APD) is a highly sensitive semiconductor electronic device that exploits the photoelectric effect to convert light to electricity. From a functional standpoint, they can be regarded as the semiconductor analog of photomultipliers. By applying a high reverse bias voltage (typically 100–200 V in silicon), APDs show an internal current gain effect (around 100) due to impact ionization (avalanche effect). However, some silicon APDs employ alternative doping and beveling techniques compared to traditional APDs that allow greater voltage to be applied (> 1500 V) before breakdown is reached and hence a greater operating gain (> 1000). In general, the higher the reverse voltage, the higher the gain. Among the various expressions for the APD multiplication factor (M), an instructive expression is given by the formula

where L is the space-charge boundary for electrons, and is the multiplication coefficient for electrons (and holes). This coefficient has a strong dependence on the applied electric field strength, temperature, and doping profile. Since APD gain varies strongly with the applied reverse bias and temperature, it is necessary to control the reverse voltage to keep a stable gain. Avalanche photodiodes therefore are more sensitive compared to other semiconductor photodiodes.

If very high gain is needed (105 to 106), certain APDs (single-photon avalanche diodes) can be operated with a reverse voltage above the APD's breakdown voltage. In this case, the APD needs to have its signal current limited and quickly diminished. Active and passive current-quenching techniques have been used for this purpose. APDs that operate in this high-gain regime are in Geiger mode. This mode is particularly useful for single-photon detection, provided that the dark count event rate and afterpulsing probability are sufficiently low.

Typical applications for APDs are laser rangefinders, long-range fiber-optic telecommunication, and quantum sensing for control algorithms. New applications include positron emission tomography and particle physics. APD arrays are becoming commercially available, also lightning detection and optical SETI may be a future application.

APD applicability and usefulness depends on many parameters. Two of the larger factors are: quantum efficiency, which indicates how well incident optical photons are absorbed and then used to generate primary charge carriers; and total leakage current, which is the sum of the dark current and photocurrent and noise. Electronic dark-noise components are series and parallel noise. Series noise, which is the effect of shot noise, is basically proportional to the APD capacitance, while the parallel noise is associated with the fluctuations of the APD bulk and surface dark currents. Another noise source is the excess noise factor, ENF. It is a multiplicative correction applied to the noise that describes the increase in the statistical noise, specifically Poisson noise, due to the multiplication process. The ENF is defined for any device, such as photomultiplier tubes, silicon solid-state photomultipliers, and APDs, that multiplies a signal, and is sometimes referred to as "gain noise".

The noise term for an APD may also contain a Fano factor, which is a multiplicative correction applied to the Poisson noise associated with the conversion of the energy deposited by a charged particle to the electron-hole pairs, which is the signal before multiplication. The correction factor describes the decrease in the noise, relative to Poisson statistics, due to the uniformity of conversion process and the absence of, or weak coupling to, bath states in the conversion process. In other words, an "ideal" semiconductor would convert the energy of the charged particle into an exact and reproducible number of electron hole pairs to conserve energy; in reality, however, the energy deposited by the charged particle is divided into the generation of electron hole pairs, the generation of sound, the generation of heat, and the generation of damage or displacement. The existence of these other channels introduces a stochastic process, where the amount of energy deposited into any single process varies from event to event, even if the amount of energy deposited is the same.

The underlying physics associated with the excess noise factor (gain noise) and the Fano factor (conversion noise) is very different. However, the application of these factors as multiplicative corrections to the expected Poisson noise is similar.

Band gap

In solid-state physics, a band gap, also called an energy gap or bandgap, is an energy range in a solid where no electron states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference (in electron volts) between the top of the valence band and the bottom of the conduction band in insulators and semiconductors. It is the energy required to promote a valence electron bound to an atom to become a conduction electron, which is free to move within the crystal lattice and serve as a charge carrier to conduct electric current. It is closely related to the HOMO/LUMO gap in chemistry. If the valence band is completely full and the conduction band is completely empty, then electrons cannot move in the solid; however, if some electrons transfer from the valence to the conduction band, then current can flow (see carrier generation and recombination). Therefore, the band gap is a major factor determining the electrical conductivity of a solid. Substances with large band gaps are generally insulators, those with smaller band gaps are semiconductors, while conductors either have very small band gaps or none, because the valence and conduction bands overlap.

Carrier generation and recombination

In the solid-state physics of semiconductors, carrier generation and carrier recombination are processes by which mobile charge carriers (electrons and electron holes) are created and eliminated. Carrier generation and recombination processes are fundamental to the operation of many optoelectronic semiconductor devices, such as photodiodes, light-emitting diodes and laser diodes. They are also critical to a full analysis of p-n junction devices such as bipolar junction transistors and p-n junction diodes.

The electron–hole pair is the fundamental unit of generation and recombination, corresponding to an electron transitioning between the valence band and the conduction band where generation of electron is a transition from the valence band to the conduction band and recombination leads to a reverse transition.

Carson D. Jeffries

Carson Dunning Jeffries (March 22, 1922 – October 18, 1995) was an American physicist.

The National Academies Press said that Jeffries "made major fundamental contributions to knowledge of nuclear magnetism, electronic spin relaxation, dynamic nuclear polarization, electron-hole droplets, nonlinear dynamics and chaos, and high-temperature superconductors."

He was noted for being the first to observe the isotropic spin-spin exchange interaction in metals (also known as the Ruderman-Kittel interaction).

He also discovered methods for the dynamic nuclear polarization by saturation of forbidden microwave resonance transitions in solids.

He also discovered the existence of giant electron-hole droplets in semiconductors.

He was a member of the U.S. National Academy of Sciences and the American Academy of Arts and Sciences.

Dropleton

A dropleton or quantum droplet is a quasiparticle comprising a collection of electrons and holes inside a semiconductor. Dropletons give the first known quasiparticle characterization where the quasiparticle behaves like a liquid. The creation of dropletons was announced on 26 February 2014 in a Nature article, which presented evidence for the creation of dropletons in an electron–hole plasma inside a gallium arsenide quantum well by ultrashort laser pulses. Their existence was not predicted before the experiment.

Despite the relatively short lifetime of about 25 picoseconds, the dropletons are stable enough to be studied and possess favorable properties for certain investigations of quantum mechanics. They are approximately 200 nanometers wide, the size of the smallest bacteria, for which reason the discoverers have expressed hope that they might one day actually see quantum droplets.

Electron-hole droplets

Electron-hole droplets are a condensed phase of excitons in semiconductors. The droplets are formed at low temperatures and high exciton densities, the latter of which can be created with intense optical excitation or electronic excitation in a p-n junction.

Elliott formula

The Elliott formula describes analytically, or with few adjustable parameters such as the dephasing constant, the light absorption or emission spectra of solids. It was originally derived by Roger James Elliott to describe linear absorption based on properties of a single electron–hole pair. The analysis can be extended to a many-body investigation with full predictive powers when all parameters are computed microscopically using, e.g., the semiconductor Bloch equations (abbreviated as SBEs) or the semiconductor luminescence equations (abbreviated as SLEs).

Exciton

An exciton is a bound state of an electron and an electron hole which are attracted to each other by the electrostatic Coulomb force. It is an electrically neutral quasiparticle that exists in insulators, semiconductors and some liquids. The exciton is regarded as an elementary excitation of condensed matter that can transport energy without transporting net electric charge. An exciton can form when a material absorbs a photon of higher energy than its bandgap . This excites an electron from the valence band into the conduction band. In turn, this leaves behind a positively charged electron hole (an abstraction for the location from which an electron was moved). The electron in the conduction band is then effectively attracted to this localized hole by the repulsive Coulomb forces from large numbers of electrons surrounding the hole and excited electron. This attraction provides a stabilizing energy balance. Consequently, the exciton has slightly less energy than the unbound electron and hole. The wavefunction of the bound state is said to be hydrogenic, an exotic atom state akin to that of a hydrogen atom. However, the binding energy is much smaller and the particle's size much larger than a hydrogen atom. This is because of both the screening of the Coulomb force by other electrons in the semiconductor (i.e., its dielectric constant), and the small effective masses of the excited electron and hole. The recombination of the electron and hole, i.e. the decay of the exciton, is limited by resonance stabilization due to the overlap of the electron and hole wave functions, resulting in an extended lifetime for the exciton.

The electron and hole may have either parallel or anti-parallel spins. The spins are coupled by the exchange interaction, giving rise to exciton fine structure. In periodic lattices, the properties of an exciton show momentum (k-vector) dependence.

The concept of excitons was first proposed by Yakov Frenkel in 1931, when he described the excitation of atoms in a lattice of insulators. He proposed that this excited state would be able to travel in a particle-like fashion through the lattice without the net transfer of charge.

Impact ionization

Impact ionization is the process in a material by which one energetic charge carrier can lose energy by the creation of other charge carriers. For example, in semiconductors, an electron (or hole) with enough kinetic energy can knock a bound electron out of its bound state (in the valence band) and promote it to a state in the conduction band, creating an electron-hole pair. For carriers to have sufficient kinetic energy a sufficiently large electric field must be applied, in essence requiring a sufficiently large voltage but not necessarily a large current.

If this occurs in a region of high electrical field then it can result in avalanche breakdown. This process is exploited in avalanche diodes, by which a small optical signal is amplified before entering an external electronic circuit. In an avalanche photodiode the original charge carrier is created by the absorption of a photon.

In some sense, impact ionization is the reverse process to Auger recombination.

Avalanche photodiodes (APD) are used in optical receivers before the signal is given to the receiver circuitry the photon is multiplied with the photocurrent and this increases the sensitivity of the receiver since photocurrent is multiplied before encountering of the thermal noise associated with the receiver circuit.

Inelastic mean free path

The inelastic mean free path (IMFP) is an index of how far an electron on average travels through a solid before losing energy.

If a monochromatic primary beam of electrons is incident on a solid surface, the majority of incident electrons lose their energy because they interact strongly with matter, leading to plasmon excitation, electron-hole pair formation, and vibrational excitation. The intensity of the primary electrons, , is damped as a function of the distance, d, into the solid. The intensity decay can be expressed as follows:

where is the intensity after the primary electron beam has traveled through the solid. The parameter , termed the inelastic mean free path (IMFP), is defined as the distance an electron beam can travel before its intensity decays to of its initial value. The inelastic mean free path of electrons can roughly be described by a universal curve, which is the same for all materials.

Light-induced voltage alteration

Light-induced voltage alteration (LIVA) is a semiconductor analysis technique that uses a laser or infrared light source to induce voltage changes in a device while scanning the beam of light across its surface. The technique relies upon the generation of electron-hole pairs in the semiconductor material when exposed to photons.

List of quasiparticles

This is a list of quasiparticles.

Photocatalysis

In chemistry, photocatalysis is the acceleration of a photoreaction in the presence of a catalyst. In catalysed photolysis, light is absorbed by an adsorbed substrate. In photogenerated catalysis, the photocatalytic activity (PCA) depends on the ability of the catalyst to create electron–hole pairs, which generate free radicals (e.g. hydroxyl radicals: •OH) able to undergo secondary reactions. Its practical application was made possible by the discovery of water electrolysis by means of titanium dioxide (TiO2).

Photodetector

Photodetectors, also called photosensors, are sensors of light or other electromagnetic radiation. A photo detector has a p–n junction that converts light photons into current. The absorbed photons make electron–hole pairs in the depletion region. Photodiodes and photo transistors are a few examples of photo detectors. Solar cells convert some of the light energy absorbed into electrical energy.

Semiconductor

A semiconductor material has an electrical conductivity value falling between that of a conductor, such as metallic copper, and an insulator, such as glass. Its resistance decreases as its temperature increases, which is behaviour opposite to that of a metal. Its conducting properties may be altered in useful ways by the deliberate, controlled introduction of impurities ("doping") into the crystal structure. Where two differently-doped regions exist in the same crystal, a semiconductor junction is created. The behavior of charge carriers which include electrons, ions and electron holes at these junctions is the basis of diodes, transistors and all modern electronics. Some examples of semiconductors are silicon, germanium, gallium arsenide, and elements near the so-called "metalloid staircase" on the periodic table. After silicon, gallium arsenide is the second most common semiconductor and is used in laser diodes, solar cells, microwave-frequency integrated circuits and others. Silicon is a critical element for fabricating most electronic circuits.

Semiconductor devices can display a range of useful properties such as passing current more easily in one direction than the other, showing variable resistance, and sensitivity to light or heat. Because the electrical properties of a semiconductor material can be modified by doping, or by the application of electrical fields or light, devices made from semiconductors can be used for amplification, switching, and energy conversion.

The conductivity of silicon is increased by adding a small amount (of the order of 1 in 108) of pentavalent (antimony, phosphorus, or arsenic) or trivalent (boron, gallium, indium) atoms. This process is known as doping and resulting semiconductors are known as doped or extrinsic semiconductors. Apart from doping, the conductivity of a semiconductor can equally be improved by increasing its temperature. This is contrary to the behaviour of a metal in which conductivity decreases with increase in temperature.

The modern understanding of the properties of a semiconductor relies on quantum physics to explain the movement of charge carriers in a crystal lattice. Doping greatly increases the number of charge carriers within the crystal. When a doped semiconductor contains mostly free holes it is called "p-type", and when it contains mostly free electrons it is known as "n-type". The semiconductor materials used in electronic devices are doped under precise conditions to control the concentration and regions of p- and n-type dopants. A single semiconductor crystal can have many p- and n-type regions; the p–n junctions between these regions are responsible for the useful electronic behavior.

Some of the properties of semiconductor materials were observed throughout the mid 19th and first decades of the 20th century. The first practical application of semiconductors in electronics was the 1904 development of the cat's-whisker detector, a primitive semiconductor diode used in early radio receivers. Developments in quantum physics in turn allowed the development of the transistor in 1947 and the integrated circuit in 1958.

Semiconductor luminescence equations

The semiconductor luminescence equations (SLEs) describe luminescence of semiconductors resulting from spontaneous recombination of electronic excitations, producing a flux of spontaneously emitted light. This description established the first step toward semiconductor quantum optics because the SLEs simultaneously includes the quantized light–matter interaction and the Coulomb-interaction coupling among electronic excitations within a semiconductor. The SLEs are one of the most accurate methods to describe light emission in semiconductors and they are suited for a systematic modeling of semiconductor emission ranging from excitonic luminescence to lasers.

Due to randomness of the vacuum-field fluctuations, semiconductor luminescence is incoherent whereas the extensions of the SLEs include the possibility to study resonance fluorescence resulting from optical pumping with coherent laser light. At this level, one is often interested to control and access higher-order photon-correlation effects, distinct many-body states, as well as light–semiconductor entanglement. Such investigations are the basis of realizing and developing the field of quantum-optical spectroscopy which is a branch of quantum optics.

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