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114-491: The company makes dicing saws and laser saws to cut semiconductor silicon wafers and other materials; grinders to process silicon and compound semiconductor wafers to ultra-thin levels; polishing machines to remove the grinding damage layer from the wafer back-side and to increase chip strength. The company was founded as Daiichi-Seitosho in May 1937, as an industrial abrasive wheel manufacturer. After World War II Japan faced

228-480: A {\displaystyle a} larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle. As we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that

342-439: A cut-off frequency of one cycle per second, too low for any practical applications, but an effective application of the available theory. At Bell Labs , William Shockley and A. Holden started investigating solid-state amplifiers in 1938. The first p–n junction in silicon was observed by Russell Ohl about 1941 when a specimen was found to be light-sensitive, with a sharp boundary between p-type impurity at one end and n-type at

456-607: A mass-production basis, which limited them to a number of specialised applications. Quantum physics Quantum mechanics is a fundamental theory that describes the behavior of nature at and below the scale of atoms . It is the foundation of all quantum physics , which includes quantum chemistry , quantum field theory , quantum technology , and quantum information science . Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary ( macroscopic and (optical) microscopic ) scale, but

570-513: A common semi-insulator is gallium arsenide . Some materials, such as titanium dioxide , can even be used as insulating materials for some applications, while being treated as wide-gap semiconductors for other applications. The partial filling of the states at the bottom of the conduction band can be understood as adding electrons to that band. The electrons do not stay indefinitely (due to the natural thermal recombination ) but they can move around for some time. The actual concentration of electrons

684-423: A completely full valence band is inert, not conducting any current. If an electron is taken out of the valence band, then the trajectory that the electron would normally have taken is now missing its charge. For the purposes of electric current, this combination of the full valence band, minus the electron, can be converted into a picture of a completely empty band containing a positively charged particle that moves in

798-399: A construction boom which also helped DISCO to boost its sales. The company's grinder discs were in high demand from utility companies, which needed them to manufacture watt-meters. In December 1968 the company developed and released an ultra-thin resinoid cutting wheel, Microncut . The wheel contained diamond powder and as a result it was capable of making sharp, precision cuts as demanded in

912-466: A definite prediction of what the quantum state ψ ( t ) {\displaystyle \psi (t)} will be at any later time. Some wave functions produce probability distributions that are independent of time, such as eigenstates of the Hamiltonian . Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example,

1026-510: A family of unitary operators parameterized by a variable t {\displaystyle t} . Under the evolution generated by A {\displaystyle A} , any observable B {\displaystyle B} that commutes with A {\displaystyle A} will be conserved. Moreover, if B {\displaystyle B} is conserved by evolution under A {\displaystyle A} , then A {\displaystyle A}

1140-410: A guide to the construction of more capable and reliable devices. Alexander Graham Bell used the light-sensitive property of selenium to transmit sound over a beam of light in 1880. A working solar cell, of low efficiency, was constructed by Charles Fritts in 1883, using a metal plate coated with selenium and a thin layer of gold; the device became commercially useful in photographic light meters in

1254-471: A loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system. Just as density matrices specify the state of a subsystem of a larger system, analogously, positive operator-valued measures (POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory. As described above, entanglement

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1368-445: A low-pressure chamber to create plasma . A common etch gas is chlorofluorocarbon , or more commonly known Freon . A high radio-frequency voltage between the cathode and anode is what creates the plasma in the chamber. The silicon wafer is located on the cathode, which causes it to be hit by the positively charged ions that are released from the plasma. The result is silicon that is etched anisotropically . The last process

1482-426: A mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples. In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector ψ {\displaystyle \psi } belonging to a ( separable ) complex Hilbert space H {\displaystyle {\mathcal {H}}} . This vector

1596-417: A measurement of its position and also at the same time for a measurement of its momentum . Another consequence of the mathematical rules of quantum mechanics is the phenomenon of quantum interference , which is often illustrated with the double-slit experiment . In the basic version of this experiment, a coherent light source , such as a laser beam, illuminates a plate pierced by two parallel slits, and

1710-519: A pair is completed. Such carrier traps are sometimes purposely added to reduce the time needed to reach the steady-state. The conductivity of semiconductors may easily be modified by introducing impurities into their crystal lattice . The process of adding controlled impurities to a semiconductor is known as doping . The amount of impurity, or dopant, added to an intrinsic (pure) semiconductor varies its level of conductivity. Doped semiconductors are referred to as extrinsic . By adding impurity to

1824-471: A probability amplitude. Applying the Born rule to these amplitudes gives a probability density function for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The Schrödinger equation relates the collection of probability amplitudes that pertain to one moment of time to

1938-431: A semiconducting material would cause it to leave thermal equilibrium and create a non-equilibrium situation. This introduces electrons and holes to the system, which interact via a process called ambipolar diffusion . Whenever thermal equilibrium is disturbed in a semiconducting material, the number of holes and electrons changes. Such disruptions can occur as a result of a temperature difference or photons , which can enter

2052-426: A semiconductor is doped by Group III elements, they will behave like acceptors creating free holes, known as " p-type " doping. 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 device crystal can have many p- and n-type regions; the p–n junctions between these regions are responsible for

2166-501: A silicon atom in the crystal, a vacant state (an electron "hole") is created, which can move around the lattice and function as a charge carrier. Group V elements have five valence electrons, which allows them to act as a donor; substitution of these atoms for silicon creates an extra free electron. Therefore, a silicon crystal doped with boron creates a p-type semiconductor whereas one doped with phosphorus results in an n-type material. During manufacture , dopants can be diffused into

2280-405: A single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus , whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an s orbital ( Fig. 1 ). Analytic solutions of

2394-551: A single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy: The general solution of the Schrödinger equation is given by which is a superposition of all possible plane waves e i ( k x − ℏ k 2 2 m t ) {\displaystyle e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}} , which are eigenstates of

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2508-783: A theory of solid-state physics , which developed greatly in the first half of the 20th century. In 1878 Edwin Herbert Hall demonstrated the deflection of flowing charge carriers by an applied magnetic field, the Hall effect . The discovery of the electron by J.J. Thomson in 1897 prompted theories of electron-based conduction in solids. Karl Baedeker , by observing a Hall effect with the reverse sign to that in metals, theorized that copper iodide had positive charge carriers. Johan Koenigsberger  [ de ] classified solid materials like metals, insulators, and "variable conductors" in 1914 although his student Josef Weiss already introduced

2622-476: A vacuum, though with a different effective mass . Because the electrons behave like an ideal gas, one may also think about conduction in very simplistic terms such as the Drude model , and introduce concepts such as electron mobility . For partial filling at the top of the valence band, it is helpful to introduce the concept of an electron hole . Although the electrons in the valence band are always moving around,

2736-567: A variety of proportions. These compounds share with better-known semiconductors the properties of intermediate conductivity and a rapid variation of conductivity with temperature, as well as occasional negative resistance . Such disordered materials lack the rigid crystalline structure of conventional semiconductors such as silicon. They are generally used in thin film structures, which do not require material of higher electronic quality, being relatively insensitive to impurities and radiation damage. Almost all of today's electronic technology involves

2850-473: Is and this provides the lower bound on the product of standard deviations: Another consequence of the canonical commutation relation is that the position and momentum operators are Fourier transforms of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of

2964-415: Is a combination of processes that are used to prepare semiconducting materials for ICs. One process is called thermal oxidation , which forms silicon dioxide on the surface of the silicon . This is used as a gate insulator and field oxide . Other processes are called photomasks and photolithography . This process is what creates the patterns on the circuit in the integrated circuit. Ultraviolet light

3078-472: Is a function of the temperature, as the probability of getting enough thermal energy to produce a pair increases with temperature, being approximately exp(− E G / kT ) , where k is the Boltzmann constant , T is the absolute temperature and E G is bandgap. The probability of meeting is increased by carrier traps – impurities or dislocations which can trap an electron or hole and hold it until

3192-478: Is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as quantum decoherence . This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic. There are many mathematically equivalent formulations of quantum mechanics. One of

3306-424: Is a valid joint state that is not separable. States that are not separable are called entangled . If the state for a composite system is entangled, it is impossible to describe either component system A or system B by a state vector. One can instead define reduced density matrices that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes

3420-404: Is called diffusion . This is the process that gives the semiconducting material its desired semiconducting properties. It is also known as doping . The process introduces an impure atom to the system, which creates the p–n junction . To get the impure atoms embedded in the silicon wafer, the wafer is first put in a 1,100 degree Celsius chamber. The atoms are injected in and eventually diffuse with

3534-405: Is conserved under the evolution generated by B {\displaystyle B} . This implies a quantum version of the result proven by Emmy Noether in classical ( Lagrangian ) mechanics: for every differentiable symmetry of a Hamiltonian, there exists a corresponding conservation law . The simplest example of a quantum system with a position degree of freedom is a free particle in

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3648-1066: Is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the action principle in classical mechanics. The Hamiltonian H {\displaystyle H} is known as the generator of time evolution, since it defines a unitary time-evolution operator U ( t ) = e − i H t / ℏ {\displaystyle U(t)=e^{-iHt/\hbar }} for each value of t {\displaystyle t} . From this relation between U ( t ) {\displaystyle U(t)} and H {\displaystyle H} , it follows that any observable A {\displaystyle A} that commutes with H {\displaystyle H} will be conserved : its expectation value will not change over time. This statement generalizes, as mathematically, any Hermitian operator A {\displaystyle A} can generate

3762-448: Is given by The operator U ( t ) = e − i H t / ℏ {\displaystyle U(t)=e^{-iHt/\hbar }} is known as the time-evolution operator, and has the crucial property that it is unitary . This time evolution is deterministic in the sense that – given an initial quantum state ψ ( 0 ) {\displaystyle \psi (0)} – it makes

3876-410: Is increased by adding a small amount (of the order of 1 in 10 ) of pentavalent ( antimony , phosphorus , or arsenic ) or trivalent ( boron , gallium , indium ) atoms. This process is known as doping, and the resulting semiconductors are known as doped or extrinsic semiconductors . Apart from doping, the conductivity of a semiconductor can be improved by increasing its temperature. This is contrary to

3990-780: Is inert, blocking the passage of other electrons via that state. The energies of these quantum states are critical since a state is partially filled only if its energy is near the Fermi level (see Fermi–Dirac statistics ). High conductivity in material comes from it having many partially filled states and much state delocalization. Metals are good electrical conductors and have many partially filled states with energies near their Fermi level. Insulators , by contrast, have few partially filled states, their Fermi levels sit within band gaps with few energy states to occupy. Importantly, an insulator can be made to conduct by increasing its temperature: heating provides energy to promote some electrons across

4104-406: Is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by ⟨ ψ , P λ ψ ⟩ {\displaystyle \langle \psi ,P_{\lambda }\psi \rangle } , where P λ {\displaystyle P_{\lambda }} is the projector onto its associated eigenspace. In

4218-726: Is known as wave–particle duality . In addition to light, electrons , atoms , and molecules are all found to exhibit the same dual behavior when fired towards a double slit. Another non-classical phenomenon predicted by quantum mechanics is quantum tunnelling : a particle that goes up against a potential barrier can cross it, even if its kinetic energy is smaller than the maximum of the potential. In classical mechanics this particle would be trapped. Quantum tunnelling has several important consequences, enabling radioactive decay , nuclear fusion in stars, and applications such as scanning tunnelling microscopy , tunnel diode and tunnel field-effect transistor . When quantum systems interact,

4332-845: Is neither a very good insulator nor a very good conductor. However, one important feature of semiconductors (and some insulators, known as semi-insulators ) is that their conductivity can be increased and controlled by doping with impurities and gating with electric fields. Doping and gating move either the conduction or valence band much closer to the Fermi level and greatly increase the number of partially filled states. Some wider-bandgap semiconductor materials are sometimes referred to as semi-insulators . When undoped, these have electrical conductivity nearer to that of electrical insulators, however they can be doped (making them as useful as semiconductors). Semi-insulators find niche applications in micro-electronics, such as substrates for HEMT . An example of

4446-444: Is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states. Instead, we can consider a Gaussian wave packet : which has Fourier transform, and therefore momentum distribution We see that as we make a {\displaystyle a} smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making

4560-628: Is not sufficient for describing them at very small submicroscopic (atomic and subatomic ) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation, valid at large (macroscopic/microscopic) scale. Quantum systems have bound states that are quantized to discrete values of energy , momentum , angular momentum , and other quantities, in contrast to classical systems where these quantities can be measured continuously. Measurements of quantum systems show characteristics of both particles and waves ( wave–particle duality ), and there are limits to how accurately

4674-815: Is part of quantum communication protocols, such as quantum key distribution and superdense coding . Contrary to popular misconception, entanglement does not allow sending signals faster than light , as demonstrated by the no-communication theorem . Another possibility opened by entanglement is testing for " hidden variables ", hypothetical properties more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory provides. A collection of results, most significantly Bell's theorem , have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics. According to Bell's theorem, if nature actually operates in accord with any theory of local hidden variables, then

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4788-540: Is postulated to be normalized under the Hilbert space inner product, that is, it obeys ⟨ ψ , ψ ⟩ = 1 {\displaystyle \langle \psi ,\psi \rangle =1} , and it is well-defined up to a complex number of modulus 1 (the global phase), that is, ψ {\displaystyle \psi } and e i α ψ {\displaystyle e^{i\alpha }\psi } represent

4902-466: Is replaced by − i ℏ ∂ ∂ x {\displaystyle -i\hbar {\frac {\partial }{\partial x}}} , and in particular in the non-relativistic Schrödinger equation in position space the momentum-squared term is replaced with a Laplacian times − ℏ 2 {\displaystyle -\hbar ^{2}} . When two different quantum systems are considered together,

5016-415: Is the reduced Planck constant . The constant i ℏ {\displaystyle i\hbar } is introduced so that the Hamiltonian is reduced to the classical Hamiltonian in cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the correspondence principle . The solution of this differential equation

5130-439: 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 different useful properties, such as passing current more easily in one direction than the other, showing variable resistance, and having sensitivity to light or heat. Because

5244-469: Is then If the state for the first system is the vector ψ A {\displaystyle \psi _{A}} and the state for the second system is ψ B {\displaystyle \psi _{B}} , then the state of the composite system is Not all states in the joint Hilbert space H A B {\displaystyle {\mathcal {H}}_{AB}} can be written in this form, however, because

5358-504: Is typically very dilute, and so (unlike in metals) it is possible to think of the electrons in the conduction band of a semiconductor as a sort of classical ideal gas , where the electrons fly around freely without being subject to the Pauli exclusion principle . In most semiconductors, the conduction bands have a parabolic dispersion relation , and so these electrons respond to forces (electric field, magnetic field, etc.) much as they would in

5472-402: Is used along with a photoresist layer to create a chemical change that generates the patterns for the circuit. The etching is the next process that is required. The part of the silicon that was not covered by the photoresist layer from the previous step can now be etched. The main process typically used today is called plasma etching . Plasma etching usually involves an etch gas pumped in

5586-532: The Annalen der Physik und Chemie in 1835; Rosenschöld's findings were ignored. Simon Sze stated that Braun's research was the earliest systematic study of semiconductor devices. Also in 1874, Arthur Schuster found that a copper oxide layer on wires had rectification properties that ceased when the wires are cleaned. William Grylls Adams and Richard Evans Day observed the photovoltaic effect in selenium in 1876. A unified explanation of these phenomena required

5700-505: The Born rule : in the simplest case the eigenvalue λ {\displaystyle \lambda } is non-degenerate and the probability is given by | ⟨ λ → , ψ ⟩ | 2 {\displaystyle |\langle {\vec {\lambda }},\psi \rangle |^{2}} , where λ → {\displaystyle {\vec {\lambda }}}

5814-429: The Pauli exclusion principle ). These states are associated with the electronic band structure of the material. Electrical conductivity arises due to the presence of electrons in states that are delocalized (extending through the material), however in order to transport electrons a state must be partially filled , containing an electron only part of the time. If the state is always occupied with an electron, then it

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5928-454: The Siege of Leningrad after successful completion. In 1926, Julius Edgar Lilienfeld patented a device resembling a field-effect transistor , but it was not practical. R. Hilsch  [ de ] and R. W. Pohl  [ de ] in 1938 demonstrated a solid-state amplifier using a structure resembling the control grid of a vacuum tube; although the device displayed power gain, it had

6042-445: The band gap , be accompanied by the emission of thermal energy (in the form of phonons ) or radiation (in the form of photons ). In some states, the generation and recombination of electron–hole pairs are in equipoise. The number of electron-hole pairs in the steady state at a given temperature is determined by quantum statistical mechanics . The precise quantum mechanical mechanisms of generation and recombination are governed by

6156-713: The canonical commutation relation : Given a quantum state, the Born rule lets us compute expectation values for both X {\displaystyle X} and P {\displaystyle P} , and moreover for powers of them. Defining the uncertainty for an observable by a standard deviation , we have and likewise for the momentum: The uncertainty principle states that Either standard deviation can in principle be made arbitrarily small, but not both simultaneously. This inequality generalizes to arbitrary pairs of self-adjoint operators A {\displaystyle A} and B {\displaystyle B} . The commutator of these two operators

6270-470: The conservation of energy and conservation of momentum . As the probability that electrons and holes meet together is proportional to the product of their numbers, the product is in the steady-state nearly constant at a given temperature, providing that there is no significant electric field (which might "flush" carriers of both types, or move them from neighbor regions containing more of them to meet together) or externally driven pair generation. The product

6384-423: The photoelectric effect . These early attempts to understand microscopic phenomena, now known as the " old quantum theory ", led to the full development of quantum mechanics in the mid-1920s by Niels Bohr , Erwin Schrödinger , Werner Heisenberg , Max Born , Paul Dirac and others. The modern theory is formulated in various specially developed mathematical formalisms . In one of them, a mathematical entity called

6498-562: The wave function provides information, in the form of probability amplitudes , about what measurements of a particle's energy, momentum, and other physical properties may yield. Quantum mechanics allows the calculation of properties and behaviour of physical systems. It is typically applied to microscopic systems: molecules, atoms and sub-atomic particles. It has been demonstrated to hold for complex molecules with thousands of atoms, but its application to human beings raises philosophical problems, such as Wigner's friend , and its application to

6612-461: The 1930s. Point-contact microwave detector rectifiers made of lead sulfide were used by Jagadish Chandra Bose in 1904; the cat's-whisker detector using natural galena or other materials became a common device in the development of radio . However, it was somewhat unpredictable in operation and required manual adjustment for best performance. In 1906, H.J. Round observed light emission when electric current passed through silicon carbide crystals,

6726-431: The Hilbert space for the spin of a single proton is simply the space of two-dimensional complex vectors C 2 {\displaystyle \mathbb {C} ^{2}} with the usual inner product. Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are Hermitian (more precisely, self-adjoint ) linear operators acting on

6840-411: The Hilbert space of the combined system is the tensor product of the Hilbert spaces of the two components. For example, let A and B be two quantum systems, with Hilbert spaces H A {\displaystyle {\mathcal {H}}_{A}} and H B {\displaystyle {\mathcal {H}}_{B}} , respectively. The Hilbert space of the composite system

6954-432: The Hilbert space. A quantum state can be an eigenvector of an observable, in which case it is called an eigenstate , and the associated eigenvalue corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a quantum superposition . When an observable is measured, the result will be one of its eigenvalues with probability given by

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7068-788: The Japan Securities Dealers' Association in October 1989, and entered the First Section of the Tokyo Stock Exchange in December 1999. Semiconductor A semiconductor is a material that is between the conductor and insulator in ability to conduct electrical current. In many cases their conducting properties may be altered in useful ways by introducing impurities (" doping ") into the crystal structure . When two differently doped regions exist in

7182-489: The Schrödinger equation are known for very few relatively simple model Hamiltonians including the quantum harmonic oscillator , the particle in a box , the dihydrogen cation , and the hydrogen atom . Even the helium atom – which contains just two electrons – has defied all attempts at a fully analytic treatment, admitting no solution in closed form . However, there are techniques for finding approximate solutions. One method, called perturbation theory , uses

7296-403: The analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak potential energy . Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion. One consequence of

7410-416: The band gap, inducing partially filled states in both the band of states beneath the band gap ( valence band ) and the band of states above the band gap ( conduction band ). An (intrinsic) semiconductor has a band gap that is smaller than that of an insulator and at room temperature, significant numbers of electrons can be excited to cross the band gap. A pure semiconductor, however, is not very useful, as it

7524-606: The basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum. Both position and momentum are observables, meaning that they are represented by Hermitian operators . The position operator X ^ {\displaystyle {\hat {X}}} and momentum operator P ^ {\displaystyle {\hat {P}}} do not commute, but rather satisfy

7638-467: The behavior of a metal, in which conductivity decreases with an 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 semiconductor is doped by Group V elements, they will behave like donors creating free electrons , known as " n-type " doping. When

7752-404: The collection of probability amplitudes that pertain to another. One consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between measurable quantities. The most famous form of this uncertainty principle says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for

7866-489: The concept of band gaps had been developed. Walter H. Schottky and Nevill Francis Mott developed models of the potential barrier and of the characteristics of a metal–semiconductor junction . By 1938, Boris Davydov had developed a theory of the copper-oxide rectifier, identifying the effect of the p–n junction and the importance of minority carriers and surface states. Agreement between theoretical predictions (based on developing quantum mechanics) and experimental results

7980-855: The construction of light-emitting diodes and fluorescent quantum dots . Semiconductors with high thermal conductivity can be used for heat dissipation and improving thermal management of electronics. They play a crucial role in electric vehicles , high-brightness LEDs and power modules , among other applications. Semiconductors have large thermoelectric power factors making them useful in thermoelectric generators , as well as high thermoelectric figures of merit making them useful in thermoelectric coolers . A large number of elements and compounds have semiconducting properties, including: The most common semiconducting materials are crystalline solids, but amorphous and liquid semiconductors are also known. These include hydrogenated amorphous silicon and mixtures of arsenic , selenium , and tellurium in

8094-626: The continuous case, these formulas give instead the probability density . After the measurement, if result λ {\displaystyle \lambda } was obtained, the quantum state is postulated to collapse to λ → {\displaystyle {\vec {\lambda }}} , in the non-degenerate case, or to P λ ψ / ⟨ ψ , P λ ψ ⟩ {\textstyle P_{\lambda }\psi {\big /}\!{\sqrt {\langle \psi ,P_{\lambda }\psi \rangle }}} , in

8208-431: The dependence in position means that the momentum operator is equivalent (up to an i / ℏ {\displaystyle i/\hbar } factor) to taking the derivative according to the position, since in Fourier analysis differentiation corresponds to multiplication in the dual space . This is why in quantum equations in position space, the momentum p i {\displaystyle p_{i}}

8322-499: The electrical properties of a semiconductor material can be modified by doping and by the application of electrical fields or light, devices made from semiconductors can be used for amplification, switching, and energy conversion . The term semiconductor is also used to describe materials used in high capacity, medium- to high-voltage cables as part of their insulation, and these materials are often plastic XLPE ( Cross-linked polyethylene ) with carbon black. The conductivity of silicon

8436-453: The electrical properties of materials. The properties of the time-temperature coefficient of resistance, rectification, and light-sensitivity were observed starting in the early 19th century. Thomas Johann Seebeck was the first to notice that semiconductors exhibit special feature such that experiment concerning an Seebeck effect emerged with much stronger result when applying semiconductors, in 1821. In 1833, Michael Faraday reported that

8550-530: The electrons in the conduction band). When ionizing radiation strikes a semiconductor, it may excite an electron out of its energy level and consequently leave a hole. This process is known as electron-hole pair generation . Electron-hole pairs are constantly generated from thermal energy as well, in the absence of any external energy source. Electron-hole pairs are also apt to recombine. Conservation of energy demands that these recombination events, in which an electron loses an amount of energy larger than

8664-474: The excess or shortage of electrons, respectively. A balanced number of electrons would cause a current to flow throughout the material. Homojunctions occur when two differently doped semiconducting materials are joined. For example, a configuration could consist of p-doped and n-doped germanium . This results in an exchange of electrons and holes between the differently doped semiconducting materials. The n-doped germanium would have an excess of electrons, and

8778-514: The fast response of crystal detectors. Considerable research and development of silicon materials occurred during the war to develop detectors of consistent quality. Detector and power rectifiers could not amplify a signal. Many efforts were made to develop a solid-state amplifier and were successful in developing a device called the point contact transistor which could amplify 20 dB or more. In 1922, Oleg Losev developed two-terminal, negative resistance amplifiers for radio, but he died in

8892-415: The general case. The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous Bohr–Einstein debates , in which the two scientists attempted to clarify these fundamental principles by way of thought experiments . In the decades after the formulation of quantum mechanics,

9006-462: The interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected photon passes through one slit (as would a classical particle), and not through both slits (as would a wave). However, such experiments demonstrate that particles do not form the interference pattern if one detects which slit they pass through. This behavior

9120-495: The invention of the transistor in 1947 and the integrated circuit in 1958. Semiconductors in their natural state are poor conductors because a current requires the flow of electrons, and semiconductors have their valence bands filled, preventing the entire flow of new electrons. Several developed techniques allow semiconducting materials to behave like conducting materials, such as doping or gating . These modifications have two outcomes: n-type and p-type . These refer to

9234-430: The light passing through the slits is observed on a screen behind the plate. The wave nature of light causes the light waves passing through the two slits to interfere , producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles. However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves;

9348-543: The material's majority carrier . The opposite carrier is called the minority carrier , which exists due to thermal excitation at a much lower concentration compared to the majority carrier. For example, the pure semiconductor silicon has four valence electrons that bond each silicon atom to its neighbors. In silicon, the most common dopants are group III and group V elements. Group III elements all contain three valence electrons, causing them to function as acceptors when used to dope silicon. When an acceptor atom replaces

9462-432: The momentum operator with momentum p = ℏ k {\displaystyle p=\hbar k} . The coefficients of the superposition are ψ ^ ( k , 0 ) {\displaystyle {\hat {\psi }}(k,0)} , which is the Fourier transform of the initial quantum state ψ ( x , 0 ) {\displaystyle \psi (x,0)} . It

9576-413: The oldest and most common is the " transformation theory " proposed by Paul Dirac , which unifies and generalizes the two earliest formulations of quantum mechanics – matrix mechanics (invented by Werner Heisenberg ) and wave mechanics (invented by Erwin Schrödinger ). An alternative formulation of quantum mechanics is Feynman 's path integral formulation , in which a quantum-mechanical amplitude

9690-412: The one-dimensional case in the x {\displaystyle x} direction, the time-independent Schrödinger equation may be written With the differential operator defined by with state ψ {\displaystyle \psi } in this case having energy E {\displaystyle E} coincident with the kinetic energy of the particle. The general solutions of

9804-455: The original quantum system ceases to exist as an independent entity (see Measurement in quantum mechanics ). The time evolution of a quantum state is described by the Schrödinger equation: Here H {\displaystyle H} denotes the Hamiltonian , the observable corresponding to the total energy of the system, and ℏ {\displaystyle \hbar }

9918-449: The other. A slice cut from the specimen at the p–n boundary developed a voltage when exposed to light. The first working transistor was a point-contact transistor invented by John Bardeen , Walter Houser Brattain , and William Shockley at Bell Labs in 1947. Shockley had earlier theorized a field-effect amplifier made from germanium and silicon, but he failed to build such a working device, before eventually using germanium to invent

10032-417: The p-doped germanium would have an excess of holes. The transfer occurs until an equilibrium is reached by a process called recombination , which causes the migrating electrons from the n-type to come in contact with the migrating holes from the p-type. The result of this process is a narrow strip of immobile ions , which causes an electric field across the junction. A difference in electric potential on

10146-508: The point-contact transistor. In France, during the war, Herbert Mataré had observed amplification between adjacent point contacts on a germanium base. After the war, Mataré's group announced their " Transistron " amplifier only shortly after Bell Labs announced the " transistor ". In 1954, physical chemist Morris Tanenbaum fabricated the first silicon junction transistor at Bell Labs . However, early junction transistors were relatively bulky devices that were difficult to manufacture on

10260-428: The position becomes more and more uncertain. The uncertainty in momentum, however, stays constant. The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere inside a certain region, and therefore infinite potential energy everywhere outside that region. For

10374-524: The principle behind the light-emitting diode . Oleg Losev observed similar light emission in 1922, but at the time the effect had no practical use. Power rectifiers, using copper oxide and selenium, were developed in the 1920s and became commercially important as an alternative to vacuum tube rectifiers. The first semiconductor devices used galena , including German physicist Ferdinand Braun's crystal detector in 1874 and Indian physicist Jagadish Chandra Bose's radio crystal detector in 1901. In

10488-574: The pure semiconductors, the electrical conductivity may be varied by factors of thousands or millions. A 1 cm specimen of a metal or semiconductor has the order of 10 atoms. In a metal, every atom donates at least one free electron for conduction, thus 1 cm of metal contains on the order of 10 free electrons, whereas a 1 cm sample of pure germanium at 20   °C contains about 4.2 × 10 atoms, but only 2.5 × 10 free electrons and 2.5 × 10 holes. The addition of 0.001% of arsenic (an impurity) donates an extra 10 free electrons in

10602-400: The question of what constitutes a "measurement" has been extensively studied. Newer interpretations of quantum mechanics have been formulated that do away with the concept of " wave function collapse " (see, for example, the many-worlds interpretation ). The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled so that

10716-629: The resistance of specimens of silver sulfide decreases when they are heated. This is contrary to the behavior of metallic substances such as copper. In 1839, Alexandre Edmond Becquerel reported observation of a voltage between a solid and a liquid electrolyte, when struck by light, the photovoltaic effect . In 1873, Willoughby Smith observed that selenium resistors exhibit decreasing resistance when light falls on them. In 1874, Karl Ferdinand Braun observed conduction and rectification in metallic sulfides , although this effect had been discovered earlier by Peter Munck af Rosenschöld ( sv ) writing for

10830-413: The result can be the creation of quantum entanglement : their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement "... the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought". Quantum entanglement enables quantum computing and

10944-566: The results of a Bell test will be constrained in a particular, quantifiable way. Many Bell tests have been performed and they have shown results incompatible with the constraints imposed by local hidden variables. It is not possible to present these concepts in more than a superficial way without introducing the mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also linear algebra , differential equations , group theory , and other more advanced subjects. Accordingly, this article will present

11058-432: 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 most 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

11172-463: The same physical system. In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space . The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex square-integrable functions L 2 ( C ) {\displaystyle L^{2}(\mathbb {C} )} , while

11286-534: The same volume and the electrical conductivity is increased by a factor of 10,000. The materials chosen as suitable dopants depend on the atomic properties of both the dopant and the material to be doped. In general, dopants that produce the desired controlled changes are classified as either electron acceptors or donors . Semiconductors doped with donor impurities are called n-type , while those doped with acceptor impurities are known as p-type . The n and p type designations indicate which charge carrier acts as

11400-472: The same way as the electron. Combined with the negative effective mass of the electrons at the top of the valence band, we arrive at a picture of a positively charged particle that responds to electric and magnetic fields just as a normal positively charged particle would do in a vacuum, again with some positive effective mass. This particle is called a hole, and the collection of holes in the valence band can again be understood in simple classical terms (as with

11514-591: The scale at which the materials are used. A high degree of crystalline perfection is also required, since faults in the crystal structure (such as dislocations , twins , and stacking faults ) interfere with the semiconducting properties of the material. Crystalline faults are a major cause of defective semiconductor devices. The larger the crystal, the more difficult it is to achieve the necessary perfection. Current mass production processes use crystal ingots between 100 and 300 mm (3.9 and 11.8 in) in diameter, grown as cylinders and sliced into wafers . There

11628-425: The semiconductor body by contact with gaseous compounds of the desired element, or ion implantation can be used to accurately position the doped regions. Some materials, when rapidly cooled to a glassy amorphous state, have semiconducting properties. These include B, Si , Ge, Se, and Te, and there are multiple theories to explain them. The history of the understanding of semiconductors begins with experiments on

11742-408: The semiconductor manufacturing process. There were no cutting machines available in the market on which ultra-thin precision wheels could be mounted and run, DISCO decided to develop its own machine in 1975. The cutting machine, DAD-2h, received instant recognition from semiconductor companies, including Texas Instruments . The company adopted the name of DISCO Corporation in May 1977, was listed with

11856-458: The silicon. After the process is completed and the silicon has reached room temperature, the doping process is done and the semiconducting wafer is almost prepared. Semiconductors are defined by their unique electric conductive behavior, somewhere between that of a conductor and an insulator. The differences between these materials can be understood in terms of the quantum states for electrons, each of which may contain zero or one electron (by

11970-625: The superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if ψ A {\displaystyle \psi _{A}} and ϕ A {\displaystyle \phi _{A}} are both possible states for system A {\displaystyle A} , and likewise ψ B {\displaystyle \psi _{B}} and ϕ B {\displaystyle \phi _{B}} are both possible states for system B {\displaystyle B} , then

12084-427: The system and create electrons and holes. The processes that create or annihilate electrons and holes are called generation and recombination, respectively. In certain semiconductors, excited electrons can relax by emitting light instead of producing heat. Controlling the semiconductor composition and electrical current allows for the manipulation of the emitted light's properties. These semiconductors are used in

12198-407: The term Halbleiter (a semiconductor in modern meaning) in his Ph.D. thesis in 1910. Felix Bloch published a theory of the movement of electrons through atomic lattices in 1928. In 1930, B. Gudden  [ de ] stated that conductivity in semiconductors was due to minor concentrations of impurities. By 1931, the band theory of conduction had been established by Alan Herries Wilson and

12312-441: The theory is that it usually cannot predict with certainty what will happen, but only give probabilities. Mathematically, a probability is found by taking the square of the absolute value of a complex number , known as a probability amplitude. This is known as the Born rule , named after physicist Max Born . For example, a quantum particle like an electron can be described by a wave function, which associates to each point in space

12426-437: The universe as a whole remains speculative. Predictions of quantum mechanics have been verified experimentally to an extremely high degree of accuracy . For example, the refinement of quantum mechanics for the interaction of light and matter, known as quantum electrodynamics (QED), has been shown to agree with experiment to within 1 part in 10 when predicting the magnetic properties of an electron. A fundamental feature of

12540-406: The use of semiconductors, with the most important aspect being the integrated circuit (IC), which are found in desktops , laptops , scanners, cell-phones , and other electronic devices. Semiconductors for ICs are mass-produced. To create an ideal semiconducting material, chemical purity is paramount. Any small imperfection can have a drastic effect on how the semiconducting material behaves due to

12654-498: The useful electronic behavior. Using a hot-point probe , one can determine quickly whether a semiconductor sample is p- or n-type. A few 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 led in turn to

12768-526: The value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the uncertainty principle ). Quantum mechanics arose gradually from theories to explain observations that could not be reconciled with classical physics, such as Max Planck 's solution in 1900 to the black-body radiation problem, and the correspondence between energy and frequency in Albert Einstein 's 1905 paper , which explained

12882-467: The years preceding World War II, infrared detection and communications devices prompted research into lead-sulfide and lead-selenide materials. These devices were used for detecting ships and aircraft, for infrared rangefinders, and for voice communication systems. The point-contact crystal detector became vital for microwave radio systems since available vacuum tube devices could not serve as detectors above about 4000 MHz; advanced radar systems relied on

12996-637: Was sometimes poor. This was later explained by John Bardeen as due to the extreme "structure sensitive" behavior of semiconductors, whose properties change dramatically based on tiny amounts of impurities. Commercially pure materials of the 1920s containing varying proportions of trace contaminants produced differing experimental results. This spurred the development of improved material refining techniques, culminating in modern semiconductor refineries producing materials with parts-per-trillion purity. Devices using semiconductors were at first constructed based on empirical knowledge before semiconductor theory provided

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