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Wannier Function
The Wannier functions are a complete set of orthogonal functions used in solid-state physics. They were introduced by Gregory Wannier in 1937. Wannier functions are the localized molecular orbitals of crystalline systems. The Wannier functions for different lattice sites in a crystal are orthogonal, allowing a convenient basis for the expansion of electron states in certain regimes. Wannier functions have found widespread use, for example, in the analysis of binding forces acting on electrons. Definition Although, like localized molecular orbitals, Wannier functions can be chosen in many different ways, the original, simplest, and most common definition in solid-state physics is as follows. Choose a single band in a perfect crystal, and denote its Bloch states by :\psi_(\mathbf) = e^u_\mathbf(\mathbf) where ''u''k(r) has the same periodicity as the crystal. Then the Wannier functions are defined by :\phi_(\mathbf) = \frac \sum_ e^ \psi_(\mathbf), where * R is any lattice ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N2 Wannier
N, or n, is the fourteenth letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages, and others worldwide. Its name in English is ''en'' (pronounced ), plural ''ens''. History One of the most common hieroglyphs, snake, was used in Egyptian writing to stand for a sound like the English , because the Egyptian word for "snake" was ''djet''. It is speculated by some, such as archeologist Douglas Petrovich, that Semitic speakers working in Egypt adapted hieroglyphs to create the first alphabet. Some hold that they used the same snake symbol to represent N, with a great proponent of this theory being Alan Gardiner, because their word for "snake" may have begun with n (an example of a possible word being ''nahash''). However, this theory has become disputed. The name for the letter in the Phoenician, Hebrew, Aramaic, and Arabic alphabets is ''nun'', which means "fish" in some of these languages. This possibly conne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Localized Molecular Orbitals
Localized molecular orbitals are molecular orbitals which are concentrated in a limited spatial region of a molecule, such as a specific bond or lone pair on a specific atom. They can be used to relate molecular orbital calculations to simple bonding theories, and also to speed up post-Hartree–Fock electronic structure calculations by taking advantage of the local nature of electron correlation. Localized orbitals in systems with periodic boundary conditions are known as Wannier functions. Standard ab initio quantum chemistry methods lead to delocalized orbitals that, in general, extend over an entire molecule and have the symmetry of the molecule. Localized orbitals may then be found as linear combinations of the delocalized orbitals, given by an appropriate unitary transformation. In the water molecule for example, ab initio calculations show bonding character primarily in two molecular orbitals, each with electron density equally distributed among the two O-H bonds. The local ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hannay Angle
In classical mechanics, the Hannay angle is a mechanics analogue of the geometric phase (or Berry phase). It was named after John Hannay of the University of Bristol, UK. Hannay first described the angle in 1985, extending the ideas of the recently formalized Berry phase to classical mechanics. Consider a one-dimensional system moving in a cycle, like a pendulum. Now slowly vary a slow parameter \lambda, like pulling and pushing on the string of a pendulum. We can picture the motion of the system as having a fast oscillation and a slow oscillation. The fast oscillation is the motion of the pendulum, and the slow oscillation is the motion of our pulling on its string. If we picture the system in phase space, its motion sweeps out a torus. The adiabatic theorem in classical mechanics states that the action variable, which corresponds to the phase space area enclosed by the system's orbit, remains approximately constant. Thus, after one slow oscillation period, the fast oscillation is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bloch's Theorem
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential can be expressed as plane waves modulated by periodic functions. The theorem is named after the Swiss physicist Felix Bloch, who discovered the theorem in 1929. Mathematically, they are written where \mathbf is position, \psi is the wave function, u is a periodic function with the same periodicity as the crystal, the wave vector \mathbf is the crystal momentum vector, e is Euler's number, and i is the imaginary unit. Functions of this form are known as Bloch functions or Bloch states, and serve as a suitable basis for the wave functions or states of electrons in crystalline solids. The description of electrons in terms of Bloch functions, termed Bloch electrons (or less often ''Bloch Waves''), underlies the concept of electronic band structures. These eigenstates are written with subscripts as \psi_, where n is a discrete index, called the band index, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbital Magnetization
In quantum mechanics, orbital magnetization, Morb, refers to the magnetization induced by orbital motion of charged particles, usually electrons in solids. The term "orbital" distinguishes it from the contribution of spin degrees of freedom, Mspin, to the total magnetization. A nonzero orbital magnetization requires broken time-reversal symmetry, which can occur spontaneously in ferromagnetic and ferrimagnetic materials, or can be induced in a non- magnetic material by an applied magnetic field. Definitions The orbital magnetic moment of a finite system, such as a molecule, is given classically by : \mathbf_ = \frac\int \mathbf\times\mathbf(\mathbf) \ d^3\mathbf where J(r) is the current density at point r. (Here SI units are used; in Gaussian units, the prefactor would be 1/2''c'' instead, where ''c'' is the speed of light.) In a quantum-mechanical context, this can also be written as : \mathbf_ = \frac \langle\Psi \vert\mathbf \vert\Psi\rangle where −''e'' and ''me'' are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Hall Effect
The spin Hall effect (SHE) is a transport phenomenon predicted by Russian physicists Mikhail I. Dyakonov and Vladimir I. Perel in 1971. It consists of the appearance of Spin (physics), spin accumulation on the lateral surfaces of an electric current-carrying sample, the signs of the spin directions being opposite on the opposing boundaries. In a cylindrical wire, the current-induced surface spins will wind around the wire. When the current direction is reversed, the directions of spin orientation is also reversed. Definition The spin Hall effect is a transport phenomenon consisting of the appearance of spin accumulation on the lateral surfaces of a sample carrying electric current. The opposing surface boundaries will have spins of opposite sign. It is analogous to the classical Hall effect, where ''charges'' of opposite sign appear on the opposing lateral surfaces in an electric-current carrying sample in a magnetic field. In the case of the classical Hall effect the charge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anomalous Photovoltaic Effect
The anomalous photovoltaic effect (APE) is a type of a photovoltaic effect which occurs in certain semiconductors and insulators. The "anomalous" refers to those cases where the photovoltage (i.e., the open-circuit voltage caused by the light) is larger than the band gap of the corresponding semiconductor. In some cases, the voltage may reach thousands of volts. Although the voltage is unusually high, the short-circuit current is unusually low. Overall, materials that exhibit the anomalous photovoltaic effect have very low power generation efficiencies, and are never used in practical power-generation systems. There are several situations in which APE can arise. First, in polycrystalline materials, each microscopic grain can act as a photovoltaic. Then the grains add in series, so that the overall open-circuit voltage across the sample is large, potentially much larger than the bandgap. Second, in a similar manner, certain ferroelectric materials can develop stripes consisting of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magneto-optic Effect
A magneto-optic effect is any one of a number of phenomena in which an electromagnetic wave propagates through a medium that has been altered by the presence of a quasistatic magnetic field. In such a medium, which is also called gyrotropic or gyromagnetic, left- and right-rotating elliptical polarizations can propagate at different speeds, leading to a number of important phenomena. When light is transmitted through a layer of magneto-optic material, the result is called the Faraday effect: the plane of polarization can be rotated, forming a Faraday rotator. The results of reflection from a magneto-optic material are known as the magneto-optic Kerr effect (not to be confused with the nonlinear Kerr effect). In general, magneto-optic effects break time reversal symmetry locally (i.e., when only the propagation of light, and not the source of the magnetic field, is considered) as well as Lorentz reciprocity, which is a necessary condition to construct devices such as optica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbital Magnetization
In quantum mechanics, orbital magnetization, Morb, refers to the magnetization induced by orbital motion of charged particles, usually electrons in solids. The term "orbital" distinguishes it from the contribution of spin degrees of freedom, Mspin, to the total magnetization. A nonzero orbital magnetization requires broken time-reversal symmetry, which can occur spontaneously in ferromagnetic and ferrimagnetic materials, or can be induced in a non- magnetic material by an applied magnetic field. Definitions The orbital magnetic moment of a finite system, such as a molecule, is given classically by : \mathbf_ = \frac\int \mathbf\times\mathbf(\mathbf) \ d^3\mathbf where J(r) is the current density at point r. (Here SI units are used; in Gaussian units, the prefactor would be 1/2''c'' instead, where ''c'' is the speed of light.) In a quantum-mechanical context, this can also be written as : \mathbf_ = \frac \langle\Psi \vert\mathbf \vert\Psi\rangle where −''e'' and ''me'' are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hall Effect
The Hall effect is the production of a voltage, potential difference (the Hall voltage) across an electrical conductor that is wikt:transverse, transverse to an electric current in the conductor and to an applied magnetic field wikt:perpendicular, perpendicular to the current. It was discovered by Edwin Hall in 1879. The ''Hall coefficient'' is defined as the ratio of the induced electric field to the product of the current density and the applied magnetic field. It is a characteristic of the material from which the conductor is made, since its value depends on the type, number, and properties of the charge carriers that constitute the current. Discovery Wires carrying current in a magnetic field experience a mechanical force perpendicular to both the current and magnetic field. In the 1820s, André-Marie Ampère observed this underlying mechanism that led to the discovery of the Hall effect. However it was not until a solid mathematical basis for electromagnetism was system ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tight Binding
In solid-state physics, the tight-binding model (or TB model) is an approach to the calculation of electronic band structure using an approximate set of wave functions based upon superposition of wave functions for isolated atoms located at each atomic site. The method is closely related to the LCAO method (linear combination of atomic orbitals method) used in chemistry. Tight-binding models are applied to a wide variety of solids. The model gives good qualitative results in many cases and can be combined with other models that give better results where the tight-binding model fails. Though the tight-binding model is a one-electron model, the model also provides a basis for more advanced calculations like the calculation of surface states and application to various kinds of many-body problem and quasiparticle calculations. Introduction The name "tight binding" of this electronic band structure model suggests that this quantum mechanical model describes the properties of tigh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berry Phase
In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Hamiltonian. The phenomenon was independently discovered by S. Pancharatnam (1956), in classical optics and by H. C. Longuet-Higgins (1958)See page 12 in molecular physics; it was generalized by Michael Berry in (1984). It is also known as the Pancharatnam–Berry phase, Pancharatnam phase, or Berry phase. It can be seen in the conical intersection of potential energy surfaces and in the Aharonov–Bohm effect. Geometric phase around the conical intersection involving the ground electronic state of the C6H3F3+ molecular ion is discussed on pages 385–386 of the textbook by Bunker and Jensen. In the case of the Aharonov–Bohm effect, the adiabatic parameter is the magnetic field enclosed by two interference paths, and it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
