Potential Well
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A potential well is the region surrounding a
local minimum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ra ...
of
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
. Energy captured in a potential well is unable to convert to another type of energy (
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
in the case of a
gravitational In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
potential well) because it is captured in the local minimum of a potential well. Therefore, a body may not proceed to the global minimum of potential energy, as it would naturally tend to do due to
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
.


Overview

Energy may be released from a potential well if sufficient energy is added to the system such that the local maximum is surmounted. In
quantum physics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
, potential energy may escape a potential well without added energy due to the probabilistic characteristics of
quantum particle In quantum field theory, the energy that a particle has as a result of changes that it causes in its environment defines self-energy \Sigma, and represents the contribution to the particle's energy, or effective mass, due to interactions between ...
s; in these cases a particle may be imagined to
tunnel A tunnel is an underground passageway, dug through surrounding soil, earth or rock, and enclosed except for the entrance and exit, commonly at each end. A pipeline is not a tunnel, though some recent tunnels have used immersed tube cons ...
''through'' the walls of a potential well. The graph of a 2D potential energy function is a
potential energy surface A potential energy surface (PES) describes the energy of a system, especially a collection of atoms, in terms of certain parameters, normally the positions of the atoms. The surface might define the energy as a function of one or more coordinates; ...
that can be imagined as the Earth's surface in a landscape of hills and valleys. Then a potential well would be a valley surrounded on all sides with higher terrain, which thus could be filled with water (e.g., be a
lake A lake is an area filled with water, localized in a basin, surrounded by land, and distinct from any river or other outlet that serves to feed or drain the lake. Lakes lie on land and are not part of the ocean, although, like the much large ...
) without any water flowing away toward another, lower minimum (e.g.
sea level Mean sea level (MSL, often shortened to sea level) is an average surface level of one or more among Earth's coastal bodies of water from which heights such as elevation may be measured. The global MSL is a type of vertical datuma standardised g ...
). In the case of
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
, the region around a mass is a gravitational potential well, unless the density of the mass is so low that
tidal force The tidal force is a gravitational effect that stretches a body along the line towards the center of mass of another body due to a gradient (difference in strength) in gravitational field from the other body; it is responsible for diverse phenomen ...
s from other masses are greater than the gravity of the body itself. A potential hill is the opposite of a potential well, and is the region surrounding a
local maximum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ra ...
.


Quantum confinement

Quantum confinement can be observed once the diameter of a material is of the same magnitude as the
de Broglie wavelength Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave ...
of the electron
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
. When materials are this small, their electronic and optical properties deviate substantially from those of bulk materials. A particle behaves as if it were free when the confining dimension is large compared to the wavelength of the particle. During this state, the
bandgap In solid-state physics, a band gap, also called an energy gap, is an energy range in a solid where no electronic states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference (i ...
remains at its original energy due to a continuous energy state. However, as the confining dimension decreases and reaches a certain limit, typically in nanoscale, the energy
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
becomes
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a g ...
. As a result, the bandgap becomes size-dependent. As the size of the particles decreases, the
electrons The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
and
electron holes In physics, chemistry, and electronic engineering, an electron hole (often simply called a hole) is a quasiparticle which 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 ...
come closer, and the energy required to activate them increases, which ultimately results in a
blueshift In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and simultaneous increase in f ...
in
light emission This is a list of sources of light, the visible part of the electromagnetic spectrum. Light sources produce photons from another energy source, such as heat, chemical reactions, or conversion of mass or a different frequency of electromagnetic ener ...
. Specifically, the effect describes the phenomenon resulting from
electrons The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
and
electron holes In physics, chemistry, and electronic engineering, an electron hole (often simply called a hole) is a quasiparticle which 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 ...
being squeezed into a dimension that approaches a critical
quantum In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...
measurement, called the exciton
Bohr radius The Bohr radius (''a''0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an ...
. In current application, a
quantum dot Quantum dots (QDs) are semiconductor particles a few nanometres in size, having light, optical and electronics, electronic properties that differ from those of larger particles as a result of quantum mechanics. They are a central topic in nanote ...
such as a small sphere confines in three dimensions, a
quantum wire In mesoscopic physics, a quantum wire is an electrically conducting wire in which quantum effects influence the transport properties. Usually such effects appear in the dimension of nanometers, so they are also referred to as nanowires. Quantum e ...
confines in two dimensions, and a
quantum well A quantum well is a potential well with only discrete energy values. The classic model used to demonstrate a quantum well is to confine particles, which were initially free to move in three dimensions, to two dimensions, by forcing them to occupy ...
confines only in one dimension. These are also known as zero-, one- and two-dimensional potential wells, respectively. In these cases they refer to the number of dimensions in which a confined particle can act as a free carrier. See
external links An internal link is a type of hyperlink on a web page to another page or resource, such as an image or document, on the same website or domain. Hyperlinks are considered either "external" or "internal" depending on their target or destination ...
, below, for application examples in biotechnology and solar cell technology.


Quantum mechanics view

The electronic and optical properties of materials are affected by size and shape. Well-established technical achievements including quantum dots were derived from size manipulation and investigation for their theoretical corroboration on quantum confinement effect. The major part of the theory is the behaviour of the exciton resembles that of an atom as its surrounding space shortens. A rather good approximation of an exciton's behaviour is the 3-D model of a
particle in a box In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypo ...
. The solution of this problem provides a sole mathematical connection between energy states and the dimension of space. Decreasing the volume or the dimensions of the available space, increases the energy of the states. Shown in the diagram is the change in electron energy level and
bandgap In solid-state physics, a band gap, also called an energy gap, is an energy range in a solid where no electronic states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference (i ...
between nanomaterial and its bulk state. The following equation shows the relationship between energy level and dimension spacing: :\psi_ = \sqrt \sin \left( \frac \right) \sin \left( \frac \right) \sin \left( \frac \right) :E_ = \frac \left \left( \frac \right)^2 + \left( \frac \right)^2 + \left( \frac \right)^2 \right/math> Research results provide an alternative explanation of the shift of properties at nanoscale. In the bulk phase, the surfaces appear to control some of the macroscopically observed properties. However, in
nanoparticles A nanoparticle or ultrafine particle is usually defined as a particle of matter that is between 1 and 100 nanometres (nm) in diameter. The term is sometimes used for larger particles, up to 500 nm, or fibers and tubes that are less than 1 ...
, surface molecules do not obey the expected configuration in space. As a result, surface tension changes tremendously.


Classical mechanics view

The
Young–Laplace equation In physics, the Young–Laplace equation () is an algebraic equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension or ...
can give a background on the investigation of the scale of forces applied to the surface molecules: :\begin \Delta P &= \gamma \nabla \cdot \hat n \\ &= 2 \gamma H \\ &= \gamma \left(\frac + \frac\right) \end Under the assumption of spherical shape R_1=R_2=R and resolving the Young–Laplace equation for the new radii R (nm), we estimate the new \Delta P(GPa). The smaller the radii, the greater the pressure is present. The increase in pressure at the nanoscale results in strong forces toward the interior of the particle. Consequently, the molecular structure of the particle appears to be different from the bulk mode, especially at the surface. These abnormalities at the surface are responsible for changes of inter-atomic interactions and
bandgap In solid-state physics, a band gap, also called an energy gap, is an energy range in a solid where no electronic states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference (i ...
.


See also

*
Quantum well A quantum well is a potential well with only discrete energy values. The classic model used to demonstrate a quantum well is to confine particles, which were initially free to move in three dimensions, to two dimensions, by forcing them to occupy ...
*
Finite potential well The finite potential well (also known as the finite square well) is a concept from quantum mechanics. It is an extension of the infinite potential well, in which a particle is confined to a "box", but one which has finite potential "walls". Unlike ...
*
Quantum dot Quantum dots (QDs) are semiconductor particles a few nanometres in size, having light, optical and electronics, electronic properties that differ from those of larger particles as a result of quantum mechanics. They are a central topic in nanote ...


References


External links

*
Semiconductor Fundamental

Quantum dots synthesisBiological application
{{DEFAULTSORT:Potential Well Quantum mechanical potentials Classical mechanics