A light-emitting diode is a semiconductor light source that emits light when current flows through it. Electrons in the semiconductor recombine with electron holes, releasing energy in the form of photons; this effect is called electroluminescence. The color of the light is determined by the energy required for electrons to cross the band gap of the semiconductor. White light is obtained by using multiple semiconductors or a layer of light-emitting phosphor on the semiconductor device. Appearing as practical electronic components in 1962, the earliest LEDs emitted low-intensity infrared light. Infrared LEDs are used in remote-control circuits, such as those used with a wide variety of consumer electronics; the first visible-light LEDs were of low intensity and limited to red. Modern LEDs are available across the visible and infrared wavelengths, with high light output. Early LEDs were used as indicator lamps, replacing small incandescent bulbs, in seven-segment displays. Recent developments have produced white-light LEDs suitable for room lighting.
LEDs have led to new displays and sensors, while their high switching rates are useful in advanced communications technology. LEDs have many advantages over incandescent light sources, including lower energy consumption, longer lifetime, improved physical robustness, smaller size, faster switching. Light-emitting diodes are used in applications as diverse as aviation lighting, automotive headlamps, general lighting, traffic signals, camera flashes, lighted wallpaper and medical devices. Unlike a laser, the color of light emitted from an LED is neither coherent nor monochromatic, but the spectrum is narrow with respect to human vision, functionally monochromatic. Electroluminescence as a phenomenon was discovered in 1907 by the British experimenter H. J. Round of Marconi Labs, using a crystal of silicon carbide and a cat's-whisker detector. Russian inventor Oleg Losev reported creation of the first LED in 1927, his research was distributed in Soviet and British scientific journals, but no practical use was made of the discovery for several decades.
In 1936, Georges Destriau observed that electroluminescence could be produced when zinc sulphide powder is suspended in an insulator and an alternating electrical field is applied to it. In his publications, Destriau referred to luminescence as Losev-Light. Destriau worked in the laboratories of Madame Marie Curie an early pioneer in the field of luminescence with research on radium. Hungarian Zoltán Bay together with György Szigeti pre-empted led lighting in Hungary in 1939 by patented a lighting device based on SiC, with an option on boron carbide, that emmitted white, yellowish white, or greenish white depending on impurities present. Kurt Lehovec, Carl Accardo, Edward Jamgochian explained these first light-emitting diodes in 1951 using an apparatus employing SiC crystals with a current source of battery or pulse generator and with a comparison to a variant, crystal in 1953. Rubin Braunstein of the Radio Corporation of America reported on infrared emission from gallium arsenide and other semiconductor alloys in 1955.
Braunstein observed infrared emission generated by simple diode structures using gallium antimonide, GaAs, indium phosphide, silicon-germanium alloys at room temperature and at 77 kelvins. In 1957, Braunstein further demonstrated that the rudimentary devices could be used for non-radio communication across a short distance; as noted by Kroemer Braunstein "…had set up a simple optical communications link: Music emerging from a record player was used via suitable electronics to modulate the forward current of a GaAs diode. The emitted light was detected by a PbS diode some distance away; this signal was played back by a loudspeaker. Intercepting the beam stopped the music. We had a great deal of fun playing with this setup." This setup presaged the use of LEDs for optical communication applications. In September 1961, while working at Texas Instruments in Dallas, James R. Biard and Gary Pittman discovered near-infrared light emission from a tunnel diode they had constructed on a GaAs substrate. By October 1961, they had demonstrated efficient light emission and signal coupling between a GaAs p-n junction light emitter and an electrically isolated semiconductor photodetector.
On August 8, 1962, Biard and Pittman filed a patent titled "Semiconductor Radiant Diode" based on their findings, which described a zinc-diffused p–n junction LED with a spaced cathode contact to allow for efficient emission of infrared light under forward bias. After establishing the priority of their work based on engineering notebooks predating submissions from G. E. Labs, RCA Research Labs, IBM Research Labs, Bell Labs, Lincoln Lab at MIT, the U. S. patent office issued the two inventors the patent for the GaAs infrared light-emitting diode, the first practical LED. After filing the patent, Texas Instruments began a project to manufacture infrared diodes. In October 1962, TI announced the first commercial LED product, which employed a pure GaAs crystal to emit an 890 nm light output. In October 1963, TI announced the first commercial hemispherical LED, the SNX-110; the first visible-spectrum LED was developed in 1962 by Nick Holonyak, Jr. while working at General Electric. Holonyak first reported his LED in the journal Applied Physics Letters on December 1, 1962.
M. George Craford, a former graduate student of Holonyak, invented the first yellow LED and improved the brightness of red and red-orange LEDs by a factor of ten in 1972. In 1976, T. P. Pearsall created the first high-brightness, high-efficiency LEDs for optical fiber telecommunicat
Density of states
In solid state physics and condensed matter physics, the density of states of a system describes the number of states per an interval of energy at each energy level available to be occupied. It is mathematically represented by a density distribution and it is an average over the space and time domains of the various states occupied by the system. A high ` DOS' at a specific energy level means. A DOS of zero means; the DOS is represented by one of the symbols g, ρ, D, n, or N. Generally, the density of the states of matter is continuous. In isolated systems however, like atoms or molecules in the gas phase, the density distribution is discrete like a spectral density. Local variations, most due to distortions of the original system, are called local density of states. If the DOS of an undisturbed system is zero, the LDOS can locally be non-zero due to the presence of a local potential. In quantum mechanical systems, waves, or wave-like particles, can occupy modes or states with wavelengths and propagation directions dictated by the system.
For example, in some systems, the interatomic spacing and the atomic charge of a material could allow only electrons of certain wavelengths to exist. In other systems, the crystalline structure of a material could allow waves to propagate in one direction, while suppressing wave propagation in another direction. Only specific states are permitted. Thus, it can happen that many states are available for occupation at a specific energy level, while no states are available at other energy levels. For example, the density of states of electrons at the bandedge between the conduction band and the valence band in a semiconductor is shown in orange in Fig. 4. For an electron in the conduction band, an increase of the electron energy causes more states to become available for occupation. Alternatively, the density of state is discontinuous for an interval of energy, which means that there are no states available for electrons to occupy within the bandgap of the material; this condition means that an electron at the conduction band edge must lose at least the bandgap energy of the material in order to transition to another state in the valence band.
Depending on the quantum mechanical system, the density of states can be calculated for electrons, photons, or phonons, can be given as a function of either energy or the wave vector k. To convert between the DOS as a function of the energy and the DOS as a function of the wave vector, the system-specific energy dispersion relation between E and k must be known. In general, the topological properties of the system have a major impact on the properties of the density of states; the most well-known systems, like neutronium in neutron stars and free electron gases in metals, have a 3-dimensional Euclidean topology. Less familiar systems, like two-dimensional electron gases in graphite layers and the quantum Hall effect system in MOSFET type devices, have a 2-dimensional Euclidean topology. Less familiar are carbon nanotubes, the quantum wire and Luttinger liquid with their 1-dimensional topologies. Systems with 1D and 2D topologies are to become more common, assuming developments in nanotechnology and materials science proceed.
In general the density of states, related to volume V and N countable energy levels, is defined by: D = 1 V ⋅ ∑ i = 1 N δ. Using d = d under the limit L → ∞, one derives the volume-related density of states for continuous energy levels D:= ∫ R d d d k d ⋅ δ. With d of the spatial dimension of the considered system and k the wave vector. Equivalently, the density of states can be understood as the derivative of the microcanonical partition function Z m with respect to the energy: D = 1 V ⋅ d Z m d E The number of stat
Quantum mechanics, including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles. Classical physics, the physics existing before quantum mechanics, describes nature at ordinary scale. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large scale. Quantum mechanics differs from classical physics in that energy, angular momentum and other quantities of a bound system are restricted to discrete values. Quantum mechanics arose from theories to explain observations which could not be reconciled with classical physics, such as Max Planck's solution in 1900 to the black-body radiation problem, from the correspondence between energy and frequency in Albert Einstein's 1905 paper which explained the photoelectric effect. Early quantum theory was profoundly re-conceived in the mid-1920s by Erwin Schrödinger, Werner Heisenberg, Max Born and others; the modern theory is formulated in various specially developed mathematical formalisms.
In one of them, a mathematical function, the wave function, provides information about the probability amplitude of position and other physical properties of a particle. Important applications of quantum theory include quantum chemistry, quantum optics, quantum computing, superconducting magnets, light-emitting diodes, the laser, the transistor and semiconductors such as the microprocessor and research imaging such as magnetic resonance imaging and electron microscopy. Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule DNA. Scientific inquiry into the wave nature of light began in the 17th and 18th centuries, when scientists such as Robert Hooke, Christiaan Huygens and Leonhard Euler proposed a wave theory of light based on experimental observations. In 1803, Thomas Young, an English polymath, performed the famous double-slit experiment that he described in a paper titled On the nature of light and colours.
This experiment played a major role in the general acceptance of the wave theory of light. In 1838, Michael Faraday discovered cathode rays; these studies were followed by the 1859 statement of the black-body radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system can be discrete, the 1900 quantum hypothesis of Max Planck. Planck's hypothesis that energy is radiated and absorbed in discrete "quanta" matched the observed patterns of black-body radiation. In 1896, Wilhelm Wien empirically determined a distribution law of black-body radiation, known as Wien's law in his honor. Ludwig Boltzmann independently arrived at this result by considerations of Maxwell's equations. However, it underestimated the radiance at low frequencies. Planck corrected this model using Boltzmann's statistical interpretation of thermodynamics and proposed what is now called Planck's law, which led to the development of quantum mechanics. Following Max Planck's solution in 1900 to the black-body radiation problem, Albert Einstein offered a quantum-based theory to explain the photoelectric effect.
Around 1900–1910, the atomic theory and the corpuscular theory of light first came to be accepted as scientific fact. Among the first to study quantum phenomena in nature were Arthur Compton, C. V. Raman, Pieter Zeeman, each of whom has a quantum effect named after him. Robert Andrews Millikan studied the photoelectric effect experimentally, Albert Einstein developed a theory for it. At the same time, Ernest Rutherford experimentally discovered the nuclear model of the atom, for which Niels Bohr developed his theory of the atomic structure, confirmed by the experiments of Henry Moseley. In 1913, Peter Debye extended Niels Bohr's theory of atomic structure, introducing elliptical orbits, a concept introduced by Arnold Sommerfeld; this phase is known as old quantum theory. According to Planck, each energy element is proportional to its frequency: E = h ν, where h is Planck's constant. Planck cautiously insisted that this was an aspect of the processes of absorption and emission of radiation and had nothing to do with the physical reality of the radiation itself.
In fact, he considered his quantum hypothesis a mathematical trick to get the right answer rather than a sizable discovery. However, in 1905 Albert Einstein interpreted Planck's quantum hypothesis realistically and used it to explain the photoelectric effect, in which shining light on certain materials can eject electrons from the material, he won the 1921 Nobel Prize in Physics for this work. Einstein further developed this idea to show that an electromagnetic wave such as light could be described as a particle, with a discrete quantum of energy, dependent on its frequency; the foundations of quantum mechanics were established during the first half of the 20th century by Max Planck, Niels Bohr, Werner Heisenberg, Louis de Broglie, Arthur Compton, Albert Einstein, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Enrico Fermi, Wolfgang Pauli, Max von Laue, Freeman Dyson, David Hilbert, Wi
Quantum dots are tiny semiconductor particles a few nanometres in size, having optical and electronic properties that differ from larger LED particles. They are a central theme in nanotechnology; when the quantum dots are illuminated by UV light, some of the electrons receive enough energy to break free from the atoms. This capability allows them to move around the nanoparticle, creating a conductance band in which electrons are free to move through a material and conduct electricity; when these electrons drop back into the outer orbit around the atom, as illustrated in the following figure, they emit light. The color of that light depends on the energy difference between the conductance band and the valence band. In the language of materials science, nanoscale semiconductor materials confine either electrons or electron holes. Quantum dots are sometimes referred to as artificial atoms, emphasizing their singularity, having bound, discrete electronic states, like occurring atoms or molecules.
Quantum dots have properties intermediate between bulk semiconductors and discrete atoms or molecules. Their optoelectronic properties change as a function of both shape. Larger QDs of 5–6 nm diameter emit longer wavelengths, with colors such as orange or red. Smaller QDs emit shorter wavelengths, yielding colors like blue and green, although the specific colors and sizes vary depending on the exact composition of the QD; because of their tunable properties, QDs are of wide interest. Potential applications include transistors, solar cells, LEDs, diode lasers and second-harmonic generation, quantum computing, medical imaging, their small size allows for QDs to be suspended in solution, which may lead to use in inkjet printing and spin-coating. They have been used in Langmuir-Blodgett thin-films; these processing techniques result in less expensive and less time-consuming methods of semiconductor fabrication. There are several ways to prepare the principal ones involving colloids. Colloidal semiconductor nanocrystals are synthesized from solutions, much like traditional chemical processes.
The main difference remains dissolved. Heating the solution at high temperature, the precursors decompose forming monomers which nucleate and generate nanocrystals. Temperature is a critical factor in determining optimal conditions for the nanocrystal growth, it must be high enough to allow for rearrangement and annealing of atoms during the synthesis process while being low enough to promote crystal growth. The concentration of monomers is another critical factor that has to be stringently controlled during nanocrystal growth; the growth process of nanocrystals can occur in two different regimes, "focusing" and "defocusing". At high monomer concentrations, the critical size is small, resulting in growth of nearly all particles. In this regime, smaller particles grow faster than large ones resulting in "focusing" of the size distribution to yield nearly monodisperse particles; the size focusing is optimal when the monomer concentration is kept such that the average nanocrystal size present is always larger than the critical size.
Over time, the monomer concentration diminishes, the critical size becomes larger than the average size present, the distribution "defocuses". There are colloidal methods to produce many different semiconductors. Typical dots are made of binary compounds such as lead sulfide, lead selenide, cadmium selenide, cadmium sulfide, cadmium telluride, indium arsenide, indium phosphide. Dots may be made from ternary compounds such as cadmium selenide sulfide; these quantum dots can contain as few as 100 to 100,000 atoms within the quantum dot volume, with a diameter of ≈10 to 50 atoms. This corresponds to about 2 to 10 nanometers, at 10 nm in diameter, nearly 3 million quantum dots could be lined up end to end and fit within the width of a human thumb. Large batches of quantum dots may be synthesized via colloidal synthesis. Due to this scalability and the convenience of benchtop conditions, colloidal synthetic methods are promising for commercial applications, it is acknowledged to be the least toxic of all the different forms of synthesis.
Plasma synthesis has evolved to be one of the most popular gas-phase approaches for the production of quantum dots those with covalent bonds. For example and germanium quantum dots have been synthesized by using nonthermal plasma; the size, shape and composition of quantum dots can all be controlled in nonthermal plasma. Doping that seems quite challenging for quantum dots has been realized in plasma synthesis. Quantum dots synthesized by plasma are in the form of powder, for which surface modification may be carried out; this can lead to excellent dispersion of quantum dots in either organic solvents or water. Self-assembled quantum dots are between 5 and 50 nm in size. Quantum dots defined by lithographically patterned gate electrodes, or by etching on two-dimensional electron gasses in semiconductor heterostructures can have lateral dimensions between 20 and 100 nm; some quantum dots are small regions of one material buried in another with a larger band gap. These can be so-called core–shell structures, e.g. with CdSe in the core and ZnS in the shell, or from special forms of silica called ormosil.
Sub-monolayer shells can be effective ways of passivating the quantum dots, such as PbS cores with sub-monolayer CdS shells. Quantum dots sometimes occur spontaneously in quantum well structures due to monolayer fluctuations in the well's thickness. Self-asse
A solar cell, or photovoltaic cell, is an electrical device that converts the energy of light directly into electricity by the photovoltaic effect, a physical and chemical phenomenon. It is a form of photoelectric cell, defined as a device whose electrical characteristics, such as current, voltage, or resistance, vary when exposed to light. Individual solar cell devices can be combined to form modules, otherwise known as solar panels. In basic terms a single junction silicon solar cell can produce a maximum open-circuit voltage of 0.5 to 0.6 volts. Solar cells are described as being photovoltaic, irrespective of whether the source is sunlight or an artificial light, they are used as a photodetector, detecting light or other electromagnetic radiation near the visible range, or measuring light intensity. The operation of a photovoltaic cell requires three basic attributes: The absorption of light, generating either electron-hole pairs or excitons; the separation of charge carriers of opposite types.
The separate extraction of those carriers to an external circuit. In contrast, a solar thermal collector supplies heat by absorbing sunlight, for the purpose of either direct heating or indirect electrical power generation from heat. A "photoelectrolytic cell", on the other hand, refers either to a type of photovoltaic cell, or to a device that splits water directly into hydrogen and oxygen using only solar illumination. Assemblies of solar cells are used to make solar modules that generate electrical power from sunlight, as distinguished from a "solar thermal module" or "solar hot water panel". A solar array generates solar power using solar energy. Multiple solar cells in an integrated group, all oriented in one plane, constitute a solar photovoltaic panel or module. Photovoltaic modules have a sheet of glass on the sun-facing side, allowing light to pass while protecting the semiconductor wafers. Solar cells are connected in series and parallel circuits or series in modules, creating an additive voltage.
Connecting cells in parallel yields a higher current. Strings of series cells are handled independently and not connected in parallel, though as of 2014, individual power boxes are supplied for each module, are connected in parallel. Although modules can be interconnected to create an array with the desired peak DC voltage and loading current capacity, using independent MPPTs is preferable. Otherwise, shunt diodes can reduce shadowing power loss in arrays with series/parallel connected cells; the photovoltaic effect was experimentally demonstrated first by French physicist Edmond Becquerel. In 1839, at age 19, he built the world's first photovoltaic cell in his father's laboratory. Willoughby Smith first described the "Effect of Light on Selenium during the passage of an Electric Current" in a 20 February 1873 issue of Nature. In 1883 Charles Fritts built the first solid state photovoltaic cell by coating the semiconductor selenium with a thin layer of gold to form the junctions. Other milestones include: 1888 – Russian physicist Aleksandr Stoletov built the first cell based on the outer photoelectric effect discovered by Heinrich Hertz in 1887.
1905 – Albert Einstein proposed a new quantum theory of light and explained the photoelectric effect in a landmark paper, for which he received the Nobel Prize in Physics in 1921. 1941 – Vadim Lashkaryov discovered p-n-junctions in Cu2O and Ag2S protocells. 1946 – Russell Ohl patented the modern junction semiconductor solar cell, while working on the series of advances that would lead to the transistor. 1954 – the first practical photovoltaic cell was publicly demonstrated at Bell Laboratories. The inventors were Daryl Chapin and Gerald Pearson. 1958 – solar cells gained prominence with their incorporation onto the Vanguard I satellite. Solar cells were first used in a prominent application when they were proposed and flown on the Vanguard satellite in 1958, as an alternative power source to the primary battery power source. By adding cells to the outside of the body, the mission time could be extended with no major changes to the spacecraft or its power systems. In 1959 the United States launched Explorer 6, featuring large wing-shaped solar arrays, which became a common feature in satellites.
These arrays consisted of 9600 Hoffman solar cells. By the 1960s, solar cells were the main power source for most Earth orbiting satellites and a number of probes into the solar system, since they offered the best power-to-weight ratio. However, this success was possible because in the space application, power system costs could be high, because space users had few other power options, were willing to pay for the best possible cells; the space power market drove the development of higher efficiencies in solar cells up until the National Science Foundation "Research Applied to National Needs" program began to push development of solar cells for terrestrial applications. In the early 1990s the technology used for space solar cells diverged from the silicon technology used for terrestrial panels, with the spacecraft application shifting to gallium arsenide-based III-V semiconductor materials, which evolved into the modern III-V multijunction photovoltaic cell used on spacecraft. Improvements were gradual over the 1960s.
This was the reason that costs remained high, becau
A semiconductor material has an electrical conductivity value falling between that of a metal, like copper, etc. and an insulator, such as glass. Their resistance decreases as their temperature increases, behaviour opposite to that of a metal, their conducting properties may be altered in useful ways by the deliberate, controlled introduction of impurities 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 and electron holes at these junctions is the basis of diodes and all modern electronics. Some examples of semiconductors are silicon and gallium arsenide. After silicon, gallium arsenide is the second most common semiconductor used in laser diodes, solar cells, microwave frequency integrated circuits, 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 in one direction than the other, showing variable resistance, 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 and energy conversion. The conductivity of silicon is increased by adding a small amount of trivalent 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 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 increases the number of charge carriers within the crystal; when a doped semiconductor contains free holes it is called "p-type", when it contains 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. Although some pure elements and many compounds display semiconductor properties, silicon and compounds of gallium are the most used in electronic devices. Elements near the so-called "metalloid staircase", where the metalloids are located on the periodic table, are used as semiconductors; 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. Variable electrical conductivity Semiconductors in their natural state are poor conductors because a current requires the flow of electrons, semiconductors have their valence bands filled, preventing the entry flow of new electrons.
There are several developed techniques that allow semiconducting materials to behave like conducting materials, such as doping or gating. These modifications have two outcomes: p-type; these refer to the shortage of electrons, respectively. An unbalanced number of electrons would cause a current to flow through the material. Heterojunctions Heterojunctions occur when two differently doped semiconducting materials are joined together. For example, a configuration could consist of 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, the p-doped germanium would have an excess of holes; the transfer occurs until 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. A product of this process is charged ions. Excited electrons A difference in electric potential on 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 the system and create electrons and holes. The process that creates and annihilates electrons and holes are called generation and recombination. Light emission In certain semiconductors, excited electrons can relax by emitting light instead of producing heat; these semiconductors are used in the construction of light-emitting diodes and fluorescent quantum dots. High thermal conductivitySemiconductors with high thermal conductivity can be used for heat dissipation and improving thermal management of electronics. Thermal energy conversion 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: Certain pure elements are found in Group 14 of the p
In physics and engineering, the envelope of an oscillating signal is a smooth curve outlining its extremes. The envelope thus generalizes the concept of a constant amplitude; the figure illustrates a modulated sine wave varying between a lower envelope. The envelope function may be a function of time, angle, or indeed of any variable. A common situation resulting in an envelope function in both space x and time t is the superposition of two waves of the same wavelength and frequency: F = sin + sin ≈ 2 cos sin which uses the trigonometric formula for the addition of two sine waves, the approximation Δλ ≪ λ: 1 λ ± Δ λ = 1 λ 1 1 ± Δ λ / λ ≈ 1 λ ∓ Δ λ λ 2. Here the modulation wavelength λmod is given by: λ m o d = λ 2 Δ λ; the modulation wavelength is double that of the envelope itself because each half-wavelength of the modulating cosine wave governs both positive and negative values of the modulated sine wave. The beat frequency is that of the envelope, twice that of the modulating wave, or 2Δf.
If this wave is a sound wave, the ear hears the frequency associated with f and the amplitude of this sound varies with the beat frequency. The argument of the sinusoids above apart from a factor 2π are: ξ C =, ξ E =, with subscripts C and E referring to the carrier and the envelope; the same amplitude F of the wave results from the same values of ξC and ξE, each of which may itself return to the same value over different but properly related choices of x and t. This invariance means that one can trace these waveforms in space to find the speed of a position of fixed amplitude as it propagates in time.