Misplaced Pages

#430569

39-472: The fractional quantum Hall effect (FQHE) is a collective behavior in a 2D system of electrons. In particular magnetic fields, the electron gas condenses into a remarkable liquid state, which is very delicate, requiring high quality material with a low carrier concentration, and extremely low temperatures. As in the integer quantum Hall effect , the Hall resistance undergoes certain quantum Hall transitions to form

78-401: A conceptual model . In order to execute the model, it needs to be implemented as a computer simulation . This requires more choices, such as numerical approximations or the use of heuristics. Despite all these epistemological and computational constraints, simulation has been recognized as the third pillar of scientific methods: theory building, simulation, and experimentation. A simulation

117-475: A logical and objective way. All models are in simulacra , that is, simplified reflections of reality that, despite being approximations, can be extremely useful. Building and disputing models is fundamental to the scientific enterprise. Complete and true representation may be impossible, but scientific debate often concerns which is the better model for a given task, e.g., which is the more accurate climate model for seasonal forecasting. Attempts to formalize

156-548: A mathematical construct is solely and precisely that it is expected to work—that is, correctly to describe phenomena from a reasonably wide area. There is also an increasing attention to scientific modelling in fields such as science education , philosophy of science , systems theory , and knowledge visualization . There is a growing collection of methods , techniques and meta- theory about all kinds of specialized scientific modelling. A scientific model seeks to represent empirical objects, phenomena, and physical processes in

195-428: A model will deal with only some aspects of the phenomenon in question, and two models of the same phenomenon may be essentially different—that is to say, that the differences between them comprise more than just a simple renaming of components. Such differences may be due to differing requirements of the model's end users, or to conceptual or aesthetic differences among the modelers and to contingent decisions made during

234-419: A perception of reality. This perception is already a model in itself, as it comes with a physical constraint. There are also constraints on what we are able to legally observe with our current tools and methods, and cognitive constraints that limit what we are able to explain with our current theories. This model comprises the concepts, their behavior, and their relations informal form and is often referred to as

273-612: A series of plateaus. Each particular value of the magnetic field corresponds to a filling factor (the ratio of electrons to magnetic flux quanta ) where p and q are integers with no common factors. Here q turns out to be an odd number with the exception of two filling factors 5/2 and 7/2. The principal series of such fractions are and Fractionally charged quasiparticles are neither bosons nor fermions and exhibit anyonic statistics. The fractional quantum Hall effect continues to be influential in theories about topological order . Certain fractional quantum Hall phases appear to have

312-450: A triangular quantum well . Electrons confined to the heterojunction of HEMTs exhibit higher mobilities than those in MOSFETs, since the former device utilizes an intentionally undoped channel thereby mitigating the deleterious effect of ionized impurity scattering . Two closely spaced heterojunction interfaces may be used to confine electrons to a rectangular quantum well. Careful choice of

351-674: A very fast coarse model with its related expensive-to-compute fine model so as to avoid direct expensive optimization of the fine model. The alignment process iteratively refines a "mapped" coarse model ( surrogate model ). One application of scientific modelling is the field of modelling and simulation , generally referred to as "M&S". M&S has a spectrum of applications which range from concept development and analysis, through experimentation, measurement, and verification, to disposal analysis. Projects and programs may use hundreds of different simulations, simulators and model analysis tools. The figure shows how modelling and simulation

390-613: Is a scientific model in solid-state physics . It is an electron gas that is free to move in two dimensions, but tightly confined in the third. This tight confinement leads to quantized energy levels for motion in the third direction, which can then be ignored for most problems. Thus the electrons appear to be a 2D sheet embedded in a 3D world. The analogous construct of holes is called a two-dimensional hole gas (2DHG), and such systems have many useful and interesting properties. Most 2DEGs are found in transistor -like structures made from semiconductors . The most commonly encountered 2DEG

429-424: Is a fundamental and sometimes intangible notion covering the recognition, observation, nature, and stability of patterns and relationships of entities. From a child's verbal description of a snowflake, to the detailed scientific analysis of the properties of magnetic fields , the concept of structure is an essential foundation of nearly every mode of inquiry and discovery in science, philosophy, and art. A system

SECTION 10

#1732868715431

468-401: Is a set of interacting or interdependent entities, real or abstract, forming an integrated whole. In general, a system is a construct or collection of different elements that together can produce results not obtainable by the elements alone. The concept of an 'integrated whole' can also be stated in terms of a system embodying a set of relationships which are differentiated from relationships of

507-539: Is a way to implement the model, often employed when the model is too complex for the analytical solution. A steady-state simulation provides information about the system at a specific instant in time (usually at equilibrium, if such a state exists). A dynamic simulation provides information over time. A simulation shows how a particular object or phenomenon will behave. Such a simulation can be useful for testing , analysis, or training in those cases where real-world systems or concepts can be represented by models. Structure

546-648: Is an activity that produces models representing empirical objects, phenomena, and physical processes, to make a particular part or feature of the world easier to understand , define , quantify , visualize , or simulate . It requires selecting and identifying relevant aspects of a situation in the real world and then developing a model to replicate a system with those features. Different types of models may be used for different purposes, such as conceptual models to better understand, operational models to operationalize , mathematical models to quantify, computational models to simulate, and graphical models to visualize

585-462: Is either impossible or impractical to create experimental conditions in which scientists can directly measure outcomes. Direct measurement of outcomes under controlled conditions (see Scientific method ) will always be more reliable than modeled estimates of outcomes. Within modeling and simulation , a model is a task-driven, purposeful simplification and abstraction of a perception of reality, shaped by physical, legal, and cognitive constraints. It

624-400: Is evaluated first and foremost by its consistency to empirical data; any model inconsistent with reproducible observations must be modified or rejected. One way to modify the model is by restricting the domain over which it is credited with having high validity. A case in point is Newtonian physics, which is highly useful except for the very small, the very fast, and the very massive phenomena of

663-452: Is still unknown, but it may be similar to modulation doping in semiconductors, with electric-field-induced oxygen vacancies acting as the dopants. Considerable research involving 2DEGs and 2DHGs has been done, and much continues to this day. 2DEGs offer a mature system of extremely high mobility electrons, especially at low temperatures. When cooled to 4 K, 2DEGs may have mobilities μ {\displaystyle \mu } of

702-420: Is task-driven because a model is captured with a certain question or task in mind. Simplifications leave all the known and observed entities and their relation out that are not important for the task. Abstraction aggregates information that is important but not needed in the same detail as the object of interest. Both activities, simplification, and abstraction, are done purposefully. However, they are done based on

741-462: Is the layer of electrons found in MOSFETs (metal–oxide–semiconductor field-effect transistors ). When the transistor is in inversion mode , the electrons underneath the gate oxide are confined to the semiconductor-oxide interface, and thus occupy well defined energy levels. For thin-enough potential wells and temperatures not too high, only the lowest level is occupied (see the figure caption), and so

780-461: The principles of the empirical sciences use an interpretation to model reality, in the same way logicians axiomatize the principles of logic . The aim of these attempts is to construct a formal system that will not produce theoretical consequences that are contrary to what is found in reality . Predictions or other statements drawn from such a formal system mirror or map the real world only insofar as these scientific models are true. For

819-495: The dawn of man. Examples from history include cave paintings , Egyptian hieroglyphs , Greek geometry , and Leonardo da Vinci 's revolutionary methods of technical drawing for engineering and scientific purposes. Space mapping refers to a methodology that employs a "quasi-global" modelling formulation to link companion "coarse" (ideal or low-fidelity) with "fine" (practical or high-fidelity) models of different complexities. In engineering optimization , space mapping aligns (maps)

SECTION 20

#1732868715431

858-555: The domain of application of the model. For example, the special theory of relativity assumes an inertial frame of reference . This assumption was contextualized and further explained by the general theory of relativity . A model makes accurate predictions when its assumptions are valid, and might well not make accurate predictions when its assumptions do not hold. Such assumptions are often the point with which older theories are succeeded by new ones (the general theory of relativity works in non-inertial reference frames as well). A model

897-406: The electrons are confined to an extreme degree. The two-dimensional electron system in graphene can be tuned to either a 2DEG or 2DHG (2-D hole gas) by gating or chemical doping . This has been a topic of current research due to the versatile (some existing but mostly envisaged) applications of graphene. A separate class of heterostructures that can host 2DEGs are oxides. Although both sides of

936-446: The heterostructure are insulators, the 2DEG at the interface may arise even without doping (which is the usual approach in semiconductors). Typical example is a ZnO/ZnMgO heterostructure. More examples can be found in a recent review including a notable discovery of 2004, a 2DEG at the LaAlO 3 /SrTiO 3 interface which becomes superconducting at low temperatures. The origin of this 2DEG

975-538: The importance of the FQHE discovered by Tsui, Stormer, and Gossard is notable for contesting old perspectives. The existence of FQH liquids suggests that there is much more to discover beyond the present symmetry breaking paradigm in condensed matter physics. Different FQH states all have the same symmetry and cannot be described by symmetry breaking theory. The associated fractional charge , fractional statistics , non-Abelian statistics, chiral edge states, etc. demonstrate

1014-646: The materials and alloy compositions allow control of the carrier densities within the 2DEG. Electrons may also be confined to the surface of a material. For example, free electrons will float on the surface of liquid helium , and are free to move along the surface, but stick to the helium; some of the earliest work in 2DEGs was done using this system. Besides liquid helium, there are also solid insulators (such as topological insulators ) that support conductive surface electronic states. Recently, atomically thin solid materials have been developed ( graphene , as well as metal dichalcogenide such as molybdenum disulfide ) where

1053-475: The modelling process. Considerations that may influence the structure of a model might be the modeler's preference for a reduced ontology , preferences regarding statistical models versus deterministic models , discrete versus continuous time, etc. In any case, users of a model need to understand the assumptions made that are pertinent to its validity for a given use. Building a model requires abstraction . Assumptions are used in modelling in order to specify

1092-424: The motion of the electrons perpendicular to the interface can be ignored. However, the electron is free to move parallel to the interface, and so is quasi-two-dimensional. Other methods for engineering 2DEGs are high-electron-mobility-transistors (HEMTs) and rectangular quantum wells . HEMTs are field-effect transistors that utilize the heterojunction between two semiconducting materials to confine electrons to

1131-433: The order of 1,000,000 cm /Vs and lower temperatures can lead to further increase of μ {\displaystyle \mu } still. Specially grown, state of the art heterostructures with mobilities around 30,000,000 cm /(V·s) have been made. These enormous mobilities offer a test bed for exploring fundamental physics, since besides confinement and effective mass , the electrons do not interact with

1170-467: The power and the fascination of emergence in many-body systems. Thus FQH states represent new states of matter that contain a completely new kind of order— topological order . For example, properties once deemed isotropic for all materials may be anisotropic in 2D planes. The new type of orders represented by FQH states greatly enrich our understanding of quantum phases and quantum phase transitions . 2DEG A two-dimensional electron gas ( 2DEG )

1209-409: The quasiparticle charge. The FQH effect shows the limits of Landau 's symmetry breaking theory. Previously it was held that the symmetry breaking theory could explain all the important concepts and properties of forms of matter. According to this view, the only thing to be done was to apply the symmetry breaking theory to all different kinds of phases and phase transitions . From this perspective,

Fractional quantum Hall effect - Misplaced Pages Continue

1248-542: The right properties for building a topological quantum computer . The FQHE was experimentally discovered in 1982 by Daniel Tsui and Horst Störmer , in experiments performed on heterostructures made out of gallium arsenide developed by Arthur Gossard . There were several major steps in the theory of the FQHE. Tsui, Störmer, and Robert B. Laughlin were awarded the 1998 Nobel Prize in Physics for their work. Experiments have reported results that specifically support

1287-528: The scientist, a model is also a way in which the human thought processes can be amplified. For instance, models that are rendered in software allow scientists to leverage computational power to simulate, visualize, manipulate and gain intuition about the entity, phenomenon, or process being represented. Such computer models are in silico . Other types of scientific models are in vivo (living models, such as laboratory rats ) and in vitro (in glassware, such as tissue culture ). Models are typically used when it

1326-809: The semiconductor very often, sometimes traveling several micrometers before colliding; this so-called mean free path ℓ {\displaystyle \ell } can be estimated in the parabolic band approximation as where n {\displaystyle n} is the electron density in the 2DEG. Note that μ {\displaystyle \mu } typically depends on n {\displaystyle n} . Mobilities of 2DHG systems are smaller than those of most 2DEG systems, in part due to larger effective masses of holes (few 1000 cm /(V·s) can already be considered high mobility ). Aside from being in practically every semiconductor device in use today, two dimensional systems allow access to interesting physics. The quantum Hall effect

1365-451: The set to other elements, and form relationships between an element of the set and elements not a part of the relational regime. There are two types of system models: 1) discrete in which the variables change instantaneously at separate points in time and, 2) continuous where the state variables change continuously with respect to time. Modelling is the process of generating a model as a conceptual representation of some phenomenon. Typically

1404-473: The subject. Modelling is an essential and inseparable part of many scientific disciplines, each of which has its own ideas about specific types of modelling. The following was said by John von Neumann . ... the sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such

1443-799: The understanding that there are fractionally-charged quasiparticles in an electron gas under FQHE conditions. In 1995, the fractional charge of Laughlin quasiparticles was measured directly in a quantum antidot electrometer at Stony Brook University , New York . In 1997, two groups of physicists at the Weizmann Institute of Science in Rehovot , Israel , and at the Commissariat à l'énergie atomique laboratory near Paris , detected such quasiparticles carrying an electric current , through measuring quantum shot noise Both of these experiments have been confirmed with certainty. A more recent experiment, measures

1482-468: The universe. However, a fit to empirical data alone is not sufficient for a model to be accepted as valid. Factors important in evaluating a model include: People may attempt to quantify the evaluation of a model using a utility function . Visualization is any technique for creating images, diagrams, or animations to communicate a message. Visualization through visual imagery has been an effective way to communicate both abstract and concrete ideas since

1521-631: Was first observed in a 2DEG, which led to two Nobel Prizes in physics , of Klaus von Klitzing in 1985, and of Robert B. Laughlin , Horst L. Störmer and Daniel C. Tsui in 1998. Spectrum of a laterally modulated 2DEG (a two-dimensional superlattice ) subject to magnetic field B can be represented as the Hofstadter's butterfly , a fractal structure in the energy vs B plot, signatures of which were observed in transport experiments. Many more interesting phenomena pertaining to 2DEG have been studied. [A] Scientific model Scientific modelling

#430569