physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

, Dirac cones are features that occur in some

electronic band structure In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or '' ...

s that describe unusual electron transport properties of materials like

graphene Graphene () is a carbon allotrope consisting of a Single-layer materials, single layer of atoms arranged in a hexagonal lattice, honeycomb planar nanostructure. The name "graphene" is derived from "graphite" and the suffix -ene, indicating ...

and

topological insulators A topological insulator is a material whose interior behaves as an electrical insulator while its surface behaves as an electrical conductor, meaning that electrons can only move along the surface of the material. A topological insulator is an ...

. In these materials, at energies near the

Fermi level The Fermi level of a solid-state body is the thermodynamic work required to add one electron to the body. It is a thermodynamic quantity usually denoted by ''μ'' or ''E''F for brevity. The Fermi level does not include the work required to re ...

, the valence band and conduction band take the shape of the upper and lower halves of a

conical surface In geometry, a conical surface is an unbounded surface in three-dimensional space formed from the union of infinite lines that pass through a fixed point and a space curve. Definitions A (''general'') conical surface is the unbounded surface ...

, meeting at what are called Dirac points. Typical examples include

bismuth Bismuth is a chemical element; it has symbol Bi and atomic number 83. It is a post-transition metal and one of the pnictogens, with chemical properties resembling its lighter group 15 siblings arsenic and antimony. Elemental bismuth occurs nat ...

antimony Antimony is a chemical element; it has chemical symbol, symbol Sb () and atomic number 51. A lustrous grey metal or metalloid, it is found in nature mainly as the sulfide mineral stibnite (). Antimony compounds have been known since ancient t ...

thin films A thin film is a layer of materials ranging from fractions of a nanometer ( monolayer) to several micrometers in thickness. The controlled synthesis of materials as thin films (a process referred to as deposition) is a fundamental step in many ...

and some other novel

nanomaterials Nanomaterials describe, in principle, chemical substances or materials of which a single unit is sized (in at least one dimension) between 1 and 100 nm (the usual definition of nanoscale). Nanomaterials research takes a materials science ...

, in which the electronic energy and momentum have a linear

dispersion relation In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the ...

such that the electronic band structure near the

takes the shape of an upper conical surface for the electrons and a lower conical surface for the holes. The two conical surfaces touch each other and form a zero-band gap semimetal. The name of Dirac cone comes from the

Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac ...

that can describe

relativistic particle In particle physics, a relativistic particle is an elementary particle with kinetic energy greater than or equal to its rest-mass energy given by Einstein's relation, E=m_0c^2, or specifically, of which the velocity is comparable to the speed of l ...

s in

quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

, proposed by

Paul Dirac Paul Adrien Maurice Dirac ( ; 8 August 1902 – 20 October 1984) was an English mathematician and Theoretical physics, theoretical physicist who is considered to be one of the founders of quantum mechanics. Dirac laid the foundations for bot ...

. Isotropic Dirac cones in graphene were first predicted by P. R. Wallace in 1947 and experimentally observed by the Nobel Prize laureates

Andre Geim Sir Andre Konstantin Geim (; born 21 October 1958; IPA1 pronunciation: ɑːndreɪ gaɪm) is a Russian-born Dutch–British physicist working in England in the School of Physics and Astronomy at the University of Manchester. Geim was awarded th ...

and Konstantin Novoselov in 2005.

Description

, Dirac cones are a kind of crossing-point which electrons avoid, where the energy of the valence and conduction bands are not equal anywhere in two dimensional lattice -space, except at the zero dimensional Dirac points. As a result of the cones, electrical conduction can be described by the movement of

charge carriers In solid state physics, a charge carrier is a particle or quasiparticle that is free to move, carrying an electric charge, especially the particles that carry electric charges in electrical conductors. Examples are electrons, ions and holes. In ...

which are massless

fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...

s, a situation which is handled theoretically by the relativistic

. The massless fermions lead to various

quantum Hall effect The quantum Hall effect (or integer quantum Hall effect) is a quantized version of the Hall effect which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance exhi ...

s, magnetoelectric effects in topological materials, and ultra high

carrier mobility In solid-state physics, the electron mobility characterizes how quickly an electron can move through a metal or semiconductor when pushed or pulled by an electric field. There is an analogous quantity for holes, called hole mobility. The term ca ...

. Dirac cones were observed in 2008-2009, using

angle-resolved photoemission spectroscopy Angle-resolved photoemission spectroscopy (ARPES) is an experimental technique used in condensed matter physics to probe the allowed energies and momenta of the electrons in a material, usually a crystalline solid. It is based on the photoel ...

(ARPES) on the potassium-

graphite intercalation compound In the area of solid state chemistry, graphite intercalation compounds are a family of materials prepared from graphite. In particular, the sheets of carbon that comprise graphite can be pried apart by the insertion ( intercalation) of ions. T ...

KC₈ and on several bismuth-based alloys. As an object with three dimensions, Dirac cones are a feature of two-dimensional materials or

surface states Surface states are electronic states found at the Surface (topology), surface of materials. They are formed due to the sharp transition from solid material that ends with a surface and are found only at the atom layers closest to the surface. The t ...

, based on a linear

between energy and the two components of the

crystal momentum In solid-state physics, crystal momentum or quasimomentum is a Momentum#Momentum in quantum mechanics, momentum-like Vector (geometric), vector associated with electrons in a Crystal structure, crystal lattice. It is defined by the associated Rec ...

_x and _y. However, this concept can be extended to three dimensions, where Dirac semimetals are defined by a linear dispersion relation between energy and _x, _y, and _z. In -space, this shows up as a

hypercone In geometry, a hypercone (or spherical cone) is the figure in the 4-dimensional Euclidean space represented by the equation :x^2 + y^2 + z^2 - w^2 = 0. It is a quadric surface, and is one of the possible 3-manifolds which are 4-dimensional eq ...

, which have doubly degenerate bands which also meet at Dirac points. Dirac semimetals contain both time reversal and spatial inversion symmetry; when one of these is broken, the Dirac points are split into two constituent Weyl points, and the material becomes a Weyl semimetal. In 2014, direct observation of the Dirac semimetal band structure using ARPES was conducted on the Dirac semimetal

cadmium arsenide Cadmium arsenide ( Cd3 As2) is an inorganic semimetal in the II-V family. It exhibits the Nernst effect. Properties Thermal Cd3As2 dissociates between 220 and 280 °C according to the reaction :2 Cd3As2(s) → 6 Cd(g) + As4(g) An e ...

Analog systems

Dirac points have been realized in many physical areas such as

plasmonics Plasmonics or nanoplasmonics refers to the generation, detection, and manipulation of signals at optical frequencies along metal-dielectric interfaces in the nanometer scale. Inspired by photonics, plasmonics follows the trend of miniaturizing op ...

phonon A phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. In the context of optically trapped objects, the quantized vibration mode can be defined a ...

ics, or

nanophotonics Nanophotonics or nano-optics is the study of the behavior of light on the nanometer scale, and of the interaction of nanometer-scale objects with light. It is a branch of optics, optical engineering, electrical engineering, and nanotechnology. I ...

(microcavities, photonic crystals).

Description

Analog systems

See also

References

Further reading