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Despite zero carrier density near the Dirac points, graphene exhibits a minimum conductivity on the order of /. The origin of this minimum conductivity is unclear. However, rippling of the graphene sheet or ionized impurities in the SiO 2 substrate may lead to local puddles of carriers that allow conduction. [3]
In k-space, this shows up as a hypercone, which have doubly degenerate bands which also meet at Dirac points. [11] 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.
The Dirac points are six locations in momentum space on the edge of the Brillouin zone, divided into two non-equivalent sets of three points. These sets are labeled K and K'. These sets give graphene a valley degeneracy of =. In contrast, for traditional semiconductors, the primary point of interest is generally Γ, where momentum is zero. [60]
The Dirac velocity gives the gradient of the dispersion at large momenta , is the mass of particle or object. In the case of massless Dirac matter, such as the fermionic quasiparticles in graphene or Weyl semimetals, the energy-momentum relation is linear,
In mathematics a Dirac structure is a geometric structure generalizing both symplectic structures and Poisson structures, and having several applications to mechanics. It is based on the notion of the Dirac bracket constraint introduced by Paul Dirac and was first introduced by Ted Courant and Alan Weinstein .
Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case.
The delta function was introduced by physicist Paul Dirac, and has since been applied routinely in physics and engineering to model point masses and instantaneous impulses. It is called the delta function because it is a continuous analogue of the Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1.
between the position operator x and momentum operator p x in the x direction of a point particle in one dimension, where [x, p x] = x p x − p x x is the commutator of x and p x , i is the imaginary unit, and ℏ is the reduced Planck constant h/2π, and is the unit operator.