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This anomalous behavior is due to graphene's massless Dirac electrons. In a magnetic field, these electrons form a Landau level at the Dirac point with an energy that is precisely zero. This is a result of the Atiyah–Singer index theorem index theorem and causes the "+1/2" term in the Hall conductivity for neutral graphene. [4] [47]
This behavior is a direct result of graphene's chiral, massless Dirac electrons. [2] [95] In a magnetic field, their spectrum has a Landau level with energy precisely at the Dirac point. This level is a consequence of the Atiyah–Singer index theorem and is half-filled in neutral graphene, [30] leading to the "+1/2" in the Hall conductivity. [33]
The name of Dirac cone comes from the Dirac equation that can describe relativistic particles in quantum mechanics, proposed by Paul Dirac. Isotropic Dirac cones in graphene were first predicted by P. R. Wallace in 1947 [6] and experimentally observed by the Nobel Prize laureates Andre Geim and Konstantin Novoselov in 2005. [7]
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,
The massless 2D case can be simulated in single-layer materials like graphene near the Dirac cones, where the eigenergies are given by [8] = where the speed of light has to be replaced with the Fermi speed v F of the material and the minus sign corresponds to electron holes.
In semiconductors with non-simple band structures, this relationship is used to define an effective mass, known as the density of states effective mass of electrons. The name "density of states effective mass" is used since the above expression for N C is derived via the density of states for a parabolic band.
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[9] [10] The electrons at the metallic structure could transfer to the graphene conduction band. This is attributed to the zero bandgap property of graphene nanosheet. Graphene plasmons can also be decoupled from their environment and give rise to genuine Dirac plasmon at low-energy range where the wavelengths exceed the damping length.