Dirac matter

The universal treatment of the class of Dirac matter in terms of an effective theory leads to a common features with respect to the density of states, the heat capacity and impurity scattering.

However, all examples of Dirac matter are unified by similarities within the algebraic structure of an effective theory describing them.

The unifying feature of all Dirac matter is the matrix structure of the equation describing the quasi-particle excitations.

However, this general definition allows the description of matter with higher order dispersion relations and in curved spacetime as long as the effective Hamiltonian exhibits the matrix structure specific to the Dirac equation.

The general features and some specific examples of conventional Dirac matter are discussed in the following sections.

For example, 2010's Nobel Prize in Physics was awarded to Andre Geim and Konstantin Novoselov "for groundbreaking experiments regarding the material graphene".

Within the official press release of the Swedish Royal Academy of Science it is stated that[6] [...] a vast variety of practical applications now appear possible including the creation of new materials and the manufacture of innovative electronics.

Graphene transistors are predicted to be substantially faster than today’s silicon transistors and result in more efficient computers.In general, the properties of massless fermionic Dirac matter can be controlled by shifting the chemical potential by means of doping or within a field effect setup.

Additionally, depending on the specific realization of the Dirac material, it may be possible to introduce a mass term

In the two-dimensional systems such as graphene and topological insulators, the density of states gives a V shape, compared with the constant value for massive particles with dispersion

Experimental measurement of the density of states near the Dirac point by standard techniques such as scanning tunnelling microscopy often differ from the theoretical form due to the effects of disorder and interactions.

[7] Therefore, for systems whose physical dimension is greater than 1, the specific heat can provide a clear signature of the underlying Dirac nature of the quasiparticles.

As a result, the charged particles can only occupy orbits with discrete energy values, called Landau levels.

This key difference plays an important role in the experimental verification of Dirac matter.

The unique transport properties and the semimetallic state of graphene are the result of the delocalized electrons occupying these pz orbitals.

[12] An energy gap in the dispersion of graphene can be obtained from a low-energy Hamiltonian of the form which now contains a mass term

[13][14] The most practical approach for creating a gap (introducing a mass term) is to break the sublattice symmetry of the lattice where each carbon atom is slightly different to its nearest but identical to its next-nearest neighbours; an effect that may result from substrate effects.

In the bulk of a non-interacting topological insulator, the Fermi level is positioned within the gap between the conduction and valence bands.

On the surface, there are special states within the bulk energy gap which can be effectively described by a Dirac Hamiltonian: where

Such Dirac cones emerging on the surface of 3-dimensional crystals were observed in experiment, e.g.: bismuth selenide (Bi

Transition metal dichalcogenide monolayers are often discussed in reference to potential applications in valleytronics.

Weyl semimetals, for example tantalum arsenide (TaAs) and related materials,[23][24][25][26][27][28] strontium silicide (SrSi

[7] Tilting of the linear cones so the Dirac velocity varies leads to type II Weyl semimetals.

[32] In crystals that are symmetric under inversion and time reversal, electronic energy bands are two-fold degenerate.

)[36] [37][38] While historic interest focussed on fermionic quasiparticles that have potential for technological applications, particularly in electronics, the mathematical structure of the Dirac equation is not restricted to the statistics of the particles.

In the case of bosons, there is no Pauli exclusion principle to confine excitations close to the chemical potential (Fermi energy for fermions) so the entire Brillouin zone must be included.

In fact, the underlying symmetry of a crystal structure strongly constrains and protects the emergence of linear band crossings.

Typical bosonic quasiparticles in condensed matter are magnons, phonons, polaritons and plasmons.

Existing examples of bosonic Dirac matter include transition metal halides such as CrX

In comparison to bosons and fermions the situation gets more complicated as translations in space do not necessarily commute.

Tuning of Dirac matter: As the density of states is well defined, it can be nicely tuned at the Fermi level (for fermionic Dirac matter) by shifting the chemical potential . Introducing a mass term leads to a gap between the two cones and the dispersion becomes quadratic near .
While they are gapped, near the and points of their hexagonal Brilliouin zone, the dispersions of transition metal dichalcogenides can be described by a massive Dirac equation with additional spin-orbit coupling terms that lead to spin-splitting in the valence band.
Dispersions for bosonic (left) and fermionic (right) Dirac materials. In contrast to the fermionic case where Pauli exclusion confines excitations close to the Fermi energy, the description of boson requires the entire Brillioun zone.