The major factor determining transport properties of solids is the number of electron states in a vicinity of Fermi level. In equilibrium, no macroscopic flow of electrons exists and, therefore, in order to create such flow, electrons must be excited over their equilibrium distribution. However, electrons with energies well below the Fermi level (compared to the characteristic energy scale *k _{B}T,* where

*k*is the Boltzmann constant and

_{B}*T*is the temperature), cannot acquire small excitation energy. Indeed, in this case, they would have energy corresponding to already occupied states, which is prohibited by the Pauli principle. In turn, well above the Fermi level, where excitation of electrons is not constrained by the Pauli principle, the electron states are not populated, thus making their contribution to the response negligible.

An elementary classification of materials as conductors and insulators is, therefore, based on the relation between the Fermi level and the energy bands characterizing electron states in the solid. In metals, such as copper or silver, the Fermi level is inside the band and, therefore, metals have high electric and thermal conductivities. In insulators, for instance, silicon dioxide (SiO_{2}) or crystal sodium chloride (NaCl), the Fermi level is inside the wide, conventionally more than 4 eV, bandgap separating valence and conduction (below and above the Fermi level, respectively) bands, for example, ∆ ≈ 8.9 eV in SiO_{2} and ∆ ≈ 8.5 eV in NaCl.

Such picture establishing a relation between the band diagram of a solid and its transport properties is sufficient for dealing with materials with sharp conducting or insulating properties. However, it requires refinement when the band gap is not too wide (∆ is smaller than 4 eV). Materials with 0 < ∆ < 4 eV are semiconductors, for example, silicon (Si, ∆ ≈ 1.1 eV) and gallium arsenide (GaAs, ∆ ≈ 1.5 eV). Since the Fermi level is essentially the electrochemical potential for electrons, its position on the band diagram noticeably depends on doping (concentration of donor or acceptor impurities) and the external electric potential. These are core properties laying in the ground of semiconductor electronics.

Further decreasing of the gap width results in overlap of the valence and conduction bands. Such materials are called semimetals. Usually, for instance in bismuth (Bi) and graphite (an allotrope of carbon, C), this occurs when the extremal points of the bands are located at points with different momenta. In this case, the direct gap, the separation between points in conduction and valence bands with the same momentum, preserves. As a result, the electron density of states at the energy range corresponding to overlapping bands is small and the materials demonstrate properties characteristic for both semiconductors and metals.

Recently, a series of new materials enriching the canonical understanding of solids started to attract significant attention. Among them, a special place is taken by semimetals with the extremal points of the conduction and valence bands residing at the same momentum and separated by a small or even absent gap. The important feature of such materials is that near the extremal points of the valence and conduction bands, the electron states with a good accuracy are described by a Dirac equation generalized to account for a possible lack of rotational symmetry. As a result, the dynamics of low-energy excitations is similar to ultrarelativistic quantum particles. This brings concepts that previously were thought of only in the context of very high energies into the “everyday” energy range.

The most famous and well-studied example of such materials is graphene, a one-atom thick carbon layer. Due to its lattice structure, the band diagram of graphene has distinctive Dirac cones touching each other at the -points of the Brillouin zone (Figure 2). Due to the linear dispersion law near these points, electrons behave like massless fermions. Such particles are called Weyl fermions after the physicist who first considered the massless limit of the Dirac equation. Respectively, the touching point of the conduction and valence bands is called the Weyl point.

*This work was done by Professor Pinaki Mazumder of the University of Michigan for the Air Force Research Laboratory.* *AFRL-0287*

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