Electronic Properties of Dirac and Weyl Semimetals

The following sections are included:

• Contents
• Preface
• Acknowledgments

The notion of quasiparticles is one of the most important concepts in condensed matter physics. In fact, the very diversity of this branch of physics is pivoted on the possibility of different energy dispersions and other characteristics of quasiparticles. Although, microscopically, the electrons in every solid-state material are described by nothing else but the standard nonrel-ativistic Schrödinger Hamiltonian with the electron–electron and electron–ion interactions, the dispersion relations of the electron Bloch states are incredibly diverse. In particular, they can have even a seemingly exotic relativistic-like form…

Although this book is devoted to the electronic properties of three-dimensional (3D) Dirac and Weyl semimetals, we find it logical to begin with the discussion of electronic properties of graphene. Indeed, its quasi-particles are governed by two-dimensional (2D) Dirac equation and demonstrate a variety of interesting properties. This makes graphene a perfect starting point.

Topology deals with the characteristics of systems that remain invariant under continuous deformations of model parameters. Therefore, a nontrivial topology often implies the robustness of physical properties with respect to perturbations, impurities, and disorder. The relevance of topology for condensed matter systems was understood a long time ago. In particular, it played a major role in the studies of numerous topological defects whose existence is often associated with spontaneous breaking of global symmetries. The magnetic domain walls, the quantization of circulation in superfluid ⁴He, and the Abrikosov vortices in type-II superconductors are just a few renowned examples [348]. The existence of such defects is usually protected by one of the nontrivial homotopy groups π_i($\mathcal{G}$/$\mathcal{H}$) (with i = 0 for domain walls, i = 1 for vortices, and i = 2 for point defects), where $\mathcal{G}$ is the symmetry group of the Hamiltonian and $\mathcal{H}$ is the symmetry subgroup of the system in the ground state. After the discovery of the integer quantum Hall effect in a two-dimensional (2D) electron gas in a strong magnetic field, it became clear that topology can be relevant also for electronic states in solids. In fact, the remarkable robustness of the experimentally observed quantization of the Hall conductivity is explained by the underlying topology [293, 295]…

The rapid increase of the computational power at the beginning of the 21st century as well as the development of effective first-principles calculation methods and software packages strongly improved the quality and significance of numerical calculations in science. Nowadays, in condensed matter physics, various properties of materials are routinely studied by using the ab initio simulations, which provide a powerful supplemental tool for the conventional theoretical and experimental research. Numerical studies and modeling of topological phases of matter are particularly fruitful because their nontrivial band topology leads to an intrinsic robustness with respect to disorder and numerical errors [17, 459]…

All crystals encountered in nature are, of course, finite. It is of interest, therefore, to discuss how an abrupt change of the periodic atomic lattice near the surface of a crystal affects an electron band structure. In this connection, one should note that there exist global bulk effects due to a finite size as well as local effects at the surface itself. First of all, in finite samples, the quasiparticle energy spectra become discrete. Since the distance between energy levels quickly decreases with the sample size, the corresponding effects are usually negligible for sufficiently large crystals, but may become important in thin films…

Without much exaggeration, the electron transport properties are among the most important characteristics of solids. Not only they carry a trove of information about the fundamental physics of electron quasiparticles but also are of immense practical value. As is well known, the electron transport is determined largely by the band structure of materials. For example, while the absence of a band gap at the Fermi level in metals makes them good conductors, a sufficiently large gap between the valence and conductance bands is one of the definitive properties of insulators. In applied physics, the understanding of widely tunable transport properties in semiconductors opened the possibility to manipulate their conductivity and led to the invention of transistors, which nowadays are the basic elements of nearly all electronics. Of similar significance was also the discovery of superconductivity, which provides an example of an unusual electron transport in solids. Superconductivity in Weyl and Dirac semimetals will be discussed in Chapter 9. It is not surprising, therefore, that the search for novel materials with unconventional transport properties is one of the key driving forces of solid-state physics…

In addition to various anomalous features connected with the condensed matter manifestation of the chiral anomaly and nontrivial topology, Dirac and Weyl semimetals have other interesting electronic transport properties. The electrical conductivity is one of the most fundamental characteristics of any material. In order to correctly describe the dissipative electron transport in Weyl and Dirac semimetals, it is crucial to know how different types of disorder affect the conductivity and how a steady state in an applied electric field is established…

Collective excitations are simple but very informative probes in all types of physical systems. They are particularly useful for the characterization of plasmas on large distance and long timescales. The studies of collective modes in plasmas are invaluable in fundamental research but also very useful for practical applications. Examples of various forms of plasmas include ionized gases, interstellar and intergalactic media, electron plasmas in solids, relativistic plasmas in heavy-ion collisions and the early Universe, as well as degenerate states of dense matter in compact stars. While chiral electron plasmas in Dirac and Weyl semimetals share many common properties with other plasmas, they do have a number of unusual features. Since electrons are electrically charged, collective modes of the corresponding plasma are accompanied by oscillating electromagnetic fields. Various types of such modes in chiral electron plasmas of Dirac and Weyl semimetals will be considered in this chapter. For simplicity, however, only waves with small amplitudes described by linearized equations will be presented. In the opposite regime of large amplitude waves, a diverse range of nonlinear phenomena can also be realized in a chiral matter. While this is also a fascinating topic, it has not been studied much in the literature yet and will not be covered here…

The phenomenon of superconductivity was observed experimentally in mercury by Heike Kamerlingh Onnes more than a century ago in 1911. In essence, superconductivity implies that an electric current flows without any resistance. The appearance of superconductivity is indicated by a phase transition at sufficiently low temperatures. This transition is marked by a jump of specific heat, which is related to a gap in the spectrum of low-energy electron excitations in a superconductor. In addition, magnetic fields are expelled from the bulk of superconducting material. This phenomenon is known as the Meissner effect [968]…

The following sections are included:

• Appendix A Green’s Functions
  • Definition
  • Green’s function in magnetic field
• Appendix B Useful Formulas
• Appendix C Fundamental Constants and Units
• Appendix D Numerical Values of Material Parameters

The following sections are included:

• Bibliography
• Index