Dirac materials are characterized by energy-momentum relations that resemble those of relativistic massless particles. Commonly denominated Dirac cones, these dispersion relations are considered to be their essential feature. These materials comprise quite diverse examples, such as graphene and topological insulators. Band-engineering techniques should aim to a full control of the parameter that characterizes the Dirac cones: the Fermi velocity. We propose a general mechanism that enables the fine-tuning of the Fermi velocity in Dirac materials in a readily accessible way for experiments. By embedding the sample in a uniform electric field, the Fermi velocity is substantially modified. We first prove this result analytically, for the surface states of a topological insulator/semiconductor interface, and postulate its universality to other Dirac materials. Then we check its correctness in carbon-based Dirac materials, namely graphene nanoribbons and nanotubes, thus showing the validity of our hypothesis in both continuum and tight-binding calculations and in different Dirac systems.
The type-II Weyl/Dirac fermions are a generalization of conventional or type-I Weyl/Dirac fermions, whose conic spectrum is tilted such that the Fermi surface becomes lines in two dimensions, and surface in three dimensions rather than discrete points of the conventional Weyl/Dirac fermions. The mass-independent renormalization group calculations show that the tilting parameter decreases monotonically with respect to the length scale, which leads to a transition from two dimensional type-II Weyl/Dirac fermions to the type-I ones. Because of the non-trivial Fermi surface, a photon gains a finit…
We numerically demonstrate that the topological corner states residing in the corners of higher-order topological insulator possess non-Abelian braiding properties. Such topological corner states are Dirac fermionic modes other than Majorana zero-modes. We claim that Dirac fermionic modes protected by nontrivial topology also support non-Abelian braiding. An analytical description on such non-Abelian braiding is conducted based on the vortex-induced Dirac-type fermionic modes. The braiding operator for Dirac fermionic modes is also analytically derivated and compared with the Majorana zero-mod…
The low energy physics of both graphene and surface states of three-dimensional topological insulators is described by gapless Dirac fermions with linear dispersion. In this work, we predict the emergence of a "heavy" Dirac fermion in a graphene/topological insulator hetero-junction, where the linear term almost vanishes and the corresponding energy dispersion becomes highly non-linear. By combining {\it ab initio} calculations and an effective low-energy model, we show explicitly how strong hybridization between Dirac fermions in graphene and the surface states of topological insulators can r…
The symmetries that protect massless Dirac fermions from a gap opening may become ineffective if the Dirac equation is discretized in space and time, either because of scattering between multiple Dirac cones in the Brillouin zone (fermion doubling) or because of singularities at zone boundaries. Here we introduce an implementation of Dirac fermions on a space-time lattice that removes both obstructions. The quasi-energy band structure has a tangent dispersion with a single Dirac cone that cannot be gapped without breaking both time-reversal and chiral symmetries. We show that this topological …
We investigate the effect of a vertical electric field on a Dirac semimetal thin film. We show that through the interplay between the quantum confinement effect and the field-induced coupling between sub-bands, the sub-band gap can be tuned and inverted, during which the system undergoes a topological phase transition between a trivial band insulator and a quantum spin Hall insulator. Consequently, one can electrically switch the topological edge channels on and off, making the system a promising platform for constructing a topological field effect transistor.
Fermi-Dirac integrals appear frequently in semiconductor problems, so a basic understanding of their properties is essential. The purpose of these notes is to collect in one place, some basic information about Fermi-Dirac integrals and their properties. We also present Matlab scripts that calculate Fermi-Dirac integrals (the "script F" defined by Dingle (1957)) in three different ways. The codes are available in Appendix and at the following website: "Notes on Fermi-Dirac Integrals (4th Edition)" by Raseong Kim, Xufeng Wang, and Mark Lundstrom at http://nanohub.org/resources/5475 In the 4th ed…
We study the quantum Hall effect of Dirac fermions on the surface of a Wilson-Dirac type topological insulator thin film in the strong topological insulating phase. Although a magnetic field breaks time reversal symmetry of the bulk, the surface states can survive even in a strong field regime. We examine how the Landau levels of the surface states are affected by symmetry breaking perturbations.
The Datta-Das-type spin field-effect transistor, using a two-dimensional electron gas in a semiconductor heterostructure as a channel, plays a key role in spintronics. Here, we theoretically present a type of spin field-effect transistor based on massless birefringent Dirac fermions in polar Dirac semimetals. The manipulation of spin arises from the existence of the strong spin-orbit coupling, polar space groups, and Dirac cones in a single phase. The oscillatory channel conductance can be controlled by the sign of gate voltage in addition to its magnitude due to the gapless band structures of…
Composite fermions in fractional quantum Hall (FQH) systems are believed to form a Fermi sea of weakly interacting particles at half filling $ν=1/2$. Recently, it was proposed (D. T. Son, Phys. Rev. X 5, 031027 (2015)) that these composite fermions are Dirac particles. In our work, we demonstrate experimentally that composite fermions found in monolayer graphene are Dirac particles at half filling. Our experiments have addressed FQH states in high-mobility, suspended graphene Corbino disks in the vicinity of $ν=1/2$. We find strong temperature dependence of conductivity $σ$ away from half fill…
The spatial discretization of the single-cone Dirac Hamiltonian on the surface of a topological insulator or superconductor needs a special "staggered" grid, to avoid the appearance of a spurious second cone in the Brillouin zone. We adapt the Stacey discretization from lattice gauge theory to produce a generalized eigenvalue problem, of the form ${\mathcal H}ψ=E {\mathcal P}ψ$, with Hermitian tight-binding operators ${\mathcal H}$, ${\mathcal P}$, a locally conserved particle current, and preserved chiral and symplectic symmetries. This permits the study of the spectral statistics of Dirac fe…
A brief survey is given of theoretical works on surface states (SSs) in Dirac materials. Within the formalism of envelope wave functions and boundary conditions for these functions, a minimal model is formulated that analytically describes surface and edge states of various (topological and non-topological) types in several systems with Dirac fermions (DFs). The applicability conditions of this model are discussed.
We study the electric-field tunable electronic properties of phosphorene thin films, using the framework of density functional theory. We show that phosphorene thin films offer a versatile material platform to study two dimensional Dirac fermions on application of a transverse electric field. Increasing the strength of the transverse electric field beyond a certain critical value in phosphorene leads to the formation of two symmetry protected gapless Dirac fermions states with anisotropic energy dispersion. The spin-orbit coupling splits each of these Dirac state into two spin- polarized Dirac…
We discuss the Schwinger pair creation process for the system of massless Dirac fermions in the presence of constant external magnetic and electric fields. The pair production rate remains finite unlike the vacuum decay rate. In the recently discovered Dirac semimetals, where the massless Dirac fermions emerge, this pair production may be observed experimentally through the transport properties. We estimate its contribution to the ordinary conductivity of the semimetals.
Recently, a new "type-II" Weyl fermion, which exhibits exotic phenomena such as angle-dependent chiral anomaly, was discovered in a new phase of matter where electron and hole pockets contact at isolated Weyl points. [Nature \textbf{527}, 495 (2015)] This raises an interesting question whether its counterpart, i.e., type-II \textit{Dirac} fermion, exists in real materials. Here, we predict the existence of symmetry-protected type-II Dirac fermions in a class of transition metal dichalcogenide materials. Our first-principles calculations on PtSe$_2$ reveal its bulk type-II Dirac fermions which …
Fermi arc surface states, a hallmark of topological Dirac semimetals, can host carriers that exhibit unusual dynamics distinct from that of their parent bulk. Here we find that Fermi arc carriers in intrinsic Dirac semimetals possess a strong and anisotropic light matter interaction. This is characterized by a large Fermi arc optical conductivity when light is polarized transverse to the Fermi arc; when light is polarized along the Fermi arc, Fermi arc optical conductivity is significantly muted. The large surface spectral weight is locked to the wide separation between Dirac nodes and persist…
Massless Dirac fermions in graphene can acquire a mass through different kinds of sublattice-symmetry-breaking perturbations, and there is a growing need to determine this mass using a conventional method. We describe how the mass caused by a staggered sublattice potential is determined using Raman spectroscopy and explain the mechanism in terms of the pseudospin polarization of massive Dirac fermions.
Quantum computation may well be performed with the use of electric circuits. Especially, the Schrödinger equation can be simulated by the lumped-element model of transmission lines, which is applicable to low-frequency electric circuits. In this paper, we show that the Dirac equation is simulated by the distributed-element model, which is applicable to high-frequency electric circuits. Then, a set of universal quantum gates (the Hadamard, phase-shift and CNOT gates) are constructed by networks made of transmission lines. We demonstrate Shor's prime factorization based on electric circuits. It …
Borophene is a monolayer materials made of boron. A perfect planar boropehene called $β_{12}$ borophene has Dirac cones and they are well reproduced by a tight-binding model according to recent experimental and first-principles calculation results. We explicitly derive a Dirac theory for them. Dirac cones are gapless when the inversion symmetry exists, while they are gapped when it is broken. In addition, three-band touching points emerge together with pseudospin triplet fermions when all transfer energy is equal and all on-site energy is equal. The three-band touching is slightly resolved oth…
The analogues of elementary particles have been extensively searched for in condensed matter systems because of both scientific interests and technological applications. Recently massless Dirac fermions were found to emerge as low energy excitations in the materials named Dirac semimetals. All the currently known Dirac semimetals are nonmagnetic with both time-reversal symmetry $\mathcal{T}$ and inversion symmetry $\mathcal{P}$. Here we show that Dirac fermions can exist in one type of antiferromagnetic systems, where $\mathcal{T}$ and $\mathcal{P}$ are broken but their combination $\mathcal{P…
We investigate the non-Fermi-liquid behaviors of the 2D and 3D Dirac/Weyl systems with low-order and higher order dispersion. The self-energy correction, symmetry, free energy, optical conductivity, density of states, and spectral function are studied. We found that, for Dirac/Weyl systems with higher order dispersion, the non-Fermi-liquid features remain even at finite chemical potential, and they are distinct from the ones in Fermi-liquid picture and the conventional non-Fermi-liquid picture. The Landau damping of the longitudinal excitations within random-phase-approximation (RPA) for the n…
Anomalous surface states with Fermi arcs are commonly considered to be a fingerprint of Dirac semimetals (DSMs). In contrast to Weyl semimetals, however, Fermi arcs of DSMs are not topologically protected. Using first-principles calculations, we predict that $β$-CuI is a peculiar DSM whose surface states form closed Fermi pockets instead of Fermi arcs. In such a fermiological Dirac semimetal, the deformation mechanism from Fermi arcs to Fermi pockets stems from a large cubic term preserving all crystal symmetries, and the small energy difference between the surface and bulk Dirac points. The c…
The coupling of monochromatic light fields and solids introduces nonequilibrium Floquet states, offering opportunities to create and explore new topological phenomena. Using combined first-principles and Floquet analysis we show that one can freely engineer Floquet-Dirac fermions (FDFs) in graphene by tuning the frequency and intensity of linearly polarized light. Not only type-II FDFs are created, but they also coexist with type-I FDFs near the Fermi level. Intriguingly, novel topologically nontrivial edge states connecting type-I and type-II Floquet-Dirac points emerge in photodriven graphen…
Superconducting and topological states are two distinct quantum states of matter. Superconductors with topologically nontrivial electronic states provide a platform to study the interplay between them. Recent study revealed that a bulk superconductor PdTe2 is a topological Dirac semimetal with type-II Dirac fermions. However, the Dirac fermions in PdTe2 have no contribution to the superconducting pair because the Dirac points reside far below the Fermi level (EF). Here, we show that in the isostructural compounds (Ir1-xPtx)Te2, the type-II Dirac points can be adjusted to EF by element substitu…
Clean two-dimensional Dirac systems have received a lot of attention for being a prime candidate to observe hydrodynamical transport behavior in interacting electronic systems. This is mostly due to recent advances in the preparation of ultrapure samples with sufficiently strong interactions. In this paper, we investigate the role of collective modes in the thermo-electric transport properties of those systems. We find that dynamical particle-hole pairs, plasmons, make a sizeable contribution to the thermal conductivity. While the increase at the Dirac point is moderate, it becomes large towar…
Recently, realizing new fermions, such as type-I and type-II Dirac/Weyl fermions in condensed matter systems, has attracted considerable attention. Here we show that the transition state from type-I to type-II Dirac fermions can be viewed as a type-III Dirac fermion, which exhibits unique characteristics, including a Dirac-line Fermi surface with nontrivial topological invariant and critical chiral anomaly effect, distinct from previously known Dirac semimetals. Most importantly, we discover Zn$_2$In$_2$S$_5$ is a type-III Dirac semimetal material, characterized with a pair of Dirac points in …
We use the framework of supersymmetric transformations in the construction of coupled systems of Dirac fermions. Its energy operator is a composite of the generators of the associated superalgebra, and the two coupled Dirac fermions acquire different Fermi velocities. We discuss in detail the peculiar spectral properties of the new system. The emergent phenomena like level crossing or generation of bound states in the continuum (BICs) are illustrated in two explicit examples.
We present a renormalization group analysis of the disorder effects on the low-energy behaviors of twodimensional tilted Dirac-fermion systems, in which the fermions have two distinct orbitals unrelated by any symmetry. Four types of disordered potential, two interorbital and two intraorbital, are considered. If there is only one type of interorbital disorder, the fermion-disorder scattering induces logarithmic or power-law corrections to the fermion density of states and specific heat. In contrast, the intraorbital disorder can turn the system into a strongly disordered phase. In this disorde…
We report quasi-two-dimensional Dirac fermions and quantum magnetoresistance in LaAgBi$_2$. The band structure shows several narrow bands with nearly linear energy dispersion and Dirac-cone-like points at the Fermi level. The quantum oscillation experiments revealed one quasi-two-dimensional Fermi pocket and another complex pocket with small cyclotron resonant mass. The in-plane transverse magnetoresistance exhibits a crossover at a critical field $B^*$ from semiclassical weak-field $B^2$ dependence to the high-field unsaturated linear magnetoresistance which is attributed to the quantum limit…
Transverse electric (TE) modes can not propagate through the conducting solids. This is because the continuum of particle-hole excitations of conductors contaminates with the TE mode and dampes it out. But in solids hosting tilted Dirac cone (TDC) that admit a description in terms of a modified Minkowski spacetime, the new spacetime structure remedies this issue and therefore a tilted Dirac cone material (TDM) supports the propagation of an undamped TE mode which is sustained by density fluctuations. The resulting TE mode propagates at fermionic velocities which strongly confines the mode to t…
We study the confinement of Dirac fermions in armchair graphene nanoribbons by means of a quantum-dot-type electrostatic potential. With the use of specific projection operators, we find exact solutions for some bound states that satisfy appropriate boundary conditions. We show that the energies of these bound states belong either to the gap of valence and conducting bands or they represent BIC's (bound states in the continuum) whose energies are embedded in the continuous spectrum.
We consider systems described by the two-dimensional Dirac equation where the Fermi velocity is inhomogeneous as a consequence of mechanical deformations. We show that the mechanical deformations can lead to deflection and focusing of the wave packets. The analogy with known reflectionless quantum systems is pointed out. Furthermore, with the use of the qualitative spectral analysis, we discuss how inhomogeneous strains can be used to create waveguides for valley polarized transport of partially dispersionless wave packets.
Recently Weyl fermions have attracted increasing interest in condensed matter physics due to their rich phenomenology originated from their nontrivial monopole charges. Here we present a theory of real Dirac points that can be understood as real monopoles in momentum space, serving as a real generalization of Weyl fermions with the reality being endowed by the $PT$ symmetry. The real counterparts of topological features of Weyl semimetals, such as Nielsen-Ninomiya no-go theorem, $2$D sub topological insulators and Fermi arcs, are studied in the $PT$ symmetric Dirac semimetals, and the underlyi…
We use the Wick-rotated time-dependent supersymmetry to construct models of two-dimensional Dirac fermions in presence of an electrostatic grating. We show that there appears omnidirectional perfect transmission through the grating at specific energy. Additionally to being transparent for incoming fermions, the grating hosts strongly localized states.
We study the confinement of Dirac fermions in graphene and in carbon nanotubes by an external magnetic field, mechanical deformations or inhomogeneities in the substrate. By applying variational principles to the square of the Dirac operator, we obtain sufficient and necessary conditions for confinement of the quasi-particles. The rigorous theoretical results are illustrated on the realistic examples of the three classes of traps.
