Two-dimensional Dirac Research Articles

Charged magnons on the surface of a topological insulator

We study a system of two-dimensional Dirac electrons (as is realized on the surface of a 3D topological insulator) coupled to an array of localized spins. The spins are coupled ferromagnetically to each other, forming an ordered ground state with low-energy spin-wave excitations (magnons). The Dirac electrons couple to the spins through a spin-dependent effective Zeeman field. The out-of-plane effective Zeeman field therefore serves as a Dirac mass that gaps the electronic spectrum. Once a spin is flipped, it creates a surrounding domain in which the sign of the Dirac mass is opposite to that of the rest of the sample. Therefore, an electronic bound state appears on the domain wall, as predicted by Jackiw and Rebbi. However, in a quantum magnet, a localized spin flip does not produce an eigenstate. Instead, the eigenstates correspond to delocalized spin waves (magnons). As in the case of the single flipped spin, the delocalized magnon also binds an in-gap electronic state. We name this excitation a `Jackiw-Rebbi-Magnon' (JRM) and study its signature in the dynamic spin susceptibility. When the sample is tunnel-coupled to an electronic reservoir, a magnon produced in a system without any electrons hybridizes with a JRM (which binds a single electron), producing magnon-JRM polaritons. For such a system, we identify a quantum phase transition when the magnon-JRM polariton energy falls below that of the fully polarized ferromagnetic ground state.

Long-range adsorbate interactions mediated by two-dimensional Dirac fermions

We provide here an analytical formalism to describe the indirect interaction between adsorbed atom or molecule pairs mediated by two-dimensional (2D) Dirac fermions. We show that in contrast to the case of traditional 2D electron gas, in the 2D Dirac system, the long-range interaction behaves as 1/r 3 decaying Friedel oscillation. This analytical formalism is fully consistent with a tight-binding numerical calculation of honeycomb lattices. Our formalism is suitable for the realistic 2D Dirac materials, such as graphene and surface states of three-dimensional topological insulators.

Topological semimetal phases in a family of monolayer X3YZ6 (X=Nb,Ta, Y=Si,Ge,Sn, Z=S,Se,Te) with abundant nodal lines and nodes

The electronic and topological properties of single-layer X3YZ6 (X=Nb,Ta, Y=Si,Ge,Sn, Z=S,Se,Te) materials have been studied with the aid of first principles calculations. This kind of materials belong to topological semimetals (TMs) with abundant nodal lines and nodes. Considering their similar properties, we focus on the analysis of Ta3SnTe6 and Ta3SiSe6. The presence of spin–orbit coupling (SOC) leads to the transition from type-I nodal lines to Dirac points as well as the disappear of type-II Dirac points. The three-dimensional (3D) plot of the band structure reproduces vividly the characteristics of nodes and nodal lines. The appearance of the flat bands in (110) edge states further confirm their nontrivial topological properties. We also explore the relationship among different nodal lines (nodes), crystal symmetry and SOC. The type-I nodal lines are protected by Mz and Mỹ symmetry in the absence of SOC. Symmetry breaking leads to band splitting even in the presence of SOC. The single-layer X3YZ6 can be used as candidates for two-dimensional (2D) Dirac materials and provide a platform for further study of interesting physical phenomena.

Third-order nonlinear Hall effect in two-dimensional Dirac systems

We theoretically investigate the third-order nonlinear Hall effect by employing the quantum kinetic equation and present an analytic formula for the third-order harmonic conductivity, where the intraband and the mixed inter-band contributions caused by the external electric field as well as the multiphoton process dressed Berry curvature are unveiled. These results are applied in a two-dimensional Dirac system. We find that the third-order harmonic conductivity exhibits a crossover from the negative to positive value due to the intraband transition of the electrons in the low-frequency region, whereas a kink or dip structure due to the multiphoton process between the conduction and valence bands in the high-frequency region. In experiment, these features can be identified by detecting the transverse current. Our work deepens the understanding of the multiphoton process in the nonlinear transport of topological materials and enlightens a possible way to its characterization in experiments.

Irregular Bloch-Zener oscillations in two-dimensional flat-band Dirac materials

When a static electrical field is applied to a two-dimensional (2D) Dirac material, Landau-Zener transition (LZT) and Bloch-Zener oscillations can occur. Employing $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathcal{T}}_{3}$ lattices as a paradigm for a broad class of 2D Dirac materials, we uncover two phenomena. First, due to the arbitrarily small energy gaps near a Dirac point that make it more likely for LZTs to occur than in other regions of the Brillouin zone, the distribution of differential LZT probability in the momentum space can form a complicated morphological pattern. Second, a change in the LZT morphology as induced by a mutual switching of the two distinct Dirac points can lead to irregular Bloch-Zener oscillations characterized by a nonsmooth behavior in the time evolution of the electrical current density associated with the oscillation. These phenomena are due to mixed interference of quantum states in multiple bands modulated by the geometric and dynamic phases. We demonstrate that the adiabatic-impulse model describing Landau-Zener-St\"uckelberg interferometry can be exploited to calculate the phases, due to the equivalence between the $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathcal{T}}_{3}$ lattice subject to a constant electrical field and strongly periodically driven two- or three-level systems. The degree of irregularity of Bloch-Zener oscillations can be harnessed by selecting the morphology pattern, which is potentially experimentally realizable.

Asymmetric transport computations in Dirac models of topological insulators

This paper presents a fast algorithm for computing transport properties of two-dimensional Dirac operators with linear domain walls, which model the macroscopic behavior of the robust and asymmetric transport observed at an interface separating two two-dimensional topological insulators. Our method is based on reformulating the partial differential equation as a corresponding volume integral equation, which we solve via a spectral discretization scheme.We demonstrate the accuracy of our method by confirming the quantization of an appropriate interface conductivity modeling transport asymmetry along the interface, and moreover, confirm that this quantity is immune to local perturbations. We also compute the far-field scattering matrix generated by such perturbations and verify that while asymmetric transport is topologically protected the absence of back-scattering is not.

Topological nodal ring semimetal in graphene

Graphene is a two-dimensional Dirac semimetal showing interesting properties as a result of its dispersion relation with both quasiparticles and quasiholes or matter and anti-matter. We introduce a topological nodal ring semimetal in graphene with a quantized quantum Hall response, a robust one-dimensional chiral edge mode and a quadratic Fermi-liquid spectrum for the quasiparticles and quasiholes in the bulk. The bulk band degeneracy at the Fermi energy is protected through a Z2 symmetry related to the two spin polarizations of an electron and a double-orthogonality structure in the sublattice and spin quantum numbers of the two crossing eigenstates. The system may have applications in nano-electronics and in quantum mechanical entanglement applied to band theory.

Substrate effects on spin relaxation in two-dimensional Dirac materials with strong spin-orbit coupling

Understanding substrate effects on spin dynamics and relaxation is of key importance for spin-based information technologies. However, the key factors that determine such effects, in particular for materials with strong spin-orbit coupling (SOC), have not been well understood. Here we performed first-principles real-time density-matrix dynamics simulations with SOC and the electron-phonon and electron-impurity scattering for spin lifetimes (τs) of supported/free-standing germanene, a prototypical strong SOC 2D Dirac material. We show that the effects of different substrates on τs can surprisingly differ by two orders of magnitude. We find that substrate effects on τs are closely related to substrate-induced modifications of the SOC-field anisotropy, which changes the spin-flip scattering matrix elements. We propose a new electronic quantity, named spin-flip angle θ↑↓, to characterize spin relaxation through intervalley spin-flip scattering. We find that {tau }_{s}^{-1} is approximately proportional to the averaged value of {sin }^{2}left({theta }^{uparrow downarrow }/2right), which serves as a guiding parameter of controlling spin relaxation.

Topological order in interacting semimetals

It has recently been demonstrated that it is possible to open a gap in a magnetic Weyl semimetal, while preserving the chiral anomaly along with the charge conservation and translational symmetries, which all protect the gapless nodes in a weakly interacting semimetal. The resulting state was shown to be a nontrivial generalization of a non-Abelian fractional quantum Hall liquid to three dimensions. Here we point out that a second fractional quantum Hall state exists in this case. This state has exactly the same electrical and thermal Hall responses as the first, but a distinct (fracton) topological order. Moreover, the existence of this second fractional quantum Hall state necessarily implies a gapless phase, which has identical topological response to a noninteracting Weyl semimetal, but is distinct from it. This may be viewed as a generalization (in a weaker form) of the known duality between a noninteracting two-dimensional Dirac fermion and ${\mathrm{QED}}_{3}$ to $3+1$ dimensions. In addition we discuss a $(3+1)$-dimensional topologically ordered state, obtained by gapping a nodal line semimetal without breaking symmetries.

Edge Currents Induced by AC Electric Field in Two-Dimensional Dirac Structures

Edges in two-dimensional structures are the source of nonlinear transport and optical phenomena which are particularly important in small-size flakes. We present a microscopic theory of the edge photogalvanic effect, i.e., the formation of DC electric current flowing along the sample edges in response to AC electric field of the incident terahertz radiation, for two-dimensional Dirac materials including the systems with massive and massless charge carriers. The edge current direction is controlled by the AC field polarization. The spectral dependence of the current is determined by the carrier dispersion and the mechanism of carrier scattering, as shown for single-layer and bilayer graphene as examples.

Two-dimensional Dirac fermions in a mass superlattice

We study two-dimensional (2D) Dirac fermions in the presence of a periodic mass term alternating between positive and negative values along one direction. This scenario could be realized for a graphene monolayer or for the surface states of topological insulators. The low-energy physics is governed by chiral Jackiw-Rebbi modes propagating along zero-mass lines, with the energy dispersion of the Bloch states given by an anisotropic Dirac cone. By means of the transfer matrix approach, we obtain exact results for a piecewise constant mass superlattice. On top of Bloch states, two different classes of boundary and/or interface modes can exist in a finite-size geometry or in a nonuniform electrostatic potential, respectively. We compute the dispersion relation for both types of boundary and interface modes, which originate either from states close to the superlattice Brillouin zone (BZ) center or, via a Lifshitz transition, from states near the BZ boundary. In the presence of a potential step, we predict that the interface modes, the Bloch wave functions, and the electrical conductance will sensitively depend on the step position relative to the mass superlattice.

First-principles study on the lattice thermal conductivity of layered Dirac semimetal BeN4

A novel two-dimensional Dirac semimetal BeN 4 was successfully synthesized in a recent experiment (Bykov et al. Phys. Rev. Lett. 126, 175501 (2021)). Here, we have studied the lattice thermal conductivities of bulk BeN 4 by the first-principles calculations. Due to the anisotropic crystal structure, the lattice thermal conductivities of BeN 4 show large anisotropy and the values at 300 K are 91.5, 42.1, and 5.0 W/m ⋅ K along the armchair, zigzag, and cross-plane directions respectively based on the three-phonon scattering. We also find that four-phonon scattering only reduces the lattice thermal conductivities of BeN 4 by about 9%. Due to the relatively weak Be-N bonds and complex crystal structure, BeN 4 has small phonon group velocities and large phonon anharmonicity, resulting in its much lower lattice thermal conductivity than that of hexagonal boron nitride and graphene. Our work would be helpful for future experimental and theoretical studies on the Dirac semimetal BeN 4 . • The lattice thermal conductivity of Dirac semi-metal BeN 4 was studied by the first-principles calculations. • BeN 4 has large anisotropic thermal conductivity due to its anisotropic crystal structure. • Despite the light elements, BeN 4 has a relatively low lattice thermal conductivity due to its large anharmonicity.

Topological tailoring-induced Dirac cone in ultrathin niobium diboride nanosheets for electrocatalytic sulfur reduction reaction

Two-dimensional topological Dirac materials with novel electronic structures have attracted increasing attention in materials science, where the Weyl fermions can emerge in topological insulating multilayers with unusual quantum transport properties. However, their practical catalytic performance is seldom reported. Here, we design ultrathin niobium diboride (NbB2) nanosheets by utilizing topological electronic states to power catalytic activity. In the lithium-sulfur batteries, the stable topological surface states of NbB2 with high carrier mobility are a recipe for high-activity catalysts to spur the sluggish electrocatalytic sulfur reduction reaction, which originates from the Dirac cone electronic structure. Most remarkably, it delivers stable capacities with 1500 cycles under the 3.6 mA/cm2 (3 C), which is far superior to most electrode catalysts. The proposed functional Weyl and/or Dirac materials would have broad applications in other related fields such as the hydrogen evolution reaction, nitrogen reduction reaction, and energy conversion.

Darboux algorithms for two-dimensional Dirac equations with upper triangular potential matrix

We construct three algorithms of Darboux transformations for the two-dimensional Dirac equation with upper triangular matrix potential. Each algorithm splits in two different variants. The potential is allowed to depend arbitrarily on the two variables, and the Darboux transformations are expressed in terms of generalized Wronskians.

Electron diffusion induced valley Hall effect and nonlinear galvanodiffusive transport in hexagonal two-dimensional Dirac monolayer materials

Diffusion currents are theoretically examined in two-dimensional Dirac materials, such as those of the transition metal dichalcogenides (TMD) family. The transversal effects are analogues of the valley Hall (VHE) and photogalvanic (PGE) transport phenomena in cases when the electron driving force is not an electric field but a gradient of electron density distribution in the sample. The latter can be created by a finite-sized laser spot or by the injection of electrons from other materials. We develop the theory of the diffusive VHE effect assuming the anisotropic electron-short-range-impurity skew scattering. The electron PGE-like transport caused by higher electron-density derivatives is analyzed assuming the trigonal warping anisotropy of electron valleys in a TMD monolayer. The nonlinear responses on an electron-density gradient are studied as well. The isotropic processes of electron scattering off the short-range and Coulomb centers are taken into account in the PGE-like transport theory.

Bulk–edge correspondence for unbounded Dirac–Landau operators

We consider two-dimensional unbounded magnetic Dirac operators, either defined on the whole plane or with infinite mass boundary conditions on a half-plane. Our main results use techniques from elliptic PDEs and integral operators, while their topological consequences are presented as corollaries of some more general identities involving magnetic derivatives of local traces of fast decaying functions of the bulk and edge operators. One of these corollaries leads to the so-called Středa formula: if the bulk operator has an isolated compact spectral island, then the integrated density of states of the corresponding bulk spectral projection varies linearly with the magnetic field as long as the gaps between the spectral island and the rest of the spectrum are not closed, and the slope of this variation is given by the Chern character of the projection. The same bulk Chern character is related to the number of edge states that appear in the gaps of the bulk operator.

Two-dimensional Dirac operators with general δ-shell interactions supported on a straight line

In this paper the two-dimensional Dirac operator with a general hermitian δ-shell interaction supported on a straight line is introduced as a self-adjoint operator and its spectral properties are investigated in detail. In particular, it is demonstrated that the singularly continuous spectrum is always empty and that by switching a certain δ-shell interaction on, it is possible to generate an eigenvalue in the gap of the spectrum of the free operator or to partially or even fully close the gap. This suggests that the studied operators may serve as interesting continuum toy-models for Dirac materials. Finally, approximations by Dirac operators with regular potentials are presented.

Gap opening mechanism for correlated Dirac electrons in organic compounds α−(BEDT−TTF)2I3 and α−(BEDT−TSeF)2I3

To determine how electron correlations open a gap in two-dimensional massless Dirac electrons in the organic compounds $\alpha$-(BEDT-TTF)$_2$I$_3$ [$\alpha$-(ET)$_2$I$_3$] and $\alpha$-(BEDT-TSeF)$_2$I$_3$ [$\alpha$-(BETS)$_2$I$_3$], we derive and analyze $ab$ $initio$ low-energy effective Hamiltonians for these two compounds. We find that the horizontal stripe charge ordering opens a gap in the massless Dirac electrons in $\alpha$-(ET)$_2$I$_3$, while an insulating phase without explicit symmetry breaking appears in $\alpha$-(BETS)$_2$I$_3$. We clarify that the combination of the anisotropic transfer integrals and the electron correlations induces a dimensional reduction in the spin correlations, i.e., one-dimensional spin correlations develop in $\alpha$-(BETS)$_2$I$_3$. We show that the one-dimensional spin correlations open a gap in the massless Dirac electrons. Our finding paves the way for opening gaps for massless Dirac electrons using strong electronic correlations.

Reversal of the chiral anomaly bulk states with periodically staggered potential

A chiral anomaly Landau level emerges when a Weyl semimetal is subjected to an external magnetic field. Recently, it was demonstrated that similar chiral anomaly bulk states exist within two-dimensional Dirac semimetals if proper boundary conditions are applied. The resulting chiral bulk states disperse linearly with the slope determined by a combination of the boundary condition and chirality of the Dirac cone, i.e., whether it is at $K$ or $K$\ensuremath{'} point. In this paper, we show that the slope (the sign of group velocity) near the $K$ and $K$\ensuremath{'} points can be reversed under a periodically staggered potential. We first analyze the parameter dependence of this phenomenon with a nearest-neighbor tight-binding model. Then, we give a semianalytical solution for the dispersion of the chiral anomaly bulk states. In the end, we provide a photonic crystal system and prove with full-wave simulations that such a phenomenon can indeed be observed.

Coherent Interlayer Coupling in Quasi-Two-Dimensional Dirac Fermions in α-(BEDT-TTF)2I3

Theoretical and experimental studies have supported that the electronic structure of $\alpha$-(BEDT-TTF)$_2$I$_3$ under pressure is described by two-dimensional Dirac fermions. When the interlayer tunneling is coherent, the electronic structure of the system becomes three-dimensional, and we expect the peak structure to appear in the interlayer resistivity under magnetic fields. We theoretically and experimentally show that the peak appears in the interlayer resistivity at low temperatures and high magnetic fields. From the experiment, we estimate that the magnitude of the interlayer tunneling is $t_1 \sim 1$ meV. Our result opens the door to investigating the three-dimensional electronic structure of $\alpha$-(BEDT-TTF)$_2$I$_3$.