Massive Dirac Fermions Research Articles

Bound states of (2+1)−dimensional massive Dirac fermions in a Lorentzian-shaped inhomogeneous perpendicular magnetic field

Abstract In this paper, we investigate massive Dirac fermions in ( 2 + 1 ) − spacetime confined by a Lorentzian-shaped inhomogeneous magnetic field and construct solutions of their bound states by both analytical and numerical methods. First, we have found a set of specific magnetic fields which allow obtaining conditionally exact solutions as well as quasi-exact ones. Second, we construct these analytical solutions explicitly and analyse the effect of Dirac fermion’s effective mass. We also calculate entire energy spectra for the considered system numerically using Feranchuk–Komarov operator method and combine them with the analytical solutions for investigating Landau quantization in the energy spectrum.

Electron Optics in Phosphorene pn Junctions: Negative Reflection and Anti-Super-Klein Tunneling.

Ballistic electrons in phosphorene pn junctions show optical-like phenomena. Phosphorene is modeled by a tight-binding Hamiltonian that describes its electronic structure at low energies, where the electrons behave in the armchair direction as massive Dirac Fermions and in the orthogonal zigzag direction as Schrödinger electrons. Applying the continuum approximation, we derive the electron optics laws in phosphorene pn junctions, which show very particular and unusual properties. Because of the anisotropy of the electronic structure, these laws depend strongly on the orientation of the junction with respect to the sublattice. Negative and anomalous reflection are observed for tilted junctions, whereas the typical specular reflection is found only if the junction is parallel to the zigzag or armchair edges. Moreover, omni-directional total reflection, called anti-super-Klein tunneling, is observed if the junction is parallel to the armchair edge. Applying the nonequilibrium Green's function method on the tight-binding model, we calculate numerically the current flow. The good agreement of both approaches confirms the atypical transport properties, which can be used in nanodevices to collimate and filter the electron flow or to switch its direction.

Charged excitons in two-dimensional transition metal dichalcogenides: Semiclassical calculation of Berry curvature effects

We theoretically study the role of the Berry curvature on neutral and charged excitons in two-dimensional transition-metal dichalcogenides. The Berry curvature arises due to a strong coupling between the conduction and valence bands in these materials that can to great extent be described within the model of massive Dirac fermions. The Berry curvature lifts the degeneracy of exciton states with opposite angular momentum. Using an electronic interaction that accounts for non-local screening effects, we find a Berry-curvature induced splitting of $\sim 17$ meV between the 2$p_{-}$ and 2$p_{+}$ exciton states in WS$_2$, consistent with experimental findings. Furthermore, we calculate the trion binding energies in WS$_2$ and WSe$_2$ for a large variety of screening lenghts and different dielectric constants for the environment. Our approach indicates the prominent role played by the Berry curvature along with non-local electronic interactions in the understanding of the energy spectra of neutral and charged excitons in transition-metal dichalcogenides and in the the interpretation of their optical properties.

Light-induced anomalous Hall effect in massless Dirac fermion systems and topological insulators with dissipation

Employing the quantum Liouville equation with phenomenological dissipation, we investigate the transport properties of massless and massive Dirac fermion systems that mimics graphene and topological insulators, respectively. The massless Dirac fermion system does not show an intrinsic Hall effect, but it shows a Hall current under the presence of circularly-polarized laser fields as a nature of a optically-driven nonequilibrium state. Based on the microscopic analysis, we find that the light-induced Hall effect mainly originates from the imbalance of photocarrier distribution in momentum space although the emergent Floquet–Berry curvature also has a non-zero contribution. We further compute the Hall transport property of the massive Dirac fermion system with an intrinsic Hall effect in order to investigate the interplay of the intrinsic topological contribution and the extrinsic light-induced population contribution. As a result, we find that the contribution from the photocarrier population imbalance becomes significant in the strong field regime and it overcomes the intrinsic contribution. This finding clearly demonstrates that intrinsic transport properties of materials can be overwritten by external driving and may open a way to ultrafast optical-control of transport properties of materials.

Engineering Zero-Dimensional Quantum Confinement in Transition-Metal Dichalcogenide Heterostructures.

Achieving robust, localized quantum states in two-dimensional (2D) materials like graphene is desirable for optoelectronics and quantum information yet challenging due to the difficulties in confining Dirac fermions. Traditional colloidal nanoparticle and epitaxially grown quantum dots are also impractical for solid-state devices, due to either complex surface chemistry, unreliable spatial positioning, or lack of electrical and optical access. In this work, we design and optimize nanoscale monolayer transition-metal dichalcogenide (TMD) heterostructures to natively host massive Dirac fermion bound states. We develop an integrated multiscale approach to translate first-principles electronic structure to higher length scales, where we apply a continuum model to consider arbitrary 2D quantum dot geometries and sizes. Focusing on a model system of an MoS2 quantum dot in a WS2 matrix (MoS2/WS2), we find discrete bound states in triangular dots with side lengths up to 20 nm. We propose figures of merit that, when optimized for, result in heterostructure configurations engineered for maximally isolated bound states at room temperature. These design principles apply to the entire family of semiconducting TMD materials, and we predict 6.5 nm MoS2/WS2 (quantum dot/matrix) triangular dots and 4.5 nm MoSe2/WSe2 triangular dots as ideal systems for confining massive Dirac fermions.

Quantum Interference Theory of Magnetoresistance in Dirac Materials.

Magnetoresistance in many samples of Dirac semimetals and topological insulators displays nonmonotonic behavior over a wide range of magnetic fields. Here a formula of magnetoconductivity is presented for massless and massive Dirac fermions in Dirac materials due to quantum interference of Dirac fermions in scalar impurity scattering potentials. It reveals a striking crossover from positive to negative magnetoresistivity, uncovering strong competition between weak localization and weak antilocalization in multiple Cooperon channels at different chemical potentials, effective masses, and finite temperatures. This work sheds light on the important role of strong coupling of the conduction and valence bands in the quantum interference transport in topological nontrivial and trivial Dirac materials.

Casimir force variability in one-dimensional QED systems

The Casimir force between two short-range charge sources, embedded in a background of one dimensional massive Dirac fermions, is explored by means of the original $\ln\text{[Wronskian]}$ contour integration techniques. For identical sources with the same (positive) charge, we find that in the non-perturbative region the Casimir interaction between them can reach sufficiently large negative values and simultaneously reveal the features of a long-range force in spite of nonzero fermion mass, that could significantly influence the properties of such quasi-one-dimensional QED systems. For large distances $s$ between sources we recover that their mutual interaction is governed first of all by the structure of the discrete spectrum of a single source, in dependence on which it can be tuned to give an attractive, a repulsive, or an (almost) compensated Casimir force with various rates of the exponential fall-down, quite different from the standard $\exp (-2 m s)$ law. By means of the same $\ln\text{[Wronskian]}$ techniques, the case of two $\delta$-sources is also considered in a self-consistent manner with similar results for the variability of the Casimir force. A quite different behavior of the Casimir force is found for the antisymmetric source-anti-source system. In particular, in this case, there is no possibility for a long-range interaction between sources. The asymptotics of the Casimir force follows the standard $\exp (-2 m s)$ law. Moreover, for small separations between sources, the Casimir force for symmetric and antisymmetric cases turns out to be of opposite sign.

Topological massive Dirac fermions inβ-tungsten

Ultrathin films of $\ensuremath{\beta}$-tungsten $(\ensuremath{\beta}\text{\ensuremath{-}}\mathrm{W})$ is attracting extensive attentions because of its promising application in spintronics due to giant spin Hall effect and its fundamental interest in superconductivity. By means of first-principles calculations, we have elucidated its nontrivial topological nature with multiple Dirac nodal lines in the $\ensuremath{\beta}\text{\ensuremath{-}}\mathrm{W}$ metal when spin-orbit coupling (SOC) is ignored. The analysis of electronic structures reveal that its topology is associated with the band inversion between the ${d}_{xz}+{d}_{yz}$ and ${d}_{{z}^{2}}$ orbitals at the $M$ point in its Brillouin zone. By switching on the SOC, these Dirac nodal lines become gapped, as accompanied with the occurrence of 24 massive Dirac fermions at the energy of 87 meV below the Fermi level. Different from the exact gapless Dirac fermions with the linear band crossings, we have identified that these massive Dirac fermions all open a tiny gap of 7 meV. In addition, we have observed the topologically protected nontrivial surface arc states connecting these massive Dirac fermions. The existence of these massive Dirac fermions in the $\ensuremath{\beta}\text{\ensuremath{-}}\mathrm{W}$ phase would be important to understand its giant spin Hall effect according to a recent study [Nature 555, 638 (2018)] where massive Dirac fermions were identified to generate Berry-curvature-induced Hall conductivity.

Electronic structure, magnetoexcitons and valley polarized electron gas in 2D crystals

We describe here recent work on the electronic properties, magnetoexcitons and valley polarised electron gas in 2D crystals. Among 2D crystals, monolayer MoS2 has attracted significant attention as a direct-gap 2D semiconductor analogue of graphene. The crystal structure of monolayer MoS2 breaks inversion symmetry and results in K valley selection rules allowing to address individual valleys optically. Additionally, the band nesting near Q points is responsible for enhancing the optical response of MoS2.We show that at low energies the electronic structure of MoS2 is well approximated by the massive Dirac Fermion model. We focus on the effect of magnetic field on optical properties of MoS2. We discuss the Landau level structure of massive Dirac fermions in the two non-equivalent valleys and resulting valley Zeeman splitting. The effects of electron-electron interaction on the valley Zeeman splitting and on the magneto-exciton spectrum are described. We show the changes in the absorption spectrum as the self-energy, electron-hole exchange and correlation effects are included. Finally, we describe the valley-polarised electron gas in WS2 and its optical signature in finite magnetic fields.

Casimir interactions between two short-range Coulomb sources

We calculate the Casimir interaction between two short range extended charge sources, embedded in a background of one dimensional massive Dirac fermions. We explore both the induced polarization density ρvac(x) and the corresponding integral charge Qvac, as well as the Casimir energy Evac and its contribution to the interaction between sources. For sources with the same charge, by means of the novel ln[Wronskian] contour integration techniques we find that the interaction energy between two sources can exceed sufficiently large negative values and simultaneously reveal the features of a long-range force in spite of non-zero fermion mass, which could significantly influence the properties of such quasi-one-dimensional systems. For large distances between sources we recover that their mutual interaction is governed first of all by the structure of the discrete spectrum of a single source, in dependence on which it can be tuned to give an attractive, a repulsive, or an (almost) compensated Casimir force with various rates of the exponential fall-down, quite different from the standard exp(−2ms) law.

Mirror anomaly and anomalous Hall effect in type-I Dirac semimetals

In addition to the well known chiral anomaly, Dirac semimetals have been argued to exhibit mirror anomaly, close analogue to the parity anomaly of ($2+1$)-dimensional massive Dirac fermions. The observable response of such anomaly is manifested in a singular step-like anomalous Hall response across the mirror-symmetric plane in the presence of a magnetic field. Although this result seems to be valid in type-II Dirac semimetals (strictly speaking, in the linearized theory), we find that type-I Dirac semimetals do not possess such an anomaly in anomalous Hall response even at the level of the linearized theory. In particular, we show that the anomalous Hall response continuously approaches zero as one approaches the mirror symmetric angle in a type-I Dirac semimetal as opposed to the singular Hall response in a type-II Dirac semimetal. Moreover, we show that, under certain condition, the anomalous Hall response may vanish in a linearized type-I Dirac semimetal, even in the presence of time reversal symmetry breaking.

Dirac parameters and topological phase diagram of Pb1−xSnxSe from magnetospectroscopy

Pb1-xSnxSe hosts 3D massive Dirac fermions across the entire composition range for which the crystal structure is cubic. In this work, we present a comprehensive experimental mapping of the 3D band structure parameters of Pb1-xSnxSe as a function of composition and temperature. We cover a parameter space spanning the band inversion that yields its topological crystalline insulator phase. A non-closure of the energy gap is evidenced in the vicinity of this phase transition. Using magnetooptical Landau level spectroscopy, we determine the energy gap, Dirac velocity, anisotropy factor and topological character of Pb1-xSnxSe epilayers grown by molecular beam epitaxy on BaF2 (111). Our results are evidence that Pb1-xSnxSe is a model system to study topological phases and the nature of the phase transition.

Intraband divergences in third order optical response of 2D systems

The existence of large nonlinear optical coefficients is one of the preconditions for using nonlinear optical materials in nonlinear optical devices. For a crystal, such large coefficients can be achieved by matching photon energies with resonant energies between different bands, and so the details of the crystal band structure play an important role. Here we demonstrate that large third-order nonlinearities can also be generally obtained by a different strategy. As any of the incident frequencies or the sum of any two or three frequencies approaches zero, the doped or excited populations of electronic states lead to divergent contributions in the induced current density. We refer to these as intraband divergences, by analogy with the behavior of Drude conductivity in linear response. Physically, such resonant processes can be associated with a combination of intraband and interband optical transitions. Current-induced second order nonlinearity, coherent current injection, and jerk currents are all related to such divergences, and we find similar divergences in degenerate four wave mixing and cross-phase modulation under certain conditions. These divergences are limited by intraband relaxation parameters and lead to a large optical response from a high quality sample; we find that they are very robust with respect to variations in the details of the band structure. To clearly track all of these effects, we analyze gapped graphene, describing the electrons as massive Dirac fermions; under the relaxation time approximation, we derive analytic expressions for the third order conductivities and identify the divergences that arise in describing the associated nonlinear phenomena.

Magnetoresistance of a three-dimensional Dirac gas

We study the transversal magnetoconductivity and magnetoresistance of a massive Dirac fermion gas. This can be used as a simple model for gapped Dirac materials. In the zero-mass limit, the case of gapless Dirac semimetals is also studied. In the case of Weyl semimetals, to reproduce the nonsaturating linear magnetoresistance seen in experiments, the use of screened charged impurities is inevitable. In this paper, these are included using the first Born approximation for the self-energy. The screening wave number is calculated using the random phase approximation with the polarization function taking into account the electron-electron interaction. The Hall conductivity is calculated analytically in the case of no impurities and is shown to be perfectly inversely proportional to the magnetic field. Thus the magnetic field dependence of the magnetoresistance is mainly determined by $\sigma_{xx}$. We show that in the extreme quantum limit at very high magnetic fields the gapped Dirac materials are expected to have $\sigma_{xx}\propto B^{-3}$ leading to $\varrho_{xx}\propto B^{-1}$, in contrast with the gapless case where $\sigma_{xx}\propto B^{-1}$ and $\varrho_{xx}\propto B$. At lower fields, we find that the effect of the mass term is negligible and in the region of the Shubnikov-de Haas oscillations the two systems behave almost identically. We suggest a phenomenological scattering rate that is able to reproduce the linear behavior at the oscillating region. We show that in the case of the scattering rate calculated using the Born approximation, the strength of the relative permittivity and the density of impurities affects the magnetic field dependence of the conductivity significantly.

Topological Insulator-Based van der Waals Heterostructures for Effective Control of Massless and Massive Dirac Fermions.

Three dimensional (3D) topological insulators (TIs) are an important class of materials with applications in electronics, spintronics and quantum computing. With the recent development of truly bulk insulating 3D TIs, it has become possible to realize surface dominated phenomena in electrical transport measurements e.g. the quantum Hall (QH) effect of massless Dirac fermions in topological surface states (TSS). However, to realize more advanced devices and phenomena, there is a need for a platform to tune the TSS or modify them e.g. gap them by proximity with magnetic insulators, in a clean manner. Here we introduce van der Waals (vdW) heterostructures in the form of topological insulator/insulator/graphite to effectively control chemical potential of the TSS. Two types of gate dielectrics, normal insulator hexagonal boron nitride (hBN) and ferromagnetic insulator Cr2Ge2Te6 (CGT) are utilized to tune charge density of TSS in the quaternary TI BiSbTeSe2. hBN/graphite gating in the QH regime shows improved quantization of TSS by suppression of magnetoconductivity of massless Dirac fermions. CGT/graphite gating of massive Dirac fermions in the QH regime yields half-quantized Hall conductance steps and a measure of the Dirac gap. Our work shows the promise of the vdW platform in creating advanced high-quality TI-based devices.

Shubnikov-de Haas oscillations of massive Dirac fermions in a Dirac antiferromagnet SrMnSb2

We investigate the physical properties of massive Dirac fermions in SrMnSb2 using transport, specific heat, electronic structure calculations, and Shubnikov-de Haas (SdH) oscillations. SrMnSb2 is a candidate Dirac antiferromagnet, consisting of the MnSb layers and the distorted square net of Sb atoms with a zigzag chain structure. This structural distortion leads to gap opening at the band crossing point found in the square lattice of the sister compound SrMnBi2 but leaves another Dirac band crossing near the Brillouin zone boundary. The small 2D Fermi surface with a light electron mass and a small Fermi energy is confirmed by the large resistivity anisotropy, the large Seebeck coefficient, and also the angle and temperature dependent SdH oscillations. The Berry phase obtained from the SdH oscillations is trivial, in contrast to the case of SrMnBi2. The relatively large spin orbit coupling gap and the small Fermi energy in SrMnSb2 is found to be essential to understand this contrasting behavior of the massive Dirac fermions as compared to SrMnBi2. Our observations demonstrate that the Berry phase of the mobile electrons in SrMnSb2 is sensitive to the Fermi level change and can be tuned by doping or deficiency.

Massive Dirac fermions in one-dimensional inhomogeneous nanorings

Within the framework of a one-dimensional model, we calculate electronic spectra and persistent currents of imperfect (containing one or several defects) nanoscale Aharonov-Bohm rings made of Dirac materials like gapped graphene, silicene or germanene. The calculations show that at zero electric field, in ideal structures, such as the defect-free ring or the ring with a defect superlattice, the electron spectrum consists of touching each other magnetic bands or minibands, respectively, with intersecting energy levels at the band (or miniband) edges. Introduction of some irregularity in the system, e.g., a single defect or a defect sequence with arbitrary barrier heights, turns the crossings into anti-crossings at the center and the boundary of the magnetic flux Brillouin zone. In silicene (germanene) rings in axially-directed electric field, extra crossings arise at some internal points of the Brillouin zone due to breaking the spin-valley symmetry. Such crossings do not disappear even in the presence of irregularity. All crossings in the ring spectrum cause the discontinuities in the persistent current depending on the band filling. Replacement of crossings by anti-crossings due to the defect existence in the ring leads to replacement of the current jumps by a smooth dependence on the magnetic flux at low degree of filling. As the Fermi energy rises, the smooth function gradually turns into the discontinuous one completely coinciding with that of the defect-free ring, i.e., the current becomes independent of the presence of inhomogeneity in the ring in the limiting case of high filling.

Anisotropic conductivity in 2D massive Dirac Fermions: an effect of time reversal symmetry breaking in the surface states of a topological insulator

We calculate the conductivity tensor for massive Dirac Fermions within the semiclassical Boltzmann approach. We consider the effect of two different types of scattering mechanism, namely scalar and magnetic, that act incoherently and use the symmetries of the transition rate to exactly solve the Boltzmann equation. We prove that the conductivity can be anisotropic depending on the strength of the magnetic scatterers in each direction. In the particular situation of magnetic impurities polarised in the x-direction, the conductivity is three times larger in y-direction as compared with the conductivity in the x-direction, for white noise scattering correlation function. We compare the approach we apply with the most commonly used way of dealing with more than one source of scattering, namely with Matthiessen’s rule, and show that the approach we applied is more general and suitable for anisotropic scattering.

Boson–fermion duality in a gravitational background

We study the 2+1 dimensional boson–fermion duality in the presence of background curvature and electromagnetic fields. The main players are, on the one hand, a massive complex |ϕ|4 scalar field coupled to a U(1) Maxwell–Chern–Simons gauge field at level 1, representing a relativistic composite boson with one unit of attached flux, and on the other hand, a massive Dirac fermion. We show that, in a curved background and at the level of the partition function, the relativistic composite boson, in the infinite coupling limit, is dual to a short-range interacting Dirac fermion. The coupling to the gravitational spin connection arises naturally from the spin factors of the Wilson loop in the Chern–Simons theory. A non-minimal coupling to the scalar curvature is included on the bosonic side in order to obtain agreement between partition functions. Although an explicit Lagrangian expression for the fermionic interactions is not obtained, their short-range nature constrains them to be irrelevant, which protects the duality in its strong interpretation as an exact mapping at the IR fixed point between a Wilson–Fisher–Chern–Simons complex scalar and a free Dirac fermion. We also show that, even away from the IR, keeping the |ϕ|4 term is of key importance as it provides the short-range bosonic interactions necessary to prevent intersections of worldlines in the path integral, thus forbidding unknotting of knots and ensuring preservation of the worldline topologies.

Shubnikov–de Haas oscillations in the anomalous Hall conductivity of Chern insulators

The Haldane model on a honeycomb lattice is a paradigmatic example of a system featuring quantized Hall conductivity in the absence of an external magnetic field, that is, a quantum anomalous Hall effect. Recent theoretical work predicted that the anomalous Hall conductivity of massive Dirac fermions can display Shubnikov-de Haas (SdH) oscillations, which could be observed in topological insulators and honeycomb layers with strong spin--orbit coupling. Here, we investigate the electronic transport properties of Chern insulators subject to high magnetic fields by means of accurate spectral expansions of lattice Green's functions. We find that the anomalous component of the Hall conductivity displays visible SdH oscillations at low temperature. \textcolor{black}{The effect is shown to result from the modulation of the next-nearest neighbour flux accumulation due to the Haldane term,} which removes the electron--hole symmetry from the Landau spectrum. To support our numerical findings, we derive a long-wavelength description beyond the linear ('Dirac cone') approximation. Finally, we discuss the dependence of the energy spectra shift for reversed magnetic fields with the topological gap and the lattice bandwidth.