Dirac Cone Dispersion Research Articles

Dirac Cones and Room Temperature Polariton Lasing Evidenced in an Organic Honeycomb Lattice.

Artificial 1D and 2D lattices have emerged as a powerful platform for the emulation of lattice Hamiltonians, the fundamental study of collective many-body effects, and phenomena arising from non-trivial topology. Exciton-polaritons, bosonic part-light and part-matter quasiparticles, combine pronounced nonlinearities with the possibility of on-chip implementation. In this context, organic semiconductors embedded in microcavities have proven to be versatile candidates to study nonlinear many-body physics and bosonic condensation, and in contrast to most inorganic systems, they allow the use at ambient conditions since they host ultra-stable Frenkel excitons. A well-controlled, high-quality optical lattice is implemented that accommodates light-matter quasiparticles. The realized polariton graphene presents with excellent cavity quality factors, showing distinct signatures of Dirac cone and flatband dispersions as well as polariton lasing at room temperature. This is realized by filling coupled dielectric microcavities with the fluorescent protein mCherry. The emergence of a coherent polariton condensate at ambient conditions are demonstrated, taking advantage of coupling conditions as precise and controllable as in state-of-the-art inorganic semiconductor-based systems, without the limitations of e.g. lattice matching in epitaxial growth. This progress allows straightforward extension to more complex systems, such as the study of topological phenomena in 2D lattices including topological lasers and non-Hermitian optics.

Formation of well-ordered surfaces of Bi2-xSbxTe3-ySey topological insulators using wet chemical treatment

The surface preparation of topological insulators (TIs) is a critical task in order to realize their efficient applications. A chemical treatment in an anhydrous solution of hydrogen chloride in isopropanol (HCl-iPA) and a subsequent annealing at relatively low temperature in ultrahigh vacuum (UHV) was successfully used for the surface preparation of bulk 3D (0001) TIs Bi2Te3, Sb2Te3, Bi2Se3 and an MBE - grown Bi2-xSbxTe3-ySey (BSTS) thin films. The surface treatment showed a significant modification of the initial TI surfaces, which was free from the structural disorder, oxidation and chemical impurities determined by X-ray photoelectron spectroscopy and low-energy electron diffraction. The insulating nontrivial bulk gap and well resolved gapless surface states with a linear dispersion of a massless Dirac cone were observed by angle-resolved photoelectron spectroscopy (ARPES). In the BSTS film, the Fermi level is located within the bulk band gap. The negative magnetoconductance corresponding to weak antilocalization demonstrated the contribution of the surface states of BSTS, that promise to be protected from backscattering. The surface treatment method proposed in this work is highly efficient for both bulk and thin TI films that can be useful for the deposition of an insulators/metals to fabricate transistor and spin valve systems.

Inverse design of polymorphic Dirac-like cone dispersion relationship in photonic crystals

Dirac-like cone linear dispersion relations in photonic crystals (PhCs) often endow them with unique properties, yet searching for such relations can be challenging. We introduce a generalized inverse design system that, given the dielectric constants and lattice of two-dimensional PhCs, can efficiently determine its structural parameters to obtain its Dirac-like cone dispersion. Employing this inverse design strategy, we design three types of Dirac cone PhCs, including triple degenerate, quadruple degenerate, and triple degenerate under dual polarization with the same frequency. Further investigations reveal a systematic relationship between the radius of the dielectric rods in these PhCs and their corresponding Dirac frequencies across varying dielectric constants. The zero refractive index characteristic is validated in two of the three PhCs studied, as confirmed through numerical simulations. Additionally, by leveraging our proposed inverse design method, we introduce an innovative shell dielectric rod model, which encapsulates a dielectric material, achieving a quadruple degenerate dispersion structure with dual Dirac cones. This research provides a potent tool for the inverse design of PhCs and expands its application in Dirac cone dispersion design.

Half‐Integer Topological Charge Polarization of Quasi‐Dirac Bound States in the Continuum

AbstractThe non‐trivial polarization topology of bound states in the continuum (BICs) provides new strategies in nanophotonics. The polarization topology depends on the geometric parameters and energy‐momentum dispersion of the system and can be engineered to add specific functionalities for light molding. Herein, such a possibility is investigated by studying the topology of the polarization states associated with the optical field radiated by BICs when Dirac‐cone‐degeneracy is lifted. The opening of a pseudogap in the Dirac cone dispersion of square‐lattice dielectric photonic crystal slabs is achieved by tuning the slab thickness. First, the emergence of half‐integer topological charges without the requirement of BIC annihilation is theoretically shown, which instead occurs when in‐plane inversion symmetry is broken. Then, using spin‐to‐orbital angular momentum conversion, the theory of half‐integer topological charges mediated by BICs is demonstrated and experimentally proved. The same device is able to give rise to vortices with different orbital angular momentum depending on the way it is illuminated, thus improving the potential of optical multiplexing. In addition, the additive character of the topology‐induced phase‐vortex generation is finally demonstrated for both integer and half‐integer charges using also vortex states as input beams, which is of relevance for information delivery.

Emergence of Dirac cone in CsPbBr3 perovskite-based two-dimensional photonic crystals

In this paper, it has been proposed that all-inorganic perovskite materials can be used to design and fabricate two-dimensional (2D) photonic crystals. The theoretical study performed here shows that these perovskite-based 2D photonic crystals exhibit a Dirac cone dispersion. In this study, cesium lead bromide (CsPbBr3) having rectangular shapes of varying widths (50, 100, and 150 nm) and a fixed height of 300 nm are arranged in a regular manner with a lattice constant of 500 nm. The pattern of these photonic crystals exhibits C4v symmetry. It has been observed when these perovskite materials, arranged in a rectangular shape, are having a certain dimension; they display a Dirac cone dispersion at the centre of the Brillouin zone at a frequency close to peta-hertz. Further, these photonic crystals are used to manipulate the propagation of electromagnetic (EM) waves of frequency ∼4 × 1014 Hz. Near this frequency, the photonic crystals respond as a nearly zero refractive index material, which allows passing EM waves even in presence of an obstacle. This study can be expanded to design and fabricate two-dimensional photonic crystals based on various perovskite materials.

Interaction between interface and massive states in multivalley topological heterostructures

Topological interface states (TISs) in multivalley systems are studied to unravel their valley sensitivity. For this purpose, multivalley IV--VI topological crystalline insulator (TCI) heterostructures are explored using magnetooptical Landau level spectroscopy up to 34 teslas. We characterize the TISs emerging from the distinct $L$ valleys in ${\mathrm{Pb}}_{1\ensuremath{-}x}{\mathrm{Sn}}_{x}\mathrm{Se}$ multiquantum wells grown along the [111] direction. It is shown that the shape of the two-dimensional (2D) Fermi surfaces of TISs residing at the TCI-trivial insulator interfaces are strongly affected by the valley anisotropy of topologically trivial ${\mathrm{Pb}}_{1\ensuremath{-}y}{\mathrm{Eu}}_{y}\mathrm{Se}$ barriers. This phenomenon is shown to be due to the deep penetration of the TISs into the barriers. For the valleys tilted with respect to the confinement direction, a significant interaction between topological states and the conventional massive quantum well states is observed, evidenced by the resulting large anticrossings between Landau levels. These are theoretically well described by a $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ model that considers tilt and anisotropy of the valleys in 2D. Therefore, in this paper, we provide a precise characterization of the TIS valley splitting as well as an accurate determination of the anisotropy of their Dirac cone dispersion.

Acoustic Tunneling Study for Hexachiral Phononic Crystals Based on Dirac-Cone Dispersion Properties

Acoustic tunneling is an essential property for phononic crystals in a Dirac-cone state. By analyzing the linear dispersion relations for the accidental degeneracy of Bloch eigenstates, the influence of geometric parameters on opening the Dirac-cone state and the directional band gaps’ widths are investigated. For two-dimensional hexachiral phononic crystals, for example, the four-fold accidental degenerate Dirac point emerges at the center of the irreducible Brillouin zone (IBZ). The Dirac cone properties and the band structure inversion problem are discussed. Finally, to verify acoustic transmission properties near the double-Dirac-cone frequency region, the numerical calculation of the finite-width phononic crystal structure is carried out, and the acoustic transmission tunneling effect is proved. The results enrich and expand the manipulating method in the topological insulator problem for hexachiral phononic crystals.

Kagome Flatbands for Coherent Exciton-Polariton Lasing

Kagome lattices supporting Dirac cone and flatband dispersions are well known as a highly frustrated, two-dimensional lattice system. Particularly the flatbands therein are attracting continuous interest based on their link to topological order, correlations and frustration. In this work, we realize coupled microcavity implementations of Kagome lattices hosting exciton-polariton quantum fluids of light. We demonstrate precise control over the dispersiveness of the flatband as well as selective condensation of exciton-polaritons into the flatband. Subsequently, we focus on the spatial and temporal coherence properties of the laser-like emission from these polariton condensates that are closely connected to the flatband nature of the system. Notably, we find a drastic increase in coherence time due to the localization of flatband condensates. Our work illustrates the outstanding suitability of the exciton-polariton system for detailed studies of flatband states as a platform for microlaser arrays in compact localized states, including strong interactions, topology and non-linearity.

Quantum electrodynamics in anisotropic and tilted Dirac photonic lattices

One of the most striking predictions of quantum electrodynamics is that vacuum fluctuations of the electromagnetic field can lead to spontaneous emission of atoms as well as photon-mediated interactions among them. Since these processes strongly depend on the nature of the photonic bath, a current burgeoning field is the study of their modification in the presence of photons with non-trivial energy dispersions, e.g. the ones confined in photonic crystals. A remarkable example is the case of isotropic Dirac-photons, which has been recently shown to lead to non-exponential spontaneous emission as well as dissipation-less long-range emitter interactions. In this work, we show how to further tune these processes by considering anisotropic Dirac cone dispersions, which include tilted, semi-Dirac, and the recently discovered type II and III Dirac points. In particular, we show how by changing the anisotropy of the lattice one can change both the spatial shape of the interactions as well as its coherent/incoherent nature. Finally, we theoretically analyze a possible implementation based on subwavelength atomic arrays where these energy dispersions can be engineered and interfaced with quantum emitters.

Transmission and reflection cloaking by using a zero-refractive-index photonic crystal in the microwave region.

We propose a rectangular column two-dimensional square lattice photonic crystal to realize zero refractive index. Through analysis of the energy band structure of the photonic crystal structure, the lattice constant and side length of the rectangular columns can be optimized, and the Dirac cone dispersion appears at the center of the Brillouin zone. The Dirac cone is formed by the interaction of a monopolar eigenstate and a dipolar eigenstate to form a triple accidental degenerate state. The effective medium theory is used to invert the effective electromagnetic parameters of the photonic crystal with a double zero refractive index. The zero-phase change and the focusing characteristic of the concave lens of this kind of zero-refractive-index material are verified. Importantly, we have achieved transmission and reflection cloaking with this zero-index medium. Through the analysis of the amplitude and phase distribution characteristics of the electromagnetic field, it is proved that the designed cloaking devices have obvious cloaking effect.

Band dynamics accompanied by bound states in the continuum at the third-order Γ point in leaky-mode photonic lattices

Bound states in the continuum (BICs) and Fano resonances in planar photonic lattices, including metasurfaces and photonic-crystal slabs, have been studied extensively in recent years. Typically, the BICs and Fano resonances are associated with the second stop bands open at the second-order Γ point. This paper addresses the fundamental properties of the fourth stop band accompanied by BICs at the third-order Γ point in one-dimensional leaky-mode photonic lattices. At the fourth stop band, one band edge mode suffers radiation loss, thereby generating a Fano resonance, while the other band edge mode becomes a nonleaky BIC. The fourth stop band is controlled primarily by the Bragg processes associated with the first, second, and fourth Fourier harmonic components of the periodic dielectric constant modulation. The interplay between these three major processes closes the fourth band gap and induces a band flip whereby the leaky and BIC edges transit across the fourth band gap. At the fourth stop band, an accidental BIC is formed owing to the destructive interplay between the first and second Fourier harmonics. When the fourth band gap closes with strongly enhanced radiative Q factors, Dirac cone dispersions can appear at the third-order Γ point.

Orbital Complexity in Intrinsic Magnetic Topological Insulators MnBi_{4}Te_{7} and MnBi_{6}Te_{10}.

Using angle-resolved photoelectron spectroscopy (ARPES), we investigate the surface electronic structure of the magnetic van der Waals compounds MnBi_{4}Te_{7} and MnBi_{6}Te_{10}, the n=1 and 2 members of a modular (Bi_{2}Te_{3})_{n}(MnBi_{2}Te_{4}) series, which have attracted recent interest as intrinsic magnetic topological insulators. Combining circular dichroic, spin-resolved and photon-energy-dependent ARPES measurements with calculations based on density functional theory, we unveil complex momentum-dependent orbital and spin textures in the surface electronic structure and disentangle topological from trivial surface bands. We find that the Dirac-cone dispersion of the topologial surface state is strongly perturbed by hybridization with valence-band states for Bi_{2}Te_{3}-terminated surfaces but remains preserved for MnBi_{2}Te_{4}-terminated surfaces. Our results firmly establish the topologically nontrivial nature of these magnetic van der Waals materials and indicate that the possibility of realizing a quantized anomalous Hall conductivity depends on surface termination.

Phononic crystals as a platform for experimental physics: The direct observation of Klein tunneling

Phononic crystals consist of periodic Mie scatterers that modulate the band structure through multi-scattering to control acoustic wave propagation. Recently, the Dirac cone dispersion of phononic crystals with triangular or hexagonal lattices has been used to demonstrate interesting physics such as topologically protected edge states and Zitterbewegung effects. In this work, we present phononic crystals with triangular lattices that can be used as a platform to directly observe Klein tunneling. Klein tunneling is an important quantum mechanical physics proposed in 1929 for high-energy electrons. Klein suggested that electrons that are accelerated to the relativistic regime will tunnel through potential barriers with 100% probability regardless the width and height of the potential barrier. However, Klein tunneling has never been directly observed in quantum mechanics and solid state physics due to the difficulties in satisfying the stringent requirements. Here, we designed phononic crystals with Dirac cone dispersion at which the phonons behave as quasi-particles in the relativistic regime. Near total transmissions of the phonons through potential barriers were measured that are independent from the width and height of the potential barrier. Our results provide the first direct observation of Klein tunneling. This work will inspire a new type of applications that uses phononic crystals as a platform for the experimental studies of wave mechanics and quantum physics.

Thermodynamics of the Ideal Two-Dimensional Magnetoexciton Gas with Linear Dispersion Law

The Bose–Einstein condensation of the two-dimensional magnetoexciton gas with Dirac cone dispersion law is possible at different from zero critical temperature. The partition function, the thermodynamic function such as the free and full energies, entropy and heat capacity were calculated in both gaseous and degenerate phases. The second order phase transition takes place. The jump of the heat capacity at critical temperature does not exist.

Effective medium theory for a photonic pseudospin- 12 system

Photonic pseudospin-1/2 systems, which exhibit Dirac cone dispersion at Brillouin zone corners in analogy to graphene, have been extensively studied in recent years. However, it is known that a linear band crossing of two bands cannot emerge at the center of Brillouin zone in a two-dimensional photonic system respecting time reversal symmetry. Using a square lattice of elliptical magneto-optical cylinders, we constructed an unpaired Dirac point at the Brillouin zone center as the intersection of the second and third bands corresponding to the monopole and dipole excitations. Effective medium theory can be applied to the two linearly crossed bands with the effective constitutive parameters numerically calculated using the boundary effective medium approach. It is shown that only the effective permittivity approaches zero while the determinant of the nonzero effective permeability vanishes at the Dirac point frequency, showing a different behavior from the double-zero index metamaterials obtained from the pseudospin-1 triply degenerate points for time reversal symmetric systems. Exotic phenomena, such as the Klein tunneling and Zitterbewegung, in the pseudospin-1/2 system can be well understood from the effective medium description. When the Dirac point is lifted, the edge state dispersion near the $\Gamma$ point can be accurately predicted by the effective constitutive parameters. We also further realized magneto-optical complex conjugate metamaterials for a wide frequency range by introducing a particular type of non-Hermittian perturbations which make the two linear bands coalescence to form exceptional points at the real frequency.

Angle-resolved reflection spectra of Dirac cones in triangular-lattice photonic crystal slabs.

The dispersion relation and the angle-resolved reflection spectra of triangular-lattice photonic crystal slabs of the C6v symmetry were examined by the finite element method. The Dirac-cone dispersion relation on the Γ point of the reciprocal space was confirmed. The reflection spectra showed unique selection rules that agreed with the analytical calculation by the k · p perturbation theory. The distortion of the liner dispersion relation of the Dirac cones due to diffraction loss was also reproduced well by the numerical calculation, while we found distortion-free Dirac cones materialized with E2-symmetric modes.

Mid-IR Dirac-cone dispersion relation materialized in SOI photonic crystal slabs.

We materialized the isotropic Dirac-cone dispersion relation in the mid-infrared range by fabricating photonic crystal slabs of the C4v symmetry in SOI (silicon-on-insulator) wafers by electron beam lithography. The dispersion relation was examined by the angle-resolved reflection spectra with our home-made high-resolution apparatus, which showed a good agreement with the dispersion relation and the reflection spectra calculated by the finite element method. The reflection spectra also agreed with the selection rules derived from the spatial symmetry of the Dirac-cone modes, which proved to be a powerful tool for the mode assignment.

Two Dimensional Bright and Dark Magnetoexcitons Interacting with Quantum Point Vortices

The theory of the two-dimensional (2D) magnetoexcitons was enlarged in two aspects. One of them takes into account the electron-hole (e–h) exchange Coulomb interaction, which appears when the conduction and the valence electrons belong partially to both bands. The exchange Coulomb interaction gives rise to the Dirac cone dispersion law in the range of small in-plane wave vectors k|| obeying to the condition |k|||l0 < 1, where l0 is the magnetic length. Such dispersion law may change essentially the properties of the high density magnetoexcitons opening the possibility of their Bose-Einstein condensation at different from zero temperatures [1]. Another aspect concerns the high density magnetoexcitons with fractional filling factors in conditions of fractional quantum Hall effects. The Chern-Simons gauge field was introduced into the Hamiltonian of the 2D coplanar e–h system by the unitary transformation. It gives rise to the additional gauge vector potential, the influence of which leads to the anisotropic corrections to the magnetoexciton magnetic mass [2].

Two-dimensional magnetoexciton superposition states with Dirac cone dispersion law and quantum interference effects in optical transitions

The essential influence of the electron-hole (e-h) exchange Coulomb interaction on the properties of the two-dimensional magnetoexcitons was revealed. The main of them is the creation of two new symmetric and asymmetric superposition states formed by two bright magnetoexciton states with electron structure determined only by the direct Coulomb interaction and with the total angular momentum projections F=±1. The symmetric state due to the exchange e-h Coulomb interaction acquires a Dirac cone dispersion law in the range of small in-plane wave vectors with the group velocity proportional to the magnetic field strength B, with equal probabilities of the quantum transitions from the ground state of the crystal in both light circular polarizations but with maximum probability in the Faraday geometry of the light propagation and vanishing probability in the Voigt one. In difference on it, the asymmetric superposition state remains with the usual dispersion law inherited from the bare magnetoexciton states and has a dipole-active quantum transitions in both circular polarizations, indifferent on the direction of the light propagation. The both symmetric and asymmetric superposition states revealed the quantum interference effects in the case of the light with two linear polarizations, the vectors of which have different parities as regards the inversion of the light wave vector k→. The symmetric (asymmetric) state is allowed in the case of linear polarization vector with positive (negative) parity and is forbidden in the case of linear polarization vector with negative (positive) parity. The obtained optical results open the possibility to investigate the thermodynamic properties of the 2D Bose gas with Dirac cone dispersion law.

Polarization tensor for tilted Dirac fermion materials: Covariance in deformed Minkowski spacetime

The rich structure of solid state physics provides us with Dirac materials the effective theory of which enjoys the Lorentz symmetry. In non-symmorphic lattices, the Lorentz symmetry will be deformed in a way that the null energy-momentum vectors will correspond to on-shell condition for tilted Dirac cone dispersion. In this sense, tilted Dirac/Weyl materials can be viewed as solid state systems where the effective spacetime is non-Minkowski. In this work, we show that the polarization tensor for tilted Dirac cone systems acquires a covariant from only when the spacetime is considered to be an appropriate deformation of the Minkowski spacetime. As a unique consequence of the deformation of the geometry of the spacetime felt by the electrons in tilted Dirac cone materials, the Coulomb density-density interactions will generate corrections in both longitudinal and transverse channels. Therefore the transverse photons also participate in mediating the Coulomb forces, implying emergent Amperean forces associated with the tilt of the spacetime.