Relativistic Dirac Equation Research Articles

Experimental Realization of Type-II Dirac Fermions in a PdTe_{2} Superconductor.

A Dirac fermion in a topological Dirac semimetal is a quadruple-degenerate quasiparticle state with a relativistic linear dispersion. Breaking either time-reversal or inversion symmetry turns this system into a Weyl semimetal that hosts double-degenerate Weyl fermion states with opposite chiralities. These two kinds of quasiparticles, although described by a relativistic Dirac equation, do not necessarily obey Lorentz invariance, allowing the existence of so-called type-II fermions. The recent theoretical discovery of type-II Weyl fermions evokes the prediction of type-II Dirac fermions in PtSe_{2}-type transition metal dichalcogenides, expecting experimental confirmation. Here, we report an experimental realization of type-II Dirac fermions in PdTe_{2} by angle-resolved photoemission spectroscopy combined with abinitio band calculations. Our experimental finding shows the first example that has both superconductivity and type-II Dirac fermions, which turns the topological material research into a new phase.

Cross sections for electrons scattering from silver at low energies

SynopsisDifferential cross sections for scattering of electrons from silver atoms are calculated in the low energy region using the relativistic Dirac equation. The projectile target interaction is represented as sum of three local and real terms i.e. the static, a parameter free correlation polarization potential and the modified semi classical exchange potentials. We compare our results for differential cross sections with the available calculations and experimental measurements.

Band-gap generation and excitation of conduction electrons in graphene by THz laser fields

ABSTRACTGraphene is a one-atom-thin 2D layer of hexagonal rings of carbon atoms (or, a giant 2D ‘carbon-molecule’) whose low-energy quasi-particles are charged and massless Dirac fermions which follow the relativistic Dirac equation in 2D. Using a recently developed theory of interaction of graphene with intense laser fields, we briefly discuss two phenomena of interest: (1) the lifting of degeneracy of graphene at the K-points by THz laser interaction and (2) induction of anisotropic electron distributions in the conduction band from the occupied valence band by absorption of one-, two-, or multiple THz photons. The first process is analogous to a phase transition of a pristine graphene from a semi-metal into a semiconductor within a time scale determined by the ultrashort pulse duration of the laser. The second process may permit control of electron currents in graphene whose magnitude and direction may be controlled simultaneously by controlling the THz laser parameters (e.g. pulse duration, field strength or polarisation).

Remove Degeneracy in Relativistic Symmetries for Manning—Rosen Plus Quasi-Hellman Potentials by Tensor Interaction

The relativistic Dirac equation under spin and pseudo-spin symmetries is investigated for Manning—Rosen plus quasi-Hellman potentials with tensor interaction. For the first time we consider the Hulthen plus Yukawa for tensor interaction. The Formula method is used to obtain the energy eigen-values and wave functions. We also discuss about the energy eigen-values and the Dirac spinors for the Manning—Rosen plus quasi-Hellman potentials for the spin and pseudo-spin symmetry with Formula method. To show the accuracy of the present model, some numerical results are shown in both pseudo-spin and spin symmetry limits.

A generalized interaction in noncommutative space: Both relativistic and nonrelativistic fields

We investigate nonrelativistic Schrodinger and relativistic Dirac equations with a rather general interaction, which includes harmonic, linear, Coulomb and inverse square term, in the noncommutative space (NCS) and report the spectrum of the system as well as the associated wave function in a quasi-exact analytical manner.

SPECTROSCOPY OF ATOMS IN A STRONG LASER FIELD: NEW METHOD TO SENSING AC STARK EFFECT, MULTIPHOTON RESONANCES PARAMETERS AND IONIZATION CROSS-SECTIONS

The resonant multiphoton resonances shifts and widths and ionization cross-sections for multielectron atoms in a intense laser radiation field are studied. It is carried out a new consistent approach to atom in a strong realistic laser field, based on the relativistic energy formalism (S-matrix adiabatic formalism), relativistic Dirac equation Green function method and relativistic many-body perturbation theory with the Dirac-Kohn-Sham zeroth approximation. In relativistic theory, the Gell-Mann and Low adiabatic formula for energy shift is connected with electrodynamical scattering matrice, which includes interaction with as a laser field as a photon vacuum field (radiative decay). The optimized basis of the relativistic orbitals is generated with using a minimization procedure for the gauge-non-invariant contribution (the fourth-order of the QED perturbation theory) to the radiation width of atomic state. An approach to treating the multiphoton atomic processes is outlined on example of H, Cs, Kr,Mg etc. Analysis shows that the shift and width of the multi-photon resonance line for the interaction “atom- multimode laser pulse” is greater than the corresponding shift and width for a case of the “atom- single-mode pulse” (the Lorenz pulse model) interaction. From the physical point of view it is naturally provided by action of the photon-correlation effects and influence of the laser pulse multi-modity.

Photoexcitations of many-electron atoms and ions by twisted light

SynopsisThe photoexcitation of (multi-electron) atoms and ions by twisted light has been explored within the framework of the density matrix theory and Dirac's relativistic equation. It is shown that the population of the excited atoms and, hence, their fluorescence emission are sensitive with regard to the transverse linear momentum and the (projection of the) total angular momentum of the incident light.

Scattering of twisted relativistic electrons by atoms

The Mott scattering of high-energetic twisted electrons by atoms is investigated within the framework of the first Born approximation and Dirac's relativistic equation. Special emphasis is placed on the angular distribution and longitudinal polarization of the scattered electrons. In order to evaluate these angular and polarization properties we consider two experimental setups in which the twisted electron beam collides with either a single well-localized atom or macroscopic atomic target. Detailed relativistic calculations have been performed for both setups and for the electrons with kinetic energy from 10 to 1000 keV. The results of these calculations indicate that the emission pattern and polarization of outgoing electrons differ significantly from the scattering of plane-wave electrons and can be very sensitive to the parameters of the incident twisted beam. In particular, it is shown that the angular- and polarization-sensitive Mott measurements may reveal valuable information about both the transverse and longitudinal components of the linear momentum and the projection of the total angular momentum of twisted electron states. Thus, the Mott scattering emerges as a diagnostic tool for the relativistic vortex beams.

Simulation of Dirac Electron Tunneling Current in Armchair Graphene Nanoribbon Tunnel Field-Effect Transistors Using a Transfer Matrix Method

We simulate quantum mechanical tunneling current in armchair graphene nanoribbon tunnel field-effect transistors (AGNR-TFETs). The relativistic Dirac equation is used to determine electron wave functions in the AGNRs, while the potential profile is solved by the Poisson equation. We use a transfer matrix method (TMM) to calculate the electron transmittance and the Dirac electron tunneling current in the AGNR-TFETs. The results show that the Dirac electron tunneling current increases with increasing the drain and gate voltages. Moreover, the AGNR width and the thickness of insulator affect the characteristics of the Dirac electron tunneling currents.

Highly relativistic spin-gravity coupling for fermions

Descriptions of highly relativistic fermions in a gravitational field in the classical (nonquantum) and quantum approaches are discussed. The results following from the Mathisson-Papapetrou equations for a fast spinning particle in Schwarzschild's and Kerr's background are considered. Numerical estimates for electron, proton and neutrino in the gravitational field of black holes are presented.The general relativistic Dirac equation is analyzed from the point of view it is using for the adequate description of highly relativistic fermions in a gravitational field, in the linear and nonlinear spin approximation. It is necessary to have some corrected Dirac equation for a highly relativistic fermion with strong spin-gravity coupling.

Particles of Spin Zero and 1/2 in Electromagnetic Field with Confining Scalar Potential in Modified Heisenberg Algebra

In this paper, we propose to solve the relativistic Klein Gordon and Dirac equations subjected to the action of a uniform electomagnetic field confining scalar potential yin the presence of a minimal length in the momentum space. In both cases, the energy eigenvalues and their corresponding eigenfunctions are obtained. The limiting cases is then deduced for a small parameter of deformation.

Dirac solitons in square binary waveguide lattices

We study optical analogs of two-dimensional (2D) Dirac solitons in square binary waveguide lattices with two different topologies in the presence of Kerr nonlinearity. These 2D solitons turn out to be quite robust. We demonstrate that with the found 2D solitons, the coupled mode equations governing light dynamics in square binary waveguide lattices can be converted into the nonlinear relativistic 2D Dirac equation with the four-component bispinor. This paves the way for using binary waveguide lattices as a classical simulator of quantum nonlinear effects arising from the 2D Dirac equation, something that is thought to be impossible to achieve in conventional (i.e., linear) quantum field theory.

Dirac equations with confining potentials

This paper is devoted to a study of relativistic eigenstates of Dirac particles which are simultaneously bound by a static Coulomb potential and added linear confining potentials. Under certain conditions, despite the addition of radially symmetric, linear confining potentials, specific bound-state energies surprisingly preserve their exact Dirac–Coulomb values. The generality of the "preservation mechanism" is investigated. To this end, a Foldy–Wouthuysen transformation is used to calculate the corrections to the spin-orbit coupling induced by the linear confining potentials. We find that the matrix elements of the effective operators obtained from the scalar, and time-like confining potentials mutually cancel for specific ratios of the prefactors of the effective operators, which must be tailored to the preservation mechanism. The result of the Foldy–Wouthuysen transformation is used to verify that the preservation is restricted (for a given Hamiltonian) to only one reference state, rather than traceable to a more general relationship among the obtained effective low-energy operators. The results derived from the nonrelativistic effective operators are compared to the fully relativistic radial Dirac equations. Furthermore, we show that the preservation mechanism does not affect antiparticle (negative-energy) states.

Interaction of twisted light with many-electron atoms and ions

The excitation of many-electron atoms and ions by twisted light has been studied within the framework of the density-matrix theory and Dirac's relativistic equation. Special attention is paid to the magnetic sublevel population of excited atomic states as described by means of the alignment parameters. General expressions for the alignment of the excited states are obtained under the assumption that the photon beam, prepared as a coherent superposition of two twisted Bessel states, irradiates a macroscopic target. We demonstrate that for this case the population of excited atoms can be sensitive to both the transverse momentum and the (projection of the) total angular momentum of the incident radiation. While the expressions are general and can be employed to describe the photoexcitation of any atom, independent on its shell structure and number of electrons, we performed calculations for the $3s\ensuremath{\rightarrow}3p$ transition in sodium. These calculations indicate that the ``twistedness'' of incoming radiation can lead to a measurable change in the alignment of the excited $^{2}P_{3/2}$ state as well as the angular distribution of the subsequent fluorescence emission.

Electron-spin dynamics in elliptically polarized light waves

We investigate the coupling of the spin angular momentum of light beams with elliptical polarization to the spin degree of freedom of free electrons. It is shown that this coupling, which is of similar origin as the well-known spin-orbit coupling, can lead to spin precession. The spin-precession frequency is proportional to the product of the laser-field's intensity and its spin density. The electron-spin dynamics is analyzed by employing exact numerical methods as well as time-dependent perturbation theory based on the fully relativistic Dirac equation and on the nonrelativistic Pauli equation that is amended by a relativistic correction that accounts for the light's spin density.

Screening Constant by Unit Nuclear Charge Calculations of Energies and Fine Structure Splitting for Li-Like Ions

We present in this paper non-relativistic energies of the 1s^2ns^2S (n = 3-20) and 1s^2np ^2P (n = 2-20) excited states of the lithium-like ions using the Screening Constant by Unit Nuclear Charge method. Calculations are performed up to Z = 20. In addition, accurate results for the fine structure splitting of the 1s^2np ^2P (n = 3, 4, and 5) levels of Z = 3-20 including quantum-electrodynamics (QED) corrections are obtained, turning into account the solution of the Dirac relativistic equation of the hydrogen-like systems with the effective nuclear charge Z* = Z(1-β), β being the screening constant by unit nuclear charge. It is shown that QED corrections are very significant for large Z systems, and then they should be included in any calculation for obtaining more accurate results. The present data agree very well with the experimental values from NIST and with various calculations.

Angular and polarization analysis for two-photon decay of2shyperfine states of hydrogenlike uranium

The amplitude of two-photon transitions between hyperfine states in hydrogenlike ions is derived based on relativistic Dirac equation and second order perturbation theory. We study angular and linear polarization properties of the photon pair emitted in the decay of $2s$ states, where spin-flip and non-spin-flip transitions are highlighted. We pay particular attention to hydrogenlike uranium, since it is an ideal candidate for investigating relativistic and high-multipole effects, such as spin-flip transitions. Two types of emission patterns are identified: i) non-spin-flip transitions are found to be characterized by an angular distribution of the type $W(\theta)\sim1+\cos^2\theta$ while the polarizations of the emitted photons are parallel; ii) spin-flip transitions have somewhat smaller decay rates and are found to be characterized by an angular distribution of the type $W(\theta)\sim1-1/3\cos^2\theta$ while the polarizations of the emitted photons are orthogonal, where $\theta$ is the angle between photons directions. Deviations due to non-dipole and relativistic contributions are evaluated for both types of transitions. This work is the first step toward exploring the effect of nucleus over the the angular and polarization properties of the photon pairs emitted by two-photon transitions.

Approximate solutions of Dirac equation for Tietz and general Manning-Rosen potentials using SUSYQM

In this paper, we consider the relativistic Dirac equation with Tietz and general Manning-Rosen potential. By using appropriate approximation, we obtained the approximate analytical solutions of the Dirac equation for the combined potential via the supersymmetric quantum mechanics (SUSYQM). Within the framework of spin and pseudospin symmetry limits, we obtained the relativistic energy eigenvalus and the corresponding components of the wave functions for Tietz and Manning-Rosen potential using the SUSYQM. We have also reported some numerical results and figures to show the effect of the tensor interactions.

The Linear Interaction in Noncommutative Space; Both Relativistic and Nonrelativistic Cases

We investigate the effects of the Aharonov-Bohm field and a linear interaction on the quantum dynamics of both nonrelativistic and relativistic (Dirac equation) particles in noncommutative formulation.

Strong anisotropy of Dirac cones in SrMnBi2 and CaMnBi2 revealed by angle-resolved photoemission spectroscopy.

The Dirac materials, such as graphene and three-dimensional topological insulators, have attracted much attention because they exhibit novel quantum phenomena with their low energy electrons governed by the relativistic Dirac equations. One particular interest is to generate Dirac cone anisotropy so that the electrons can propagate differently from one direction to the other, creating an additional tunability for new properties and applications. While various theoretical approaches have been proposed to make the isotropic Dirac cones of graphene into anisotropic ones, it has not yet been met with success. There are also some theoretical predictions and/or experimental indications of anisotropic Dirac cone in novel topological insulators and AMnBi2 (A = Sr and Ca) but more experimental investigations are needed. Here we report systematic high resolution angle-resolved photoemission measurements that have provided direct evidence on the existence of strongly anisotropic Dirac cones in SrMnBi2 and CaMnBi2. Distinct behaviors of the Dirac cones between SrMnBi2 and CaMnBi2 are also observed. These results have provided important information on the strong anisotropy of the Dirac cones in AMnBi2 system that can be governed by the spin-orbital coupling and the local environment surrounding the Bi square net.