Solving Dirac Equation Research Articles

PHOTON-ASSISTED TRANSPORT IN CARBON NANOTUBE MESOSCOPIC DEVICE

The aim of the present paper is to investigate the quantum transport properties of a mesoscopic device under the influence of gate voltage and photon energy. A model for such mesoscopic devices is proposed as two metal contacts are deposited on the carbon nanotube quantum dot to serve as source and drain electrodes. The conducting substrate is the gate electrode in this three-terminal mesoscopic device. Another metallic gate is used to govern the electrostatics and the switching of carbon nanotube channel. The substrate at the carbon nanotube quantum dot contacts are controlled by the back gate. Both effects of the photons energy and gate voltage are investigated. The photon-assisted tunneling probability is deduced by solving Dirac equation. Then the current is deduced according to Landauer–Buttiker formula. The quantum capacitance for the device is deduced in terms of density of states. Oscillatory behavior of the current is observed which is due to the Coulomb blockade oscillations. It was found, also, that the peak heights of the dependence of the current on the parameters under study are strongly affected by the interplay between the tunneled electrons and the photon energy. This interplay affects the sidebands resonance. The results obtained in this study are found to be in concordant with those in the literature, which confirm the correctness of the proposed model. This study is valuable for nanotechnology applications, e.g., photodetector devices and solid state quantum computing systems and quantum information processes.

Solving Dirac Equation with New Ring-Shaped Non-Spherical Harmonic Oscillator Potential

A new ring-shaped non-harmonic oscillator potential is proposed. The precise hound solution of Dirac equation with the potential is gained when the scalar potential is equal to the vector potential. The angular equation and radial equation are obtained through the variable separation method. The results indicate that the normalized angle wave function can be expressed with the generalized associated-Legendre polynomial, and the normalized radial wave function can be expressed with confluent hypergeometric function. And then the precise energy spectrum equations are obtained. The ground state and several low excited states of the system are solved. And those results are compared with the non-relativistic effect energy level in Phys. Lett. A 340 (2005) 94. The positive energy states of system are discussed and the conclusions are made properly.

Elastic scattering of electrons from Cs and Fr atoms

We have calculated the results for differential cross sections and spin parameters for the elastic scattering of electrons from Cs and Fr atoms in the range of intermediate and high incident electron energies. The theoretical results are obtained by solving Dirac equation in the field of phenomenological potential and utilizing partial wave phase shift analysis. We have compared our results for Cs with the available experimental and theoretical results. Theoretical calculations for e-Fr elastic scattering are reported here for the first time. We expect that the importance of relativistic effects will be substantial for such heavy atoms like Cs and Fr. We hope new experimental results will be reported in the light of our calculations for the e-Fr elastic scattering.

BPS Monopoles in Moduli Space under SU(2) Gauge Potential

Solving Dirac equation for a BPS monopole moving in the field of another BPS monopole in moduli space, it has been shown that spin momentum of the interacting monopole behaves as an extra energy source. The possibilities of splitting of the energy levels of the system have been explored.

First-principles calculations of parameters of electron-vibrational interaction and estimations of Jahn–Teller stabilization energy for Cr 3+ ion in elpasolites

Detailed microscopic study of the crystal field strength 10 Dq, constants of electron-vibrational interaction and Huang–Rhys factors for Cs 2NaYX 6:Cr 3+ (X = F, Cl, Br) crystals is presented. The method used in the calculations is the first-principles fully relativistic discrete-variational Dirac–Slater (DV–DS) method based on solving Dirac equations with the local density approximation. The basic features of the method are: (1) the first-principles method without any phenomenological parameters; (2) wide area of applications: to any atom or ion in any symmetry from O h to C 1 for any energy interval from IR to X-ray; (3) possibility to take into account all effects of chemical bond formation. The key idea of the method is that the molecular orbitals (MO) consisting of the wave functions of an impurity ion and ligands are used throughout the calculations rather than atomic wave functions. From the calculated values of the Huang–Rhys factors and available in the literature frequencies of normal vibrations for the a 1 g and e g modes for Cs 2NaYX 6:Cr 3+ (X = F, Cl, Br) crystals the Jahn–Teller stabilization energy for Cr 3+ ion in the 4T 2g excited state was estimated. The obtained results are compared with estimations from the available experimental data.

Microscopic analysis of the crystal field strength and lowest charge transfer energies in the elpasolite crystalsCs2NaYX6(X=F,Cl,Br)doped withCr3+

Detailed microscopic study of the crystal field strength $10Dq$ and lowest charge transfer (CT) energies for different interionic distances in ${\mathrm{Cs}}_{2}\mathrm{NaY}{X}_{6}:{\mathrm{Cr}}^{3+}\phantom{\rule{0.3em}{0ex}}(X=\mathrm{F},\phantom{\rule{0.3em}{0ex}}\mathrm{Cl},\phantom{\rule{0.3em}{0ex}}\mathrm{Br})$ crystals is performed in the present paper. The method used in the calculations is the first-principles fully relativistic discrete variational multielectron (DVME) method [K. Ogasawara et al., Phys. Rev. B 64, 115413 (2001)] based on solving Dirac equations with the local density approximation. From the results of performed calculations, the functions describing the dependencies of $10Dq$ and lowest CT energy on the metal-ligand distance $R$ were obtained without introducing any fitting or semiempirical parameters. It was shown that 10Dq depends on $R$ as $1∕{R}^{n}$, with $n=4.4634$, 4.3742, and 4.3532 for ${\mathrm{Cs}}_{2}{\mathrm{NaYF}}_{6}:{\mathrm{Cr}}^{3+}$, ${\mathrm{Cs}}_{2}{\mathrm{NaYCl}}_{6}:{\mathrm{Cr}}^{3+}$, and ${\mathrm{Cs}}_{2}{\mathrm{NaYBr}}_{6}:{\mathrm{Cr}}^{3+}$, respectively. The lowest CT energies $E(\mathrm{CT})$ are linear functions of $R$ and decrease with increasing $R$ with $dE(\mathrm{CT})∕dR=\ensuremath{-}953\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}∕\mathrm{pm}$, $\ensuremath{-}621\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}∕\mathrm{pm}$, and $\ensuremath{-}520\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}∕\mathrm{pm}$ for ${\mathrm{Cs}}_{2}{\mathrm{NaYF}}_{6}:{\mathrm{Cr}}^{3+}$, ${\mathrm{Cs}}_{2}{\mathrm{NaYCl}}_{6}:{\mathrm{Cr}}^{3+}$, and ${\mathrm{Cs}}_{2}{\mathrm{NaYBr}}_{6}:{\mathrm{Cr}}^{3+}$, respectively. Using these results, the constants of the electron-vibronic interaction, Huang-Rhys parameters, Stokes shifts, and Gruneisen constants for the ${a}_{1g}$ normal mode in all considered hosts were calculated. The obtained results are in good agreement with the experimental data and can be readily applied for analysis of the optical spectra and electron-vibronic interaction of the $[\mathrm{Cr}{X}_{6}{]}^{3\ensuremath{-}}\phantom{\rule{0.3em}{0ex}}(X=\mathrm{F},\phantom{\rule{0.3em}{0ex}}\mathrm{Cl},\phantom{\rule{0.3em}{0ex}}\mathrm{Br})$ units in other hosts.

Pseudospin symmetry and the relativistic ring-shaped non-spherical harmonic oscillator

A generalized relativistic harmonic oscillator for spin 1/2 particles is studied. The eigenfunctions and eigenenergies are obtained for the ring-shaped non-spherical harmonic oscillator by solving Dirac equation with equal mixture of vector and scalar potentials in opposite signs, for which pseudospin symmetry is exact. Several particular cases such as the ring-shaped harmonic oscillator, non-spherical harmonic oscillator, and spherical harmonic oscillator are also discussed.

First Principles Calculation of Fe <i>L</i><SUB>2,3</SUB>-edge X-ray Absorption Near Edge Structures of Iron Oxides

X-ray absorption near edge structure (XANES) at L2,3-edge of 3d transition elements is dominated by strong correlation effects among 2p core hole and 3d electrons. In the present study, we have performed systematic configuration interaction (CI) calculations in order to reproduce and interpret Fe-L2,3 XANES of FeO, LaFeO3 and SrFeO3. Relativistic four components wave functions were obtained by solving Dirac equations with density functional theory. CI calculations were made using the relativistic molecular orbitals instead of atomic orbitals, which enables inclusion of the O-2p orbital contributions through covalency. The oscillator strength of the electric dipole transition was then computed. Experimental XANES spectra of three compounds were satisfactorily reproduced by the theoretical spectra obtained for (FeO6)m− clusters in octahedral symmetry. Chemical shifts between compounds were quantitatively reproduced as well. Component analysis of CI was systematically made in order to analyze the origin of differences in spectral shapes.

Exact solution of the Dirac equation for a spin-12 charged particle in two-dimensional and three-dimensional Euclidean spaces with shape invariance symmetry

It is shown that two types of shape invariance with respect to quantum numbers n and m lead us to solve Dirac equation for a spin-12 charged particle in two-dimensional and three-dimensional Euclidean spaces in the presence of magnetic fields. Also, we introduce supersymmetry algebra and shape invariance symmetry represented by two bunches of spinors obtained from the above solutions.

Study of heavy mesons with a relativistic approach

Mass spectra of heavy meson ground states D, D s , B and B s are calculated based on a relativistic quark potential model with mean field approach. Radiative and hadronic pion decays and Isgur–Wise functions of semileptonic decays of these heavy states are investigated. In our approach, both light and heavy quark wave functions are treated independently by solving Dirac equation self-consistently. Linear confinement and one-gluon exchange interaction are taken into account. The calculations show that the mass splitting between the triplet 3 S 1 and singlet 1 S 0 heavy meson states affects the Isgur–Wise functions. Our results also illustrate the success of heavy quark effective theory in the mass spectra of the heavy mesons. Our calculations for the radiative and strong decays of the heavy mesons are in agreement with the data.

Pauli equation for monopolonium

The study of the behaviour of spin-1/2 magnetic monopole moving in the field of another monopole has been undertaken by solving Dirac equation for energy eigenvalues. The Hamiltonian of this system has been shown to involve spin contributions. By introducing suitable spinors, the Pauli equation of a spin-1/2 monopole moving in the field of another monopole has been solved and it has been demonstrated thatad hoc introduction of spin perceptibly modifies the energy eigenvalue and eigenfunctions of the system. It has also been demonstrated that Bohr radius of the system is very small in comparison to the atomic Bohr radius and the energy of the system is enormously high.

Pauli equation for spin-1/2 dyonium in non-Abelian gauge theory

The study of the behaviour of a spin-1/2 non-Abelian dyon moving in the field of an Abelian dyon has been undertaken by solving Dirac equation for energy eigenvalues. The Hamiltonian of this system has been shown to involve spin as well as the contribution of massive fields associated with these particles. By introducing suitable spinors, the Pauli equation for a spin-1/2 dyon moving in the field of another dyon has been solved in non-Abelian gauge theory. It is shown that the introduction of massive fields perceptibly modifies the energy eigenvalue and eigenfunction of the bound state of the system.

Relativistic surface states of a semi-infinite chain acted on by an electric field

In this paper, we deal with the relativistic effect of surface states of a semi-infinite chain of the Kronig-Penney type potential acted on by an electric field. The crux of the problem is to solve Dirac equation with a linear potential. Combining the solution with other solutions in the various regions, known through the boundary conditions, we get a relativistic expression for energy of surface states. The form is similar to that obtained in the absence of an electric field, which the expression automatically approaches as the field vanishes.

Relativistic surface states of a semi-infinite chain acted on by an electric field

In this paper, we deal with the relativistic effect of surface states of a semi-infinite chain of the Kronig-Penney type potential acted on by an electric field. The crux of the problem is to solve Dirac equation with a linear potential. Combining the solution with other solutions in the various regions, known through the boundary conditions, we get a relativistic expression for energy of surface states. The form is similar to that obtained in the absence of an electric field, which the expression automatically approaches as the field vanishes.