Mathieu Potential Research Articles

Dirac equation in relativistic density functional theory and mapped Fourier grid method

For studying the electronic structure of solids consisting of heavy atoms, the Dirac–Kohn–Sham equation is considered in the presence of the Mathieu potential. The two-component spinors and the potential function are expanded in terms of spherical harmonics. Also, a numerical approach is presented to study the Dirac–Kohn–Sham equation in the presence of the noncentral Mathieu potential. Then, the energy eigenvalues and eigenvectors are obtained in the case of spherically expanded potential in the Brillouin zoon by using the mapped Fourier grid method.

Proposal for High-Energy Cutoff Extension of Optical Harmonics of Solid Materials Using the Example of a One-Dimensional ZnO Crystal.

We propose a novel approach based on the subcycle injection of carriers to extend the high-energy cutoff in solid-state high harmonics. The mechanism is first examined by employing the standard single-cell semiconductor Bloch equation (SC SBE) method for one-dimensional (1D) Mathieu potential model for ZnO subjected to the intense linearly polarized midinfrared laser field and extreme-ultraviolet pulse. Then, we use coupled solution of Maxwell propagation equation and SC SBE to propagate the fundamental laser field through the sample, and find that the high-harmonics pulse train from the entrance section of the sample can inject carriers to the conduction bands with attosecond timing, subsequently leading to a dramatic extension of high-energy cutoff in harmonics from the backside. We predict that for a peak intensity at 2×10^{11} W/cm^{2}, as a result of the self-seeding, the high-energy cutoff shifts from 20th (7.75eV) order to around 50th (19.38eV) order harmonics.

1-D Dirac equation in the presence of the Mathieu potential

1-D Dirac equation in the presence of the Mathieu potential

Exact solutions of the 1D Schrödinger equation with the Mathieu potential

The exact solutions of the 1D Schrödinger equation with the Mathieu potential V(x)=a2sin2⁡(bx)−ab(2c+1)cos⁡(bx)+vb2 (a>0,b>0) are presented as a confluent Heun function HC(α,β,γ,δ,η;z). The eigenvalues are calculated precisely by solving the Wronskian determinant. The wave functions for the positive and negative parameter c, which correspond to two different potential wells with symmetric axis x=0 and x=π are plotted. It is found that the wave functions are shrunk to the origin for given values of the parameters a=1,b=1 and v=2 when the potential parameter |c| increases.

Numerical solution of the relativistic single-site scattering problem for the Coulomb and the Mathieu potential

For a reliable fully-relativistic Korringa–Kohn–Rostoker Green function method, an accurate solution of the underlying single-site scattering problem is necessary. We present an extensive discussion on numerical solutions of the related differential equations by means of standard methods for a direct solution and by means of integral equations. Our implementation is tested and exemplarily demonstrated for a spherically symmetric treatment of a Coulomb potential and for a Mathieu potential to cover the full-potential implementation. For the Coulomb potential we include an analytic discussion of the asymptotic behaviour of irregular scattering solutions close to the origin ().

Three-loop correction to the instanton density. II. The sine-Gordon potential

In this second paper on quantum fluctuations near the classical instanton configurations, see {\em Phys. Rev. D \bf 92}, 025046 (2015) and arXiv:1501.03993, we focus on another well studied quantum-mechanical problem, the one-dimensional Sine-Gordon potential (the Mathieu potential). Using only the tools from quantum field theory, the Feynman diagrams in the instanton background, we calculate the tunneling amplitude (the instanton density) to the three-loop order. The result confirms (to seven significant figures) the one recently recalculated by G. V. Dunne and M. \"{U}nsal, {\it Phys. Rev. \bf D 89}, 105009 (2014) from the resurgence perspective. As in the double well potential case, we found that the largest contribution is given by the diagrams originating from the Jacobian. We again observe that in the three-loop case individual Feynman diagrams contain irrational contributions, while their sum does not.

Band structure analysis of an analytically solvable Hill equation with continuous potential

This paper concerns analytically solvable cases of Hill’s equation containing a continuously differentiable periodic potential. We outline a procedure for constructing the Floquet–Bloch fundamental system, and analyze the band structure of the system. The similarities to, and differences from, the cases of a piecewise constant periodic potential and the Mathieu potential, are illuminated.

Invisible defects in complex crystals

We show that invisible localized defects, i.e. defects that cannot be detected by an outside observer, can be realized in a crystal with an engineered imaginary potential at the defect site. The invisible defects are synthesized by means of supersymmetric (Darboux) transformations of an ordinary crystal using band-edge wavefunctions to construct the superpotential. The complex crystal has an entire real-valued energy spectrum and Bragg scattering is not influenced by the defects. An example of complex crystal synthesis is presented for the Mathieu potential.

Application of Mathieu potential to photoemission calculations: the case of a strong potential

We have applied the Mathieu potential model to photoemission calculations from metals. The case of a strong potential is considered to describe the wavefunctions which is used for the evaluation of photocurrent from tungsten, molybdenum, copper and silicon.

Application of Mathieu potential to photoemission from metals

The Mathieu potential is used to define the crystal potential from which the initial state wavefunction for the surface state is derived. The wavefunction is used for photoemission calculations in the case of free electron metals like Al and Be.

A relativistic equation for a slowly varying potential

A relativistic equation is derived for a slowly varying potential by suitably approximating the one-dimensional Dirac equation. This equation is shown to be akin to the Schrodinger equation with an effective potential and effective eigenvalues. An iterative procedure for solving this equation is indicated. As an application, the relativistic treatment of the Mathieu potential on the basis of this equation is considered and results are compared with those obtained by solving the exact one-dimensional Dirac equation. These results are likely to take adequate account of the relativistic effects on electrons near Fermi levels in metals.

A simple numerical method for the determination of relativistic one-dimensional band structures

A discretized form of the one-dimensional Dirac equation is solved to study relativistic band structures. Properties of periodic continued fractions are used to find the dispersion relation inside allowed bands. The Mathieu potential is considered and it is found that relativistic effects cause a reduction of the width of allowed bands and gaps.

Full-potential Korringa-Kohn-Rostoker band theory applied to the Mathieu potential.

The band theory of Korringa, Kohn, and Rostoker (KKR) based on the Green-function method is extended to space-filling potentials. A numerical test using the Mathieu potential shows good convergence for the bands up to 1.5 Ry with {ital l}{le}4 included in the angular-momentum expansion for the wave functions. Our results strongly support the applicability of the full-potential KKR to bulk electronic-structure problems.

Modified method of variational calculation of generalized Wannier functions for a crystal with a model surface potential

Generalized Wannier functions for a crystal described by the Mathieu potential and the model surface potential are calculated by the modified variational method previously used for calculating the Wannier functions for bulk crystals. The interaction energy integrals calculated in various approximations are compared and conclusions expected to be also valid for crystals with realistic potentials are derived. The main conclusions are the following: The trial functions with exponential tails used for constructing Wannier functions give better Wannier functions than the trial functions with Gaussian tails. The approximation of assuming that the trial function for the surface layer is the only trial function which differs from bulk trial functions is a very good approximation for the case when the potential differs from the bulk crystal potential only for the surface layer.

Modified method of variational calculation of Wannier functions

The energy functional expressed in terms of the Wannier function is varied with respect to the trial function in the region surrounding the lattice site on which the function is centred and with respect to parameters describing the exponential tail of the trial function for the remaining volume of the crystal. The differential equation for the trial function in the region surrounding the lattice site and the algebraic equation for the parameter describing the exponential tail of the function obtained in this way are simultaneously solved for the case of a one-dimensional crystal with a Mathieu potential. The band centre, the band width and the interaction integrals are compared with the exact values available for the crystal described by the Mathieu potential.

Raman-Nath equation and Schrödinger-like equation with a Mathieu potential

In this paper we present an analytical perturbed solution of a Schrödingerlike equation with a Mathieu potential taking advantage from the already known solution of the Eaman-Nath equation.

Relation, expressed in terms of elliptic integrals, for determining characteristic values and characteristic exponents within stable and unstable regions (bands and gaps) associated with the Mathieu potential

An approximate relation for the determination of characteristic values and characteristic exponents within stable and unstable regions (bands and gaps), associated with the Mathieu potential, is given. The relation, which is based on the use of the first- or the third-order phase-integral approximation, is expressed in terms of complete elliptic integrals of the first and second kind. The great accuracy of the relation is illustrated numerically.

Dispersion relation for energy bands and energy gaps derived by the use of a phase-integral method, with an application to the Mathieu equation

The quantal problem of a particle moving in a one-dimensional periodic potential with one barrier per period is treated rigorously by means of a phase-integral method. A general dispersion formula, relating real energies to real or complex wavenumbers, valid for any conveniently chosen order of the phase-integral approximations used, is derived. The formula allows for the use of modified as well as unmodified phase-integral approximations, and different possibilities are discussed. For instance, a modification introduced by Floyd in the usual first-order JWKB approximation and applicable to the interior of high-energy bands can be utilised in our scheme for the generation of higher-order phase-integral approximations. This approach opens the possibility of using this phase-integral method in situations where the parameters of the problem are such that with unmodified approximations it would not work at all. The accuracy obtainable is illustrated by calculations on the Mathieu potential and comparison with available accurate numerical results.

The von Weizsäcker Coefficient in Density Functional Theoryt

Abstract For any given potential, linear response theory tells us the size that the von Weizsäcker coefficient must assume in order to obtain the correct energy in the weak coupling limit. With this necessarily imposed choice, the quality of results is investigated as the coupling is strengthened. From the study of a number of model problems (the harmonic oscillator, the Yukawa well and the Mathieu potential) it is concluded that good results can be maintained to high couplings.

Calculation of electronic Green functions using nonorthogonal basis functions: application to crystals

Formulae are derived for the electronic Green function of a system, and in particular of a crystal, when this is expressed as a linear combination of basis functions which are not mutually orthogonal. Expressions are derived for the electronic density of states and the electrical conductivity of a system in this representation, and the differences in these properties between perfect and imperfect crystals, and between different types of crystal, are examined. Application of the method to a simple cubic crystal with the Mathieu potential shows that, as for a one dimensional system, the use of optimized in place of fixed atomic orbitals as basis functions leads to a great increase in accuracy.