Staggered grid leap-frog scheme for the Dirac equation

Hammer, René; Pötz, W.

doi:10.1016/j.cpc.2013.08.013

Cited by 26 publications

(37 citation statements)

References 43 publications

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“…For a longtime propagation it is of great importance to use an almost dispersion-preserving finite-difference scheme, since scattering at spatiotemporal potentials and at the boundary can introduce higher wave vector components even when one starts with a wave packet with its wave vector components closely centered at k = 0. Moreover, the formulation of proper boundary conditions is crucial to avoid spurious reflections and eventually instability [29]. …”

Section: ∂ψ(Rt) ∂T = H D ψ(Rt)mentioning

confidence: 99%

“…In order to solve numerically the (2 + 1) Dirac equation we use a specially developed staggered-grid leap frog scheme, which we introduced and discussed in detail in Ref. [29]. The numerical solution of the Dirac equation on a finite grid is a more subtle issue than for the nonrelativistic Schrödinger equation.…”

Section: ∂ψ(Rt) ∂T = H D ψ(Rt)mentioning

confidence: 99%

“…For this purpose we use a specially developed discretization scheme, which was introduced and discussed in detail in Ref. [29]. We study two different scenarios for inducing a coupling between the dots: (i) by conventional means, i.e., by electrostatically reducing the barrier height in between the dots, and (ii) by coupling the dots via Klein tunneling upon a hole state in the barrier, which can be shifted by a gate voltage.…”

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confidence: 99%

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Gate-defined coupled quantum dots in topological insulators

2014

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We consider electrostatically coupled quantum dots in topological insulators, otherwise confined and gapped by a magnetic texture. By numerically solving the (2 + 1) Dirac equation for the wave packet dynamics, we extract the energy spectrum of the coupled dots as a function of bias-controlled coupling and an external perpendicular magnetic field. We show that the tunneling energy can be controlled to a large extent by the electrostatic barrier potential. Particularly interesting is the coupling via Klein tunneling through a resonant valence state of the barrier. The effective three-level system nicely maps to a model Hamiltonian, from which we extract the Klein coupling between the confined conduction and valence dots levels. For large enough magnetic fields Klein tunneling can be completely blocked due to the enhanced localization of the degenerate Landau levels formed in the quantum dots. In topological insulators (TIs), according to the bulkboundary correspondence principle [1,2], topologically protected surface states are formed, which are robust against timereversal (TR) elastic perturbations. In the long-wavelength limit the two-dimensional (2D) electron states at the surfaces of three-dimensional (3D) TIs can be described as massless Dirac electrons with the peculiar property that the spin is locked to the momentum, thereby forming a helical electron gas. Charge and spin properties become strongly intertwined, opening new opportunities for spintronic [3,4] applications [5][6][7][8][9][10].To build functional nanostructures, such as quantum dots (QDs) or quantum point contacts, additional confinement of the Dirac electrons is needed. However, conventional electrostatic confinement in a massless Dirac system is ineffective due to Klein (interband) tunneling. In graphene this problem could be overcome by either mechanically cutting or etching QD islands out of graphene flakes [11][12][13] or by inducing a gap by an underlying substrate, which breaks the pseudospin symmetry [14,15]. Another promising idea to overcome the restrictions given by Klein tunneling is to use graphene strips or nanoribbons. An electrostatic confinement in such a system has been proposed in Ref. [16] by employing the transversal electron motion. Moreover, an effective spin exchange coupling of two gate-defined quantum dots becomes possible in a graphene nanoribbon by indirectly coupling the dots via the tunneling to a common continuum of delocalized states [17].In TIs a mass gap can be created by breaking the TR symmetry at the surface by applying a magnetic field. This could be achieved by proximity to a magnetic material [18,19], or by coating the surface randomly with magnetic impurities [20][21][22]. By modifying the magnetic texture of the deposited magnetic film, a spatially inhomogeneous mass term is induced, opening the possibility to define quantum dot (QD) regions [23], or waveguides formed along the magnetic domain wall regions [24]. Another interesting, possibly more feasible way of defining confinement regions, is to ind...

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Section: ∂ψ(Rt) ∂T = H D ψ(Rt)mentioning

confidence: 99%

Section: ∂ψ(Rt) ∂T = H D ψ(Rt)mentioning

confidence: 99%

mentioning

confidence: 99%

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Gate-defined coupled quantum dots in topological insulators

2014

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“…Particular to the FD approach special care must be taken so as to avoid the Fermion-doubling problem [24]. Elimination of spurious modes introduced to the solution may be accomplished by means of nonlocal approximation for the spatial derivative operator [32,38] or by staggered-grid schemes [12,13]. As remarked upon in [13] the issue of spurious modes is not particular to the Dirac equation but can occur whenever a symmetric FD approximant is used for a first derivative on a uniform grid.…”

Section: Introductionmentioning

confidence: 99%

“…This will allow for a test of the spin-weighted spectral transformations we present in the context of evolution equations. Common techniques for treating the numerical problem of the Dirac equation consist of: FD schemes formulated on a flat-lattice in configuration space [12,13], on a grid within a finite-volume in momentumspace [22] and using methods based on the split-step operator technique [7,8,20,21]. Particular to the FD approach special care must be taken so as to avoid the Fermion-doubling problem [24].…”

Section: Introductionmentioning

confidence: 99%

A spectral method for half-integer spin fields based on spin-weighted spherical harmonics

Beyer

Daszuta

Frauendiener

2015

Class. Quantum Grav.

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High‐order implicit staggered‐grid finite differences methods for the acoustic wave equation

Zapata

Balam

Montalvo-Urquizo

2017

Numerical Methods Partial

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Motivated by the idea that staggered‐grid methods give a greater stability and give energy conservation, this article presents a new family of high‐order implicit staggered‐grid finite difference methods with any order of accuracy to approximate partial differential equations involving second‐order derivatives. In particular, we numerically analyze our new methods for the solution of the one‐dimensional acoustic wave equation. The implicit formulation is based on the plane wave theory and the Taylor series expansion and only involves the solution of tridiagonal matrix equations resulting in an attractive method with higher order of accuracy but nearly the same computation cost as those of explicit formulation. The order of accuracy of the proposal staggered formulas are similar to the methods with conventional grids for a ( 2 M + 2 ) ‐point operator: the explicit formula is ( 2 M ) th‐order and the implicit formula is ( 2 M + 2 ) th‐order; however, the results demonstrate that new staggered methods are superior in terms of stability properties to the classical methods in the context of solving wave equations.

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Staggered grid leap-frog scheme for the Dirac equation

Cited by 26 publications

References 43 publications

Gate-defined coupled quantum dots in topological insulators

Gate-defined coupled quantum dots in topological insulators

A spectral method for half-integer spin fields based on spin-weighted spherical harmonics

High‐order implicit staggered‐grid finite differences methods for the acoustic wave equation

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