Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                

Perfectly matched layers for the Dirac equation in general electromagnetic texture

Walter Pötz
Phys. Rev. E 103, 013301 – Published 7 January 2021
PDFHTMLExport Citation
Abstract
Authors
Article Text
  • INTRODUCTION
  • PML FOR THE CONTINUUM DIRAC EQUATION…
  • PML FOR THE FINITE-DIFFERENCE SCHEME
  • SPECTRAL PROPERTIES UNDER CONSTANT…
  • NUMERICAL TESTS
  • SUMMARY AND CONCLUSIONS
    • References

    Abstract

    Perfectly matched layer (PML) boundary conditions are constructed for the Dirac equation and general electromagnetic potentials. A PML extension is performed for the partial differential equation and two versions of a staggered-grid single-cone finite-difference scheme. For the latter, PML auxiliary functions are computed either within a Crank-Nicholson scheme or one derived from the formal continuum solution in integral form. Stability conditions are found to be more stringent than for the original scheme. Spectral properties under spatially uniform PML confirm damping of any out-propagating wave contributions. Numerical tests deal with static and time-dependent electromagnetic textures in the boundary regions for parameters characteristic for topological insulator surfaces. When compared to the alternative imaginary-potential method, PML offers vastly improved wave absorption owing to a more efficient suppression of back-reflection. Remarkably, this holds for time-dependent textures as well, making PML a useful approach for transient transport simulations of Dirac fermion systems.

      • Received 24 September 2020
      • Accepted 18 December 2020

      DOI:https://doi.org/10.1103/PhysRevE.103.013301

      ©2021 American Physical Society

      Physics Subject Headings (PhySH)

      1. Research Areas
      Ballistic transportQuantum transportSpintronicsTopological materials
      Condensed Matter, Materials & Applied Physics

      Authors & Affiliations

      Walter Pötz

      • Institut für Physik, Karl-Franzens-Universität Graz, Universitätsplatz 5, 8010 Graz, Austria

      Article Text (Subscription Required)

      Click to Expand

      References (Subscription Required)

      Click to Expand
      Issue

      Vol. 103, Iss. 1 — January 2021

      Reuse & Permissions
      Access Options
      Author publication services for translation and copyediting assistance advertisement

      Authorization Required

      Log In
      Other Options
      ×

      Images

      • Figure 1

        Energy dispersion for the PDE under uniform PML for m=0.02 eV and ℜ{ℏω−V}≤0, Vo=0.015 eV: (a) real part, (b) imaginary part; Vo=0.15 eV: (c) real part, (d) imaginary part.

        Reuse & Permissions
      • Figure 2

        Energy dispersion E=ℏω of the VinH scheme for V=0.04 eV: (a) m=0 eV; (b) m=0.02 eV; energy difference to the GI scheme, ℏω|V=0+V−ℏω|V, for (c) m=0 eV, (d) m=0.02 eV.

        Reuse & Permissions
      • Figure 3
        Figure 3

        Energy dispersion for the GIEXP scheme under uniform PML for m=0.02 eV and ℜ{ℏω−V}≥0, Vo=0.15 eV: PML-induced energy difference for Δt from Eq. (55): (a) real part, (b) imaginary part; energy dispersion with PML and Δt=ΔtCFL: (c) real part, (d) imaginary part.

        Reuse & Permissions
      • Figure 4

        Time-evolution of the functional Eq. (23) (solid blue lines) and lattice trace (dashed red lines) under uniform PML: (a) scheme GIEX for Vo=0.08 eV, (b) scheme GIEX for Vo=0.8 eV, (c) scheme GICN for Vo=0.08 eV, and (d) scheme GICN for Vo=0.8 eV. Note the logarithmic scale along the y axis.

        Reuse & Permissions
      • Figure 5

        Time-evolution of the functional Eq. (23) (solid blue lines) and lattice trace (dashed red lines) under uniform PML: (a) scheme VinHEX for Vo=0.08 eV, (b) scheme VinHEX for Vo=0.8 eV, (c) scheme VinHCN for Vo=0.08 eV, and (d) scheme VinHCN for Vo=0.8 eV. Note the logarithmic scale along the y axis.

        Reuse & Permissions
      • Figure 6

        Comparison of PML scheme performance for free-particle propagation (m=0.02 eV) on a domain of constant scalar potential V=0.05 eV and picture frame PML Eqs. (56) and (57) for W=6(Δ/2), V=0.05 eV, and Vo=0.5 eV. Results within IPM (red dot-dashed lines labeled Vim) are given for comparison.

        Reuse & Permissions
      • Figure 7

        (a) Imaginary potential corresponding to picture frame PML Eqs. (56) and (57) for W=2(Δ/2), V=0.8 eV. (b) Time evolution under scheme GIEXP and time step ΔtEXP under variation of frame width W and Vo: dotted lines: W=2(Δ/2), dashed lines: W=4(Δ/2), solid lines: W=8(Δ/2); Vo=0.2 eV: blue lines ending after 1132 time steps, Vo=0.8 eV: red lines ending after 1867 time steps. IPM simulations for W=8(Δ/2) and Vo=0.8 eV: black dot-dashed lines.

        Reuse & Permissions
      • Figure 8

        Scattering at a static Klein step of height VK=0.05 eV, width WK=Δ: (a) electric potential profile; (b) time-evolution of functional Eq. (23) under GIEXP for Vo=0.75 eV (solid blue line) and Vo=0.5 eV (dashed red line), and under IPM for Vo=1 eV (dot-dashed black line). Wave packet evolution under GIEXP (Vo=0.5 eV): shortly after impact (c), at final time (d).

        Reuse & Permissions
      • Figure 9

        Central collision of a Dirac fermion with a planar magnetic vortex (Φo=0): (a) time-evolution of functional Eq. (23) under GIEXP for Vo=0.75 eV (solid blue line) and Vo=0.5 eV (dashed red line), and under IPM for Vo=1 eV (dot-dashed black line); wave packet prior to impact upon the boundaries (b), remnant wave packet under GIEXP (c), and remnant wave packet under IPM (d).

        Reuse & Permissions
      • Figure 10

        Dynamic Klein step under the GIEXP and IPM scheme. Simulation parameters are given in the main text.

        Reuse & Permissions
      • Figure 11

        Propagation of a Dirac fermion in see-saw potentials: (a) and (c) for zero magnetization; (b) and (d) for a hedgehog planar magnetic vortex.

        Reuse & Permissions
      ×

      Sign up to receive regular email alerts from Physical Review E

      Log In

      Cancel
      ×

      Search

      Article Lookup

      Paste a citation or DOI
      Enter a citation
      ×