Perfectly matched layers for the Dirac equation in general electromagnetic texture

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.

Figure 1

Energy dispersion for the PDE under uniform PML for $m=0.02$ eV and $\Re \{\hslash \omega -V\}\le 0, V_o=0.015$ eV: (a) real part, (b) imaginary part; $V_o=0.15$ eV: (c) real part, (d) imaginary part.

Figure 2

Energy dispersion $E=\hslash \omega$ of the VinH scheme for $V=0.04$ eV: (a) $m=0$ eV; (b) $m=0.02$ eV; energy difference to the GI scheme, ${\hslash \omega |}_{V=0}{+V-\hslash \omega |}_V$, for (c) $m=0$ eV, (d) $m=0.02$ eV.

Figure 3

Energy dispersion for the GIEXP scheme under uniform PML for $m=0.02$ eV and $\Re \{\hslash \omega -V\}\ge 0, V_o=0.15$ eV: PML-induced energy difference for ${\mathrm{\Delta }}_t$ from Eq. (55): (a) real part, (b) imaginary part; energy dispersion with PML and ${\mathrm{\Delta }}_t={\mathrm{\Delta }}_t^{\text{CFL}}$: (c) real part, (d) imaginary part.

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 $V_o=0.08$ eV, (b) scheme GIEX for $V_o=0.8$ eV, (c) scheme GICN for $V_o=0.08$ eV, and (d) scheme GICN for $V_o=0.8$ eV. Note the logarithmic scale along the $y$ axis.

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 $V_o=0.08$ eV, (b) scheme VinHEX for $V_o=0.8$ eV, (c) scheme VinHCN for $V_o=0.08$ eV, and (d) scheme VinHCN for $V_o=0.8$ eV. Note the logarithmic scale along the $y$ axis.

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 $W=6(\mathrm{\Delta }/2), V=0.05$ eV, and $V_o=0.5$ eV. Results within IPM (red dot-dashed lines labeled $V_{\mathrm{im}}$) are given for comparison.

Figure 7

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

Figure 8

Scattering at a static Klein step of height $V_K=0.05$ eV, width $W_K=\mathrm{\Delta }$: (a) electric potential profile; (b) time-evolution of functional Eq. (23) under GIEXP for $V_o=0.75$ eV (solid blue line) and $V_o=0.5$ eV (dashed red line), and under IPM for $V_o=1$ eV (dot-dashed black line). Wave packet evolution under GIEXP ($V_o=0.5$ eV): shortly after impact (c), at final time (d).

Figure 9

Central collision of a Dirac fermion with a planar magnetic vortex (${\mathrm{\Phi }}_o=0$): (a) time-evolution of functional Eq. (23) under GIEXP for $V_o=0.75$ eV (solid blue line) and $V_o=0.5$ eV (dashed red line), and under IPM for $V_o=1$ 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).

Figure 10

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

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.