Dynamic impurities in two-dimensional topological-insulator edge states

Helical edge states of two-dimensional topological insulators show a gap in the density of states (DOS) and suppressed conductance in the presence of ordered magnetic impurities. Here we will consider the dynamical effects on the DOS and transmission when the magnetic impurities are driven periodically. Using the Floquet formalism and Green's functions, the system properties are studied as a function of the driving frequency and the potential energy contribution of the impurities. We see that increasing the potential part closes the DOS gap for all driving regimes. The transmission gap is also closed, showing a pronounced asymmetry as a function of energy. These features indicate that the dynamical transport properties could yield valuable information about the magnetic impurities.

Figure 1

Schematic drawing of the impurity system of length $L$ with $N_{\mathrm{imp}}$ impurities and a detailed view of a single impurity with indication of the dynamics of the magnetic moment. $\mathit{M}\left(t\right)$ is the total magnetic moment of the impurity and $M_s$ is the ratio between the total magnetic moment and the static part pointing along the edge.

Figure 2

DOS and transmission $T$ as a function of energy for $M_s=1.0$ (fully static impurities) and different ratios of potential and magnetic part $V/M$, $\left|M\right|/\hslash {\nu }_F=0.2$, and $L/l_{\mathrm{\Delta }}=8.0$ averaged over $N_{\mathrm{runs}}=1000$ impurity configurations.

Figure 3

DOS and transmission $T$ as a function of energy for different ratios of potential and magnetic part $V/M$ and $\hslash \mathrm{\Omega }/\mathrm{\Delta }=25$, $n_{\max }=2$, $M_S=0.0$ (blue), $M_S=0.1$ (orange), $M_S=0.5$ (green), i.e., 10%/50% static part, $\left|M\right|/\hslash {\nu }_F=0.2$, and $L/l_{\mathrm{\Delta }}=8.0$ averaged over $N_{\mathrm{runs}}=1000$ impurity configurations.

Figure 4

DOS and transmission $T$ as a function of energy for different ratios of potential and magnetic part $V/M$ and $\hslash \mathrm{\Omega }/\mathrm{\Delta }=2$, $n_{\max }=6$, $M_S=0.0$ (blue), $M_S=0.1$ (orange), $M_S=0.5$ (green), i.e., 10%/50% static part, $\left|M\right|/\hslash {\nu }_F=0.2$, and $L/l_{\mathrm{\Delta }}=8.0$ averaged over $N_{\mathrm{runs}}=1000$ impurity configurations.

Figure 5

DOS and transmission $T$ as a function of energy for different ratios of potential and magnetic part $V/M$ and $\hslash \mathrm{\Omega }/M=0.2$, $n_{\max }=20$, $M_S=0.0$ (blue), $M_S=0.1$ (orange), $M_S=0.5$ (green), i.e., 10%/50% static part, $\left|M\right|/\hslash {\nu }_F=0.2$, and $L/l_{\mathrm{\Delta }}=8.0$ averaged over $N_{\mathrm{runs}}=1000$ impurity configurations.

Figure 6

Transmission $T$ as a function of energy for $V/M=1.0$ and $\hslash \mathrm{\Omega }/M=0.2$, $n_{\max }=20$, $M_S=0.0$ (blue), and $M_S=0.5$ (green) compared to the static case $M_S=1.0$ (black). $\left|M\right|/\hslash {\nu }_F=0.2$ and $L/l_{\mathrm{\Delta }}=8.0$ averaged over $N_{\mathrm{runs}}=1000$ impurity configurations.

Figure 7

DOS and transmission $T$ as a function of energy for single impurity configurations with $\hslash \mathrm{\Omega }/M=2.0$, $n_{\max }=6$, $M_S=0.0$, $\left|M\right|/\hslash {\nu }_F=0.2$, and $L/l_{\mathrm{\Delta }}=8.0$.

Figure 8

Right panel: band structure calculated directly from the Floquet-Hamiltonian (red and gray dashed lines) and directly by keeping the first two Fourier components (heat map) for a homogeneous dynamic magnetic barrier. Left panel: density of states via integration of the heat map. Floquet parameters: $\hslash \mathrm{\Omega }/\mathrm{\Delta }=0.9$, $n_{\max }=2$ (red), and $n_{\max }=10$ (gray).