Gap formation in helical edge states with magnetic impurities

Helical edge states appear at the surface of two-dimensional topological insulators and are characterized by spin up traveling in one direction and the spin down traveling in the opposite direction. Such states are protected by time-reversal symmetry and no backscattering due to scalar impurities can occur. However, magnetic impurities break time-reversal symmetry and lead to backscattering. Often their presence is unintentional, but in some cases they are introduced into the sample to open up gaps in the spectrum. We investigate the influence of random impurities on helical edge states, specifically how the gap behaves in the realistic case of impurities having both a magnetic and a scalar component. It turns out that for a fixed magnetic contribution the gap closes when either the scalar component is increased, or Fermi velocity is decreased. We compare diagrammatic techniques in the self-consistent Born approximation to numerical calculations, which yields good agreement. For experimentally relevant parameters we find that even moderate scalar components can be quite detrimental for the gap formation.

Figure 1

(a) A schematic showing the edge state that appears when the bulk TI sample is cut. For sample widths much greater than $\lambda$ the edge states are decoupled. The red crosses represent magnetic impurities distributed evenly throughout the bulk. (b) For the numerical calculations $N_{\mathrm{imp}}$ magnetic impurities are placed randomly along a length $L$. The length $L$ is chosen such that tunneling through the impurity region is suppressed, i.e., a real gap can develop.

Figure 2

Diagrammatic representation of the full Born approximation.

Figure 3

The diagrams constituting the self-consistent Born approximation. Note that the second line is the same as in Fig. 2 with full Green's functions instead of bare ones.

Figure 4

The DOS $\mathcal{D}$ as a function of energy with $x_M=0.05$ in the absence of scalar scattering ($V=0$) for different approximations discussed in the text. The DOS obtained numerically by impurity averaging shows oscillations related to finite size effects, which are absent for the SCBA and BA. The inset shows a zoom of the three curves into the gap region; note that the BA fails to show a hard gap.

Figure 5

The density of states $\mathcal{D}$ [within SCBA (red) and numerical approach for $L=8\hslash v_F/\mathrm{\Delta }$ (blue) and $L=6\hslash v_F/\mathrm{\Delta }$ (green)] shown for $x_M=0.05$ and for four different values $V/M=0$, 1, 2, and 4. The energy is shifted by $V$ to align the gap centers. The gap closes at around $V/M=4$. The black dashed line indicates where $\mathcal{D}=0.5$, which is used to determine the gap from the numerics.

Figure 6

The gap size plotted as function of $V$ for three different values of $x_M=0.02$, 0.05, and 0.1. The solid lines show the SCBA result and the open circles are the numerical results. The SCBA shows that the gap cap closes at $V/M\approx 3$, 4.3, and 6.7 for $x_M=0.1$, 0.05, and 0.02, respectively.