Mapping the effect of defect-induced strain disorder on the Dirac states of topological insulators

We provide a detailed microscopic characterization of the influence of defect-induced disorder on the Dirac spectrum of three-dimensional topological insulators. By spatially resolved Landau-level spectroscopy measurements, we reveal the existence of nanoscale fluctuations of both the Dirac point energy as well as of the Dirac-fermion velocity which is found to spatially change in opposite direction for electrons and holes, respectively. These results evidence a scenario which goes beyond the existing picture based on chemical potential fluctuations. The findings are consistently explained by considering the microscopic effects of local stain introduced by defects, which our model calculations show to effectively couple to topological states, reshaping their Dirac-like dispersion over a large energy range. In particular, our results indicate that the presence of microscopic spatially varying stain, inevitably present in crystals because of the random distribution of defects, effectively couple to topological states and should be carefully considered for correctly describing the effects of disorder.

Figure 1

(a) Scanning tunneling microscopy image acquired in the constant-current mode on ${\mathrm{Sb}}_2{\mathrm{Te}}_3$. Several defects introduced during the growth process are visible onto the surface. (b) Schematic illustration of the Landau levels condensation of topological states under high magnetic fields. (c) Scanning tunneling spectroscopy data acquired over two different sample regions. Green and blue lines correspond to the positions marked by circles in (a).

Figure 2

(a) Dirac point mapping with respect to the Fermi level obtained over the sample region displayed in Fig. 1. (b) Landau level positions for points taken along the trace identified by circles in (a). The results obtained over each point have been vertically shifted to improve clarity. (c) Dirac-fermion velocity for electron (circle) and hole (square) branches of the Dirac cone obtained by fitting the data presented in (b). The gray triangles report the average velocity, which stays constant for all positions. The bottom panel quantifies the position of the Dirac point with respect to the Fermi level. A comparison between upper and lower panels reveals that the electron-hole asymmetry increases once the Dirac point moves closer to the Fermi level. (d) Schematic illustration of the Dirac point fluctuations and electron-hole symmetry breaking.

Figure 3

Landau level energies as a function of the momenta $k_n$ defined in the text, measured at three different magnetic fields at the eleven spatial positions depicted in Fig. 2. The energies have been shifted upwards for clarity and the position of the zeroth LL has been removed.

Figure 4

Estimated values of the parameters $\overline{D}$ (upper panel) and ${\overline{v}}_F$ (middle) for the various positions in the sample shown in Fig. 2. The values are extracted from Eqs. (8) and shown for magnetic fields $B=6$, 9, 12 T. Despite the dispersion in the values of the parameter $D$, the coincidence of the trend for the different values of the magnetic field constitutes a very nontrivial test of the strain model. The bottom panel shows the variation of the Fermi energy at the different positions of the sample for different values of the magnetic field. The coincidence is remarkable.