Visualizing Landau Levels of Dirac Electrons in a One-Dimensional Potential

Using scanning tunneling spectroscopy, we study a 3D topological insulator ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ with a periodic structural deformation (buckling). The buckled surface allows us to measure the response of Dirac electrons in a magnetic field to the presence of a well-defined potential variation. We find that while the $n=0$ Landau level exhibits a 12 meV energy shift across the buckled structure at 7 T, the amplitude of this shift changes with the Landau level index. Modeling these effects reveals that the Landau level behavior encodes information on the spatial extent of their wave functions. Our findings have important implications for transport and magnetoresistance measurements in Dirac materials with engineered potential landscapes.

Figure 1

Topography of flat and striped regions. (a) Topography of a flat region; (b) 300 nm topography of striped region in the same sample obtained with a $+200\mathrm{mV}$ bias voltage. The stripes are aligned to within $5°$ of one of the high symmetry directions of the crystal and remain contiguous over micron length scales. Both striped and flat (no topographic stripes) regions exist within each sample. The stripes occupy $5–10\mu \mathrm{m}$-sized patches separated by the flat areas. The small sharp spikes in the striped topography are the naturally occurring impurities in the sample. (c) Line cut across stripes showing one-dimensional topographic modulations with $\sim 1000\AA$ periodicity and $\sim 1\AA$ height. Stripes were observed on multiple samples with multiple tips.

Figure 2

Landau levels in flat regions. (a) $dI/dV$ spectrum (see inset for full spectrum) at 7 T with a polynomial fit to the background; (b) $dI/dV$ spectrum with the background subtracted. (c) Intensity plot of background subtracted $dI/dV$ spectra across a flat surface away from impurities. (d) Peak positions of the Landau levels shown in (b).

Figure 3

Variation of Landau level energy across stripes. (a) and (b) Topography and corresponding topographic height variation (solid line, purple in online version) across the stripes along the line from point a to point b. Also shown is the Landau level energy variation across the same linecut for the $n=0$ (gray circles) and $n=1$ (black squares) levels. (c) Intensity plot of background subtracted $dI/dV$ spectra as we traverse the stripe from point (a) to point (b). (d) Comparison of LLs at the crest (point B) and the valley (point D) of the stripes, showing the relative shift in the LL energies. The dotted lines refer to the peak positions plotted in (e). (e) Peak positions of the LLs shown in (d).

Figure 4

Simulation of Landau level energy across stripes. (a) Simulation parameters and model for results shown in (d). $V_0(x)$ is the sinusoidal potential variation superimposed on the Dirac electrons. (b) Intensity plot of change in size of LL wave function represented by $r(n,B)=⟨\sqrt{x^2}⟩$ with magnetic field and LL index (c) Wave function squared for Dirac electrons in a magnetic field at 7T superimposed on periodic stripes of 100 nm periodicity showing the relative length scales. (d) Simulated intensity plot of $dI/dV$ spectra across the stripes using the model described in the Supplemental Material [27].