Probing the Penetration Depth of Topological Surface States by Magnetic Impurity Scattering in V-doped Sb$_2$Te$_3$

Abstract

Topological insulators host Dirac surface states (SS) protected by time-reversal symmetry. Inter-surface hybridization can gap the SS and give rise to the quantum spin Hall effect in films that are sufficiently thin compared to the SS penetration depth. However, quantifying the SS penetration depth typically requires painstaking synthesis of multiple films with varying thickness. Here we introduce a direct method to probe the SS penetration depth in bulk crystals, by studying the interplay between SS and magnetic impurities in \SVT. Using scanning tunneling microscopy and spectroscopy, we find that even sparse magnetic impurities ($\lesssim0.25\%$ vanadium) can gap the Dirac SS. However, a single V impurity induces only localized states, and does not form an impurity band, so the gapped Dirac dispersion is preserved away from the impurity. In high magnetic fields, we observe an energy shift of the $0^\text{th}$ Landau level and a suppression of quasiparticle lifetime at the Dirac point, indicating \newtext{magnetic} scattering of the SS. Crucially, by employing V impurities at different depths as precise scattering probes, we reveal the SS penetration depth on the sub-nanometer scale in a bulk crystal.

Figures (5)

• Figure 1: Crystal structure and topography of (V$_x$Sb$_{1-x}$)$_2$Te$_3$ with $x \approx 0.23\pm 0.03\%$. (a) Crystal structure of Sb$_2$Te$_3$, with Type I and II vanadium impurities colored in red and orange, respectively. The lattice constants, determined by x-ray diffraction at 10 K, are $a=b=4.2547(3)$ Å and $c=30.268(3)$ Å, corresponding to 3 QLs MansourJAP2014. (b) A $30\times30$ nm$^2$ STM topographic image of a cleaved Te-terminated surface, with typical Type I and II impurities boxed in red and orange, respectively. (c) Topographic images of a single Type I impurity at positive and negative sample biases. (d) Topographic images of a single Type II impurity at positive and negative sample biases. STM setpoints: $V_{\mathrm{sample}} = 300$ mV, $I_{\mathrm{set}} = 100$ pA in (b) and the top row of (c, d); $V_{\mathrm{sample}} = -300$ mV, $I_{\mathrm{set}} = 100$ pA in the bottom row of (c, d).
• Figure 2: Spectroscopic imaging of surface and impurity states. (a) A typical differential conductance $dI/dV$ spectrum obtained 3 nm away from the nearest V impurity. The bulk band edges are identified from the local maxima of the second derivative, and are shaded in purple (valence band) and orange (conduction band). (b) Zoomed view of the energy window within the bulk gap (dashed box in (a)), showing a mass gap $2\Delta$ at the DP and approximately linear dispersions on both sides. Dashed lines and arrow indicate the gap size and serve as guides to the eye. (c) DFT-calculated band structure of a pristine Sb$_2$Te$_3$ slab with 10 QLs. The marker size is proportional to the spectral weight projected onto the top QL atoms. The surface Dirac cone at $\Gamma$ connects the bulk valence and conduction bands. Here, purple and orange shading denotes the same energy intervals below and above the DP where measured bulk bands onset in (a). The bands next to the Dirac cones between the shaded regions do not show up in the measured $dI/dV$ due to their low surface spectral weight. (d) Average $dI/dV$ spectra over a clean area (black), Type I impurity (red), and Type II impurity (orange). Setpoints: $V_{\mathrm{sample}} = -100$ mV, $I_{\mathrm{set}} = 500$ pA. Lock-in zero-to-peak amplitude $V_{\mathrm{exc}} = 2$ mV in (a,b) and $V_{\mathrm{exc}} = 5$ mV in (d).
• Figure 3: Evidence of exchange scattering of Dirac SS. (a) $dI/dV$ measurement of Landau quantization in magnetic field $B$ up to 8 T, applied perpendicular to the surface. Spectra at each field are measured at different sample locations but all at least 3 nm away from the nearest V impurity. The curves are shifted vertically for clarity. The gray dashed line indicates the DP energy $E_{\mathrm{D}}$, which would be the field-independent $0^\text{th}$ LL energy without mass acquisition. A clear shift of the $0^\text{th}$ LL towards higher energy signals a mass gap opening in the Dirac SS. (b) Extracted LL energies plotted as a function of $\sqrt{nB}$ for spectra from 5--8 T. The dashed line shows a linear fit to the $|n|\geq 2$ LLs at high magnetic fields, which we use to determine $E_{\mathrm{D}}$. The energy offset of the zeroth LL relative to this linear trend yields the mass term $\Delta$. (c) FWHM of the LL peaks extracted from the 5--8 T data in (a). We obtain the LL peak positions and widths by fitting the spectra, after background removal, with a sum of Lorentzian functions. The gray dashed line indicates $E_{\mathrm{D}}$, same as in (a). Setpoint: $V_{\mathrm{sample}} = -100$ mV, $I_{\mathrm{set}}= 2$ nA, lock-in zero-to-peak amplitude $V_{\mathrm{exc}} = 3$ mV, which is smaller than the fitted peak width.
• Figure 4: Probing SS depth from LL suppression around V impurities. (a, b) LL spectra at 8 T acquired along line cuts passing through Type I (a) and Type II (b) impurities, with the background subtracted. The red and orange arrows mark the impurity centers. Top panels show the corresponding topographic images, where the white dashed arrows denote the direction of the line cuts and the colored dots indicate the impurity centers. (c) DFT-calculated real-space distribution of the averaged surface-state wave-function density (WFD) on the top 7 Sb layers (top 4 QLs). Horizontal dashed lines indicate the Sb layer index measured from the surface. The color gradient in the background indicates the decay of SS. Setpoint in (a, b): $V_{\mathrm{sample}} = -100$ mV, $I_{\mathrm{set}} = 400$ pA, lock-in zero-to-peak amplitude $V_{\mathrm{exc}} = 3$ mV. STM setpoint for the topographic insets in (a, b): $V_{\mathrm{sample}} = 200$ mV, $I_{\mathrm{set}} = 50$ pA, $B=8$ T.
• Figure S1: V impurity concentration determination. (a) The same $30 \times 30$ nm$^2$ STM topography as Fig. 1(b) in the main text. Red and orange circles mark Type I and Type II impurities, respectively. We identify 11 Type I defects (0.198%) and 9 Type II defects (0.162%). (b, c) STM topographies acquired over a $40 \times 40$ nm$^2$ region. We identify 30 Type I defects (0.3%) and 20 Type II defects (0.2%). STM setpoints: $V_{\mathrm{sample}} = +300$ mV, $I_{\mathrm{set}} = 100$ pA for (a); $V_{\mathrm{sample}} = +300$ mV, $I_{\mathrm{set}} = 30$ pA for (b); and $V_{\mathrm{sample}} = -300$ mV, $I_{\mathrm{set}} = 30$ pA for (c).