Scattering properties of the three-dimensional topological insulator Sb${}_2$Te${}_3$: Coexistence of topologically trivial and non-trivial surface states with opposite spin-momentum helicity

The discovery of topological insulators (TIs) König et al. (2007); Hsieh et al. (2008); Chen et al. (2009) led to intense research efforts towards the potential utilization of these materials in spintronic, magneto-electric, or quantum computation devices Fu and Kane (2008); Garate and Franz (2010); Pesin and MacDonald (2012); Mellnik et al. (2014). In particular, the existence of a linearly dispersing, gapless surface state is of enormous interest, because—contrary to the trivial surface states usually found at surfaces of metals and semiconductors—it cannot be destroyed by the presence of defects and adsorbates as long as time-reversal symmetry is preserved Chen et al. (2010); Okada et al. (2011); Wray et al. (2011); Valla et al. (2012); Scholz et al. (2012); Xu et al. (2012); Sessi et al. (2014). The strong spin-orbit coupling inherent to these systems perpendicularly locks the spin to the momentum, creates a helical spin texture which forbids backscattering Roushan et al. (2009); Zhang et al. (2009), and results in spin currents that are intrinsically tied to charge currents König et al. (2007).

Among the three-dimensional (3D) TIs, the binary chalcogenides Bi${}_2$Te${}_3$ Chen et al. (2009) and Bi${}_2$Se${}_3$ Xia et al. (2009) represent the most widely studied compounds. In contrast, the electronic properties of Sb${}_2$Te${}_3$ Jiang et al. (2012a) which belongs to the same material class have been studied to a much lesser extent. This deficiency can directly be linked to the intrinsic strong $p$-doping characterizing the compound, which leads to a Dirac point that lies well above the Fermi level Seibel et al. (2015a, b). As a result relevant parts of the surface electronic band structure are inaccessible to angle-resolved photoemission spectroscopy (ARPES), a technique that played a key role towards the identification and study of topologically non-trivial states of matter.

Another technique frequently applied to TIs, which gives access to the scattering properties of surfaces not only for occupied states below but also for empty electronic states above the Fermi level, is quasiparticle interference (QPI) imaged with a scanning tunneling microscope (STM) Roushan et al. (2009); Zhang et al. (2009); Alpichshev et al. (2010); Okada et al. (2011); Sessi et al. (2014). However, the observation of QPI signals in TIs depends on the deformation of the Dirac cone which is described by the so-called warping term Fu (2009). This contribution to the Hamilton operator leads to an effective nesting of equipotential surfaces and to the development of out-of-plane spin polarization components, which both result in new scattering channels and a higher QPI signal strength. Unfortunately, the warping term of Sb${}_2$Te${}_3$ is very weak as compared to Bi${}_2$Te${}_3$ Zhang et al. (2009); Alpichshev et al. (2010); Sessi et al. (2013). This together with the above mentioned $p$-doping, which requires to work at energies far above the Fermi level and results in strongly reduced lifetimes and rapidly damped standing wave patterns, makes scattering experiments on Sb${}_2$Te${}_3$ highly challenging.

This deficiency is the more annoying as the quantum anomalous Hall effect has recently been observed in magnetically doped Sb${}_2$Te${}_3$ Chang et al. (2013, 2015). Therefore, a comparison of the scattering properties with pristine Sb${}_2$Te${}_3$ would be highly interesting to identify the correlation between dissipation-less quantized transport and the onset of ferromagnetism in a topologically non-trivial material. Here, we close this gap by QPI of $p$-doped Sb${}_2$Te${}_3$ which contains defects that effectively scatter the surface state. In addition to the linearly dispersing spin-momentum–locked Dirac states, our data evidence the existence of another surface-related electronic feature with strongly directional scattering properties. Comparison with ab-initio calculations reveals that the latter originates from a trivial surface resonance which also possesses a helical spin texture suppressing backscattering. Interestingly, the helical spin texture of the Dirac state and the trivial surface resonance are opposite. This unique feature may allow for the independent tuning of spin and charge currents, for example by tuning the potentials by means of gating.

The Sb${}_2$Te${}_3$ single crystal was grown by the modified vertical Bridgman method with rotating heat field Kokh et al. (2007). Stoichiometric amounts of Sb and Te were loaded to a carbon-coated quartz ampoule. After evacuation to ${10}^{-4}$ torr the ampoule translation rate and axial temperature gradient were set to 10 mm/day and $\approx {15}^{\circ }$/cm, respectively. Single crystals of about 10 mm in diameter and 60 mm in length were obtained as shown in Fig. 1(a).

After growth, crystals have been cut in sizes suitable for STM experiments, cleaved at room temperature in ultra-high vacuum, and immediately inserted into a cryogenic STM. Because of the Sb${}_2$Te${}_3$ structure depicted in Fig. 1(b), which consist of quintuple layers that are weakly bound by van der Waals forces, the surface is always Te-terminated. Spectroscopic data have been obtained by low-temperature scanning tunneling microscopy at $T=\mathrm{1.5...4.8}$ K under ultrahigh vacuum conditions ($p\le {10}^{-10}$ mbar) using the lock-in technique ($f=793$ Hz) with a modulation voltage $U_{mod}=\mathrm{10...20}$ meV. Unless otherwise stated experiments were performed at zero magnetic field. While maps of the differential conductance $dI/dU$ have been acquired simultaneously with topographic images in the constant-current mode, scanning tunneling spectroscopy (STS) curves are measured by ramping the bias voltage after deactivation of the feedback-loop, i.e. at constant tip–sample separation.

Figure 1(c) shows a topographic image of the Sb${}_2$Te${}_3$ surface with defects typical for binary chalcogenides Jiang et al. (2012b). To our experience the existence of the defect type indicated by an arrow, which probably corresponds to a Te surface vacancy, is crucial for the formation of a QPI pattern. We speculate that this may be related to the fact that only this defect exhibits a potential that sufficiently overlaps with the wave function of the topological surface state. The electronic structure is investigated by analyzing the local density of states as probed by STS. The bottom curve of Fig. 1(d) shows a typical STS spectrum obtained on pristine Sb${}_2$Te${}_3$ by positioning the tip away from defects (blue line). The minimum corresponds to the position of the Dirac point Jiang et al. (2012a, b) which is found at about 170 meV above the Fermi level, i.e. in the empty electronic states, confirming the intrinsic $p$-doping characterizing the material Seibel et al. (2015a). This assignment is also corroborated by the position of the zeroth-order Landau level measured at a magnetic field $B=10$ T (red line).

To investigate the scattering properties of Sb${}_2$Te${}_3$, we performed energy-dependent QPI experiments. QPI mapping makes use of the standing-wave pattern generated by elastic scattering of electronic states at surface defects and is known as a powerful method to study scattering mechanisms. While originally applied to noble metal surfaces Crommie et al. (1993), its use has been recently extended to investigate non-degenerate spin-polarized bands, e.g. on surfaces with strong contributions from spin-orbit coupling Pascual et al. (2004); El-Kareh et al. (2013) and topological insulators Roushan et al. (2009); Zhang et al. (2009); Alpichshev et al. (2010); Okada et al. (2011); Sessi et al. (2014). Fourier transformation (FT) translates real-space information into reciprocal space, thereby providing a convenient way to visualize scattering vectors, which correspond to points of a constant energy cut (CEC) connected by nesting vectors Liu et al. (2012).

The shape of equipotential surfaces of TIs and the spin texture are both strongly energy-dependent. In Sb${}_2$Te${}_3$, similar to other TIs hosting a single Dirac cone centered around the $\overline{\Gamma }$-point of the surface Brillouin zone, one finds circular CECs at energies close to the Dirac point supporting only one nesting vector, i.e. backscattering ($q_1$). As schematically represented by red/blue arrows in the upper panel of Fig. 2(a), however, the spin polarization in this energy range is perpendicularly locked to the crystal momentum $k$ and completely in-plane. In this case backscattering is forbidden by time-reversal symmetry which explains the absence of any scattering channel close to the Dirac point.

In contrast, at higher energies the introduction of the warping term progressively deforms the circular CEC, eventually giving rise to a snowflake-like shape, as shown in the lower panel of Fig. 2(a). At the same time the warping leads to an increasing out-of-plane modulation of the spin polarization. Although backscattering remains forbidden, it is well-known from both theoretical Fu (2009) and experimental Roushan et al. (2009); Zhang et al. (2009); Sessi et al. (2013) investigations that the warping term opens new scattering channels along $\overline{\Gamma M}$ directions by connecting next-nearest neighbor concavely warped sides that are centered at the $\overline{K}$ point of the hexagram [schematically represented by the red arrow labeled $q_2$ in Fig. 2(a), lower panel]. This scattering channel can be experimentally visualized by QPI. For example, Fig. 2(b) shows the $dI/dU$ map (upper panel) and the corresponding FT (lower panel) at an energy $E=550$ meV above the Fermi level. The FT exhibits six distinct maxima that represent $q_2$ (one of which is labeled A). With increasing energy it progressively moves away from the center of the FT, thereby reflecting the band dispersion relation of the Dirac cone [Fig. 2(c)].

Careful inspection of the FT of $dI/dU$ maps taken at higher energy reveals the emergence of another, unexpected scattering channel not previously observed on TIs. This scattering channel, which is labeled B in Fig. 2(d), becomes detectable at about 750 meV above the Fermi level. It also shows a well-defined directionality with maxima pointing along the $\overline{\Gamma M}$ direction. Since its length is shorter than $q_2$ we can safely exclude an additional scattering channel involving topological states. While it initially coexists with $q_2$, the new scattering channel completely dominates the FT-QPI signal at higher energies, as shown in Fig. 2(e) at $E=850$ meV.

To identify the electronic states that lead to the appearance of this additional scattering channel on pristine Sb${}_2$Te${}_3$ we have quantitatively analyzed a larger series of FT-QPI maps that is only partially presented in Fig. 2(b)-(e). The result is shown in Fig. 3(a). The linear dispersion relation of signal A (green dots) that represents the topological state is clearly visible. A fit to the data provides a Dirac point at $E_D=(202\pm 17)$ meV above the Fermi level. This value is slightly higher than the one obtained from the STS data presented in Fig. 1(d), a deviation we attribute to the warping term, which introduces a subtle deviation from a linear dispersion relation also visible in the calculations. The slope of the linear dispersion gives a Fermi velocity $v_F=(4.3\pm 0.2)\times {10}^5$ ms${}^{-1}$, in reasonable agreement with previous reports on binary chalcogenides Jiang et al. (2012a); Pauly et al. (2012); Sessi et al. (2014). In contrast, scattering channel B (red dots) appears in a narrow energy and crystal momentum range around $E-E_F=800$ meV and $k_{exp}=1.3$ nm${}^{-1}$ only and cannot be reasonably fit. Correspondingly, the red line serves as a guide to the eye only.

These experimental data have been compared with ab-initio DFT calculations (obtained as described in Ref. [Pauly et al., 2012]). We would like to emphasize, that DFT calculations are always performed for ideal crystals and, therefore, always result in a Fermi level positioned at or near the valence band maximum (VBM) which is close to the Dirac point (DP) here. For this reason the experimentally observed doping-induced shift of the chemical potential cannot be simulated in DFT. Any comparison between the experiment and theory must be performed on a relative level, i.e. by comparing the position of electronic bands with respect to certain reference points. In our experiments the Fermi level is unambiguously defined by the zero bias condition in STS ($U=0$ V). Furthermore, the energetic position of the Dirac point relative to the Fermi level can be determined with an accuracy we estimate to about $\pm 10$ meV. Therefore, we have chosen to shift the theoretical Dirac point energy to the experimental value, i.e. $+170$ meV [cf. Fig. 1(d)].

The band structure obtained from DFT is reported in Fig. 3(b). In good agreement with the experimental data presented in Fig. 3(a), DFT evidences the existence of an additional surface resonance with a charge density centered around $E-E_F=800$ meV, but at a slightly lower value of the crystal momentum, $k_{theo}\approx 1.0$ nm${}^{-1}$. To shed light on the strongly directional character of the scattering events involving from this band, its spin texture has been calculated and compared with the topological state. Results for the spin polarization component along the $x$-direction is reported in Fig. 3(c). Interestingly, the two electronic states under discussion here, which are energetically located at around 0.42 eV and 0.82 eV above the Fermi level, exhibit opposite spin helicities. Furthermore, a detailed analysis of the other directional components of the spin polarization (not shown here) reveals a helical texture with significant out-of-plane components. This is expected for the warped Dirac-cone, but it is also observed for the higher lying state.

To understand this surprising result we analyze the charge density in more detail, following the model of Nagano et al. Nagano et al. (2009). This model unfolded that the major effect of spin-orbit coupling, that determines the spin orientation, originates from a region close to the nucleus where the potential gradient $V\left(r\right)$ is strongest Bihlmayer et al. (2006). The charge density plots shown in Fig. 4 reveal that both states are predominantly localized at subsurface Sb atoms and exhibit $p_z$ character. In spite of their similar shape, the strength of the spin-orbit coupling and the resulting spin polarization of both electronic features is found to sensitively depend on subtle variations of the exact spatial distribution of the charge density. If the state is located on average a bit above the position of the heavy Sb nucleus (i.e. shifted to positive $z$-direction) it will be influenced by regions where ${\partial }_{z}V$ is positive, giving rise to an effective electric field acting on the state. Due to the $\text{boldmath}\sigma \cdot (k\times \nabla V)$ term of the spin-orbit coupling, this favors one particular spin direction for a given $k$ vector leading to a certain helicity of the spin texture. If the state is shifted a bit in the negative $z$-direction, however, the other spin direction (and helicity) will be favored. In the middle panel of Fig. 4, the charge densities of the states at 0.42 eV and 0.82 eV, both in-plane–averaged around the subsurface Bi atoms, are shown as thin black and red lines, respectively. It appears that the state at 0.42 eV is a bit more displaced towards the vacuum. To quantify the effect, we multiply the charge density with the $z$-derivative of an atomic ($1/|r|$) potential and integrate the product (dotted line in the middle panel of Fig. 4) in the $z$-direction. Indeed, the integrals (thick full lines) converge to values of opposite sign for large $z$, thereby confirming our hypothesis that spin-orbit coupling arising from this Sb atom induces opposite spin orientations of the two states.

In conclusion, we have investigated the scattering properties of the three-dimensional topological insulator Sb${}_2$Te${}_3$. While no scattering events occur close to the Dirac point, scattering vectors connecting next-nearest neighbor concavely warped sides of the hexagram-shaped constant energy cuts appear as soon as the warping term introduces a substantial out-of-plane spin polarization. A new surface-related electronic band with strongly directional scattering channels has been revealed at higher energy. Although its nature is not topological, it shows a well-defined spin texture, with a spin-momentum locking opposite to the Dirac fermion.

This work was supported by the Deutsche Forschungsgemeinschaft within SPP 1666 (Grant Nos. BO1468/21-1 and BI). K.A.K. and O.E.T. acknowledge the financial support by the RFBR (Grant nos. 14-08-31110 and 15-02-01797 ) and Petersburg State University (project no. 11.50.202.2015).