Strong and Weak Three-Dimensional Topological Insulators Probed by Surface Science Methods

We review the contributions of surface science methods to discover and improve 3D topological insulator materials, while illustrating with examples from our own work. In particular, we demonstrate that spin-polarized angular-resolved photoelectron spectroscopy is instrumental to evidence the spin-helical surface Dirac cone, to tune its Dirac point energy towards the Fermi level, and to discover novel types of topological insulators such as dual ones or switchable ones in phase change materials. Moreover, we introduce procedures to spatially map potential fluctuations by scanning tunneling spectroscopy and to identify topological edge states in weak topological insulators.

contributions form bulk bands, other surface bands and inelastic scattering (for details see [50]).

II. IDENTIFYING TOPOLOGICAL SURFACE STATES
Soon after establishing 2DTIs experimentally [5] based on theoretical predictions [25], an extension of the formalism to 3D was proposed [23,24]. It results in two types of 3D topological insulators (3DTIs). One exhibits an odd number of spin-helical TSSs on each surface and is dubbed strong 3DTI, while the other one has an even number of topological surface states on every surface except one and is dubbed weak 3DTI [23,24]. After identifying a first strong 3DTI in a BiSb alloy by ARPES [28], DFT calculations predicted stoichiometric materials to be strong 3DTIs, namely Bi 2 Te 3 , Sb 2 Te 3 , and Bi 2 Se 3 [30]. These three materials share the same crystal structure of quintuple layers (QL) that are stacked on each other by van-der-Waals forces ( Fig. 1(a)). Hence, these materials can be cleaved in-situ and can be exfoliated as thin films [63,64]. Moreover, they have been predicted to exhibit a single TSS on the cleavage plane with the Dirac point located in the center of the Brillouin zone at the so-called Γ point ( Fig. 1(b)) [30].
These properties enable a simple investigation by SARPES provided that the Dirac cone (TSS) is below E F . Indeed, the first ARPES measurements of a TSS on Bi 2 Se 3 (0001) have been published [65] back-to-back with the DFT based predictions [30]. First SARPES measurements appeared only three month later [29]. It turned out that the cleaved bulk samples of Bi 2 Se 3 and Bi 2 Te 3 are n-doped, being beneficial for the ARPES mapping of the TSS, but detrimental for electric transport. In contrast, Sb 2 Te 3 is usually p-doped [66,67] impeding ARPES mapping. Luckily, we obtained a twenty year old Sb 2 Te 3 crystal that enabled mapping of the lower part of the Dirac cone via ARPES ( Fig. 1(c)) [50]. This part of the Dirac cone encloses states of the bulk valence band in k space in quantitative accordance with DFT calculations ( Fig. 1(b)). Since the doping is caused by point defects of the material [68,69], we speculate that the particular defect distribution within this material is responsible for establishing the favorable E D E F . Similar results exhibiting Dirac cones within the band gap close to E F have also been found for Sb 2 Te 3 , Bi 2 Se 3 , and Bi 2 Te 3 after careful optimization of growth conditions in UHV [44,68,70].
Figure 1(e)−(g) show SARPES data recorded via a Mott detector, that probes the two in-plane directions of the spin. Two peaks at opposite k are recorded corresponding to the two opposite sides of the Dirac cone. The spin polarization is found to be exclusively perpendicular to the k vector as expected from the (Rashba-type) spin-orbit interaction. It W2H W3H x Sb =-70 meV (a) x Sb  is, moreover, helical, i.e., it switches sign when inverting k. These are the typical fingerprints of a Dirac cone type TSS [30]. Out-of-plane spin polarizations have also been observed, in particular, further away from E D and are traced back to distortions of the simple Dirac cone, e.g., via warping, i.e., by influences of the crystal structure [72]. It is important to realize that SARPES does not probe the spin polarization of the initial state exclusively, but that the photoemission process is an excitation to unoccupied states extending into the vacuum that can change the spin polarization either by matrix effects or by spin polarization of the final state [73]. This can be captured by calculations within the so-called fully relativistic one-step model based on DFT calculations [74]. In particular, at low photon energies, it turns out that the detected spin polarization can even be inverted with respect to the initial state depending on the polarization direction of the exciting light [75]. At higher energies in the deep UV regime, this is less relevant, since excited states are well above the vacuum level. Hence, the helicity of the TSS can be deduced being counterclockwise for the lower part of the Dirac cone of Sb 2 Te 3 ( Fig. 1(d)). This is in accordance with the DFT calculations ( Fig. 1(b)). The absolute value of the spin polarization of the TSS is not extracted directly from our SARPES data due to the limited angular and energy resolution. The reduced resolution during SARPES with respect to ARPES is caused by the low efficiency of the Mott detector. Novel approaches improve this efficiency considerably via spin dependent, k conserving reflections of the photoelectrons at single crystals [76]. Hence, resolution can be much better, but such apparatus was not available during the measurements presented in Fig. 1. Consequently, spin polarization had to be extracted rather indirectly by carefully subtracting the inelastic background, the background originating from the also measured spin-polarized surface states at lower energy (visible in Fig. 1(b) at −0.4 to −0.8 eV), and the background from the overlapping, enclosed bulk states. Nevertheless, the accordingly best fit of the SARPES data revealed a spin polarization of the TSS of 80 − 95 % ( Fig. 1(g)) matching the DFT result of 90 % surprisingly well [50]. Obviously, the TSS is not 100 % spinpolarized, albeit it is spin-helical. This is a natural consequence of spin-orbit interaction, that strongly mixes the spin with orbital degrees of freedom via the heavy atoms involved.
Thus, spin is not a good quantum number in these materials.

III. TUNING THE DIRAC POINT ENERGY
One main task after the experimental discovery of 3DTIs was to tune their Dirac point energy E D , that mostly turned out to be far away from E F [8]. Hence, literally speaking, the first 3DTIs were not even insulators in their interior. More importantly, the transport properties of the 3DTIs could not be probed without rendering the bulk of the material sufficiently insulating. A rather obvious, initial approach was to exploit the opposite p-type doping of Sb 2 Te 3 and n-type doping of Bi 2 Te 3 or Bi 2 Se 3 . Two main strategies have been pursued. Either, the two materials are mixed in a way such that they exhibit a similar  density of acceptors and donors [78,79]. This approach eventually led to the observation of the quantum Hall effect within thin films of BiSbTeSe 2 as a clear signature of dominating 2D-type transport [80,81]. Detailed analysis of the filling factor dependence of the Hall conductance identified the TSSs on bottom and top surface as the origin of the half integer quantum Hall effect [81]. The respective tuning of the Dirac cone, respectively E D , with respect to E F can be monitored by ARPES in detail [82]. This is particularly important for the protection of Majorana states within vortices of a topological superconductor against conventional single-particle excitations by an effective gap E gap,eff reading [83][84][85] E gap,eff ∆ 2 with ∆ being the excitation gap of the surrounding topological superconductor.  from fitting intensity profiles I(k ) (Fig. 2(c)) at different energies that are subsequently extrapolated linearly to determine the crossing point as E D (Fig. 2(d)). Alternatively, the full width at half maximum (FWHM) of I(k ) closer to E D (Fig. 2(e)) is employed via identifying E D as the energy with lowest FWHM (Fig. 2(f)). In both cases, E F has to be carefully calibrated as well. For the particular sample, we found E F E D within 5 meV [71]. Since no time dependent band shifts were observed, the value is likely robust as long as the sample is in UHV. However, ex-situ Hall measurements on identically prepared samples exhibit a transition form p-type to n-type bulk conduction at much lower Sb concentration (x Sb 60 %) [87]. Hence, rescuing the precise tuning for electric devices requires additional efforts and investigations.
Another approach uses the electric field at interfaces between p-type and n-type 3DTIs [77,88]. As well known for semiconductor p-n junctions, a depletion region forms at the interface such that a thin enough overlayer can maintain in the depletion region. This implies that E F remains in the band gap up to the surface. The approach has the general advantage that it avoids ternary or quarternary alloys that potentially induce additional scattering centers for electrons via alloying. Figure 3 To determine E D including the thicknesses, where it is above E F , DFT results of 6 QL Sb 2 Te 3 are overlaid after rigidly shifting them to reproduce the ARPES data. It turned out that the best anchor point for shifting is the surface state at lower energy ( Fig. 1(b) at −0.4 to −0.8 eV). This state is vertically stronger confined to the surface area and, hence, is more intense in ARPES and less prone to the averaging by the vertical band bending [50] (details in [77]). The resulting E D − E F has been compared with the result of a 1D Poisson-Schrödinger model revealing reasonable agreement (Fig. 3(c)). The model is based on the charge carrier densities of MBE grown films of Sb 2 Te 3 and Bi 2 Te 3 as determined by Hall measurements, while assuming the same density of dopants and charge carriers. An intermixing at the interface is additionally taken into account that is deduced from Auger electron spectroscopy depth profiling [77]. Obviously, depletion method via p-n junction is also able to tune E F E D for a thickness of ∼ 20 QL Sb 2 Te 3 on top of Bi 2 Te 3 .  [90,91], Dirac semimetals [92][93][94] and Weyl semimetals [95,96]. Interestingly, topological properties of different kind can be combined in a single material, if the topo-logical indices belong to different symmetries of the Hamiltonian [97]. For example, 3DTIs protected by time-reversal symmetry can be combined with TCIs protected by a crystal symmetry such as n-fold rotation or mirroring [98]. This raises the perspective to break one of the symmetries, hence, switching between different topology types [86,97]. The first material that experimentally showed dual topology was Bi 1 Te 1 [86]. It consists of stacked Bi bilayers (BLs) and Bi 2 Te 3 QLs in a ratio of 1 : 2 as evidenced by TEM (Fig. 4(a)−(b)).
Bi BLs are well known to be 2DTIs [53,99,100]  This indicates a weak 3DTI with its dark surface perpendicular to the (001) direction [86].
However, the reasoning via stacked 2DTIs is too simple, when analyzing the DFT data in more detail. Interlayer hybridizations mix up the 2D bands strongly, such that the weak Bi 1 Te 1 (001) (Fig. 4(c), lower row). In order to achieve this agreement, the Bi 1 Te 1 film had to be terminated by a single QL and had to be downshifted by 100 meV with respect to E F , Both is reasonable with the latter accounting for n-type doping as expected from the well known n-type doping of Bi 2 Te 3 . The good agrement between ARPES and DFT data, also found for multiple other bands of Bi 1 Te 1 , is the central evidence for the dual topological character of Bi 1 Te 1 [86]. b. Topological phase change materials Another interesting class of 3DTI materials layer is the central ingredient for the conductivity of the metastable PCM phase [89].
The TSS above the broadened valence bands has been found by two-photon ARPES, i.e., a first light pulse transfers electrons into the initially unoccupied TSS and a second light pulse with time delay ∆t extracts photoelectrons from the now occupied TSS. Figure 5(e) shows data for several energies above E F exhibiting a rather isotropic circle in k space.
The circle shrinks in diameter with decreasing energy. Extrapolation of the radius to lower energies ( Fig. 5(f)) implies vanishing diameter at about 160 meV above E F that represents E D . Hence, the well-established conducting phase of the PCM Ge 2 Sb 2 Te 5 is a strong 3DTI, at least, after the preparation by MBE as probed in this study [89]. This is appealing for 3DTI-based applications via exploiting the established expertise for upscaling conventional Ge 2 Sb 2 Te 5 devices [115]. Counteracting the unfavorable p-doping of Ge 2 Sb 2 Te 5 is possible by replacement of Ge with the heavier Sn [116], where, however, 3DTI properties still have to be demonstrated experimentally [117].

V. DISORDER CHARACTERIZATION
As described in the introduction, a central task for improving the electric transport properties of 3DTIs (and 2DTIs) is the reduction of disorder. Disorder can lead to additional transport channels concealing the features of the TSS as well as to scattering of the TSS electrons [8,32]. STS is the tool of choice for probing the disorder at the surface due to its unprecedented spatial and energy resolution in probing the LDOS. It has only the minor drawback that it is exclusively measuring the surface disorder and not the disorder within deeper layers of the bulk of the crystal [118].
One possibility by STS is to track characteristic features of the energy dependent LDOS [43,44]. One measures dI/dV (V ) curves with I being the tunnel current and V being the voltage applied between tip and sample. Mostly, such curves are measured by lockin technique, i.e., the tip-surface distance is stabilized at voltage V stab and current I stab .
Afterwards, the feedback loop is switched off, such that the tip surface distance remains constant, while the voltage is changed linearly and overlapped with an oscillating voltage of amplitude V mod that enables the phase sensitive detection of dI/dV via a lock-in amplifier.
In first order, the resulting dI/dV (V ) represents the LDOS(E − E F ) [118][119][120][121]. This gives direct access, e.g., to spatial variations of the band gap for a semiconductor or insulator.  (111) surface of the strong 3DTI Ge 2 Sb 2 Te 5 exhibiting Te as the top layer with hexagonal atomic structure [45]. Several, largely triangular bright protrusions appear on top of the atomic lattice ( Fig. 6(b)−(d)). They have been identified as subsurface defects by comparison with DFT data [45]. The lateral size of the triangle increases with the depth of the defect below the surface. The particular sample grown by MBE exhibits a defect density of ∼ 1.5 · 10 12 /cm 2 . This implies a potential disorder due to the positive charging of most of the defects, in particular, vacancies [45,114,122]. The dI/dV (V ) curves (Fig. 6(e)) show a band gap of about 0.5 eV with the valence band onset being close to E F in agreement with optical absorption [123] and ARPES data ( Fig. 5(b)), respectively. The band gap onset is spatially varying. It is quantified via the peak position of the numerically determined dI 3 /dV 3 (V ) curves leading to a nearly Gaussian distribution of the spatially varying valence band onset with σ width of 20 meV (Fig. 6(f)). We compare this with a simple model calculation randomly distributing positive point charges with a density identical to the charge carrier density determined by Hall measurements (Fig. 6(g)). This leads to potential fluctuations on the surface with the same σ width as in the experiment (Fig. 6(h)). It implies that the Coulomb centers of the charged acceptors (vacancies) dominate disorder in this sample. Interestingly, the LDOS does not vanish within the band gap ( Fig. 6(e)) indicating the presence of in-gap surface states in agreement with the two-photon ARPES revealing a TSS (Fig. 5(f)).
Another possibility to map potential disorder is Landau level spectroscopy, however, requiring a magnetic field. It exploits the Dirac type spin chirality of the TSS implying a so called zeroth Landau level (LL0) that is tied to E D [124,125]. Hence, tracking LL0 across the surface maps the potential disorder as seen by the TSS, i.e., averaged across some of the upper QLs [47][48][49]. The lateral spatial resolution of the method is largely given by the magnetic length [126]. Figure 6(i) shows STM data of in-situ cleaved Sb 2 Te 3 (0001) featuring a few defects that have been identified previously by comparison with DFT calculations as Sb substitutional in the upper Te layer (Sb Te , bright) and vacancies in the Sb layer directly below the surface (Vac Sb , dark) [69]. We find a defect density of 4·10 12 /cm 2 with all apparent defects attributed to the upper QL [49]. Figure 6(j)-(k) show Landau level spectra recorded at two different locations of the sample. It is apparent that the energy of LL0 does not shift with B field (Fig. 6(k)). Moreover, LL0 appears at the same energy as the minimum in dI/dV (V ) curves at B = 0 T. Finally, LL0 deviates by ∼ 40 meV between the two probed areas indicating the potential fluctuations. We found that the deduced LL0 energy correlates with the local density of defects visible in the STM data (not shown) [49]. Weak 3DTIs have initially barely been studied due to the wrong conjecture that they are unstable with respect to most type of perturbations [24]. More detailed studies, however, revealed that the only detrimental perturbation is a strong dimerization of adjacent layers along the surface normal of the dark surface leading to a doubling of the unit cell [128,129].

VI. EDGE STATES OF WEAK TOPOLOGICAL INSULATORS
Hence, also weak 3DTIs typically exhibit robust spin-helical surface states protected from backscattering. The most simple way to construct a weak 3DTI is stacking 2DTIs without interlayer interaction [24,129]. This naturally implies that single-layer terraces on the dark surface are patches of 2DTIs that consequently must host one-dimensional topological edge states at its step edges. These edge states are spin helical and, hence, ideal conductors as long as time-reversal symmetry is not broken [130]. It turns out that such edge states appear generally for weak 3DTIs even if constructed differently [130]. This implies the possibility to scratch a network of ideal conductors into the surface of a weak 3DTI [54].
The first experimental realization of a weak 3DTI was Bi 14 Rh 3 I 9 [131]. It consists of alternating layers of the 2DTI (Bi 4 Rh) 3 I [132] and the trivial insulator Bi 2 I 8 ( Fig. 7(a)).
The 2DTI exhibits a honeycomb unit cell such as graphene, but is made of the heavy atoms Bi, I and Rh ( Fig. 7(b)). It, thus, mimics the initial idea of a 2DTI in a honeycomb lattice [25], but provides a much stronger spin-orbit interaction (∼ 1 eV) leading to a sizable inverted band gap of 200 − 300 meV [132]. This gap is much larger than in graphene with inverted band gap of ∼ 20 µeV [133]. Hence, the idea to construct the 3D material is to stack 2DTI honeycomb structures [25] that are separated by trivial insulators as spacers impeding interactions between the 2DTI layers. However, it turned out that the strong spin-orbit interaction shifts much more bands across E F than only the initial Dirac cone of the honeycomb lattice that appears at E F without spin-orbit interaction [125]. Thus, the topological indices of a weak 3DTI again appear rather accidentally via inversion of several bands at the TRIMs of the Brillouin zone [132]. Nevertheless, topological edge states at each step edge are expected and have been found by STS. They are directly visible as enhanced LDOS intensity at step edges ( Fig. 7(c), background). In dI/dV (V ) curves, the band gap region of the material (−0.15 to −0.35 eV) exhibits strong intensity exclusively at the step edges ( Fig.7(d)). The edge states appear continuously along all edges [54] and are only ∼ 1 nm wide perpendicular to the edge (Fig. 7(e)). Moreover, the edge states did not exhibit any fingerprints of standing waves, but only intensity modulations periodic with the unit cell as expected for Bloch states. Thus, backscattering is largely impeded. Networks of topological edge states can indeed be scratched into the surface either by the tip of an atomic force microscope (AFM) (Fig. 7(f)) with separation down to 25 nm [54] or by the tip of an STM. The resulting scratches indeed show an increased LDOS within the band gap ( Fig.7(g)), but not at energies outside the gap (Fig. 7(h)). Unfortunately, E F is not within the band gap and, thus, the edge states are not accessible by electric transport. Four-tip STM measurements in UHV (Fig. 8(a)) [138], however, revealed that the resistance as a function of distance between the tips is not described by a 3D transport model only, but required a sizable contribution from a parallel 2D transport channel ( Fig. 8(b)−(c)). The best fit of the experimental data (red curve in Fig. 8(c), [134,139]) implies conductances for the 2D and 3D contribution σ 2D = 0.064 ± 0.005 S and σ 3D = 9200 ± 800 S/m, respectively. Thus, the 2D conductance corresponds to a ∼ 7 µm thick layer with the 3D conductance σ 3D .
This implies that the surface region of Bi 14 Rh 3 I 9 is significantly more conductive than the bulk. The encouraging finding is corroborated by DFT calculations of bulk Bi 14 Rh 3 I 9 ( Fig.8(d), bottom, orange curve) showing E F within the band gap. Additional calculations of a thin film revealed that the surface is strongly n-doped ( Fig.8(d), yellow curve) with the band gap at similar energies as found in the STS data ( Fig.8(d), top, red curve) [137]. This is in line with the strong 2D conductivity found by 4-tip STM. The band gap favorably moves quickly towards its bulk position already for the subsurface layer ( Fig.8(d), pink curve). To explain the surface n-doping, we consider the charging of the individual layers. It turns out that the 2DTI layer (Bi 4 Rh) 3 I transfers about one electron per unit cell to each of its neighboring spacer layers Bi 2 I 8 such that it is positively charged by about 2 electrons per unit cell in equilibrium. Under these circumstances, E F is in the band gap of the 2DTI layer. At the surface, however, one neighboring spacer layer is missing, such that about one electron per unit cell remains on the 2DTI layer making it strongly n-doped [137]. In principle, this could be counteracted by adding acceptors such as iodine onto the surface, but a relatively large amount of about one iodine atom per unit cell is required [140].
Using the four-tip STM, we also performed scanning tunneling potentiometry [141]. This method measures the tip voltage V that is required to nullify the current between tip and sample. Consequently, it maps the local potential, typically while current is flowing laterally.
With four-tip STM, two tips can be used to inject the current, while a third tip is scanned in between to probe the nullifying voltage [138,142]. Consequently, the current induced potential is mapped. Figure 8