# Correlated and non-correlated topological insulators

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^{1}Max-Planck-Institut für Chemische Physik fester Stoffe, 01187 Dresden, Germany

Topological insulators (TI) form a novel state of matter that open up new opportunities to create unique quantum particles. Many exciting experiments have been proposed by theory, but still await their experimental verification, not to mention their implementation into applications. Today, the main obstacle is material quality and cleanliness of the experimental conditions. The charge carrier concentration of the topological surface states (SS) is only about 10^{12} cm^{-2}. Thus the extrinsic conductivity arising from the presence of tiny amounts of vacancies and defects in the bulk, and the contamination of the surface at ambient atmosphere, will mask the special properties of the TI.

For Bi_{2}Te_{3} thin films we were able to identify the conditions under which **molecular beam epitaxy (MBE)** gives stable and reproducible results for fabrication single domain and bulk insulating samples **[Höfer2014].** Hereby, the key factors are to ensure a full distillation of the Te while maintaining a Te/Bi ratio of ~8, and the usage of substrates with negligible lattice mismatch, namely highly insulating BaF_{2} (111). The optimal conditions for MBE growth of Bi_{2}Te_{3} were found at a substrate temperature of 250^{∘}C with a Bi flux of ~1 Å/min. Angle resolved photoemission spectroscopy (ARPES) of a 10 QL Bi_{2}Te_{3} film, recorded in the vicinity of the Fermi level, clearly displays the characteristic linear dispersion of the Dirac states, cf. Fig. 1.

_{2}Te

_{3}film at room temperature. The dashed line indicates the position of the Fermi level, calibrated using a silver reference sample. Clearly, only the topological surface states intersect the Fermi level, revealing insulating bulk behavior of the film.

Dirac states, cf. Fig. 1. The ARPES results provide clear evidence for the Fermi level being located well within the band gap and being only intersected by the topological surface states. The absence of a bulk conduction band signal in the ARPES spectra recorded at room temperature ensures that the electrical conductance at low temperatures is solely given by the Dirac surface states.

Special efforts have been made to provide the capability to carry out temperature dependent four-point conductance measurements in situ, i.e. under UHV conditions, in addition to ARPES. Only in this way is the integrity of the topological surface states preserved and reliable results can be achieved. The in situ electrical contacts are realized through a homemade, non-permanent point contact mechanism, using spring-loaded probes. Bi_{2}Te_{3} films with thicknesses ranging from 10 to 50 QL have been fabricated in an effort to unambiguously clarify whether the conductivity is truly arising from the surface. The results are displayed in Fig. 2. The inset of Fig. 2A shows the voltage-current characteristics at selected temperatures. The linear relation demonstrates that the prepared contacts are ohmic over the entire temperature range measured. All samples display a metallic-like behavior with temperature. Furthermore, no significant changes between the results obtained during cool-down and warm-up of the samples are observable, demonstrating the excellent stability of the film and the fabricated contacts. Comparing the sheet resistance at different film thicknesses (Fig. 2B), a scattering with the tendency of lower resistance with thicker films can be noticed.

_{2}Te

_{3}thin films. (A) temperature-dependent sheet resistance of the thin films ranging from 10 to 50 QL. Results for cool-down (blue) and warm-up (red) are shown for the different thicknesses. (Inset) Exemplary I–V characteristics of a 10 QL Bi

_{2}Te

_{3}thin film. The linear relation demonstrates ohmic contacts within the whole temperature range. (B) Variation of sheet resistance with film thickness at low (blue dots) and room temperature (red dots). (C) Charge carrier concentrations (green triangles) calculated from the ARPES spectra and resulting mobility values for the different film thicknesses at room temperature (red dots) and at 14 K (blue dots). The dashed lines are guides to the eye.

However, the resistance is not inversely proportional to the film thickness; instead, it varies only by a factor of 1.3 at 14 K and 1.5 at 295 K, upon investigating film thicknesses from 10 to 50 QL. This provides evidence for the fact that the surface indeed dominates the transport properties of the Bi_{2}Te_{3} films. The observation of a resistance that is practically constant for thicknesses beyond 20 – 30 QL hints towards the possibly of some minute amounts of defects which may have formed during the initial stages of the growth. The obtained charge carrier concentrations, estimated from the ARPES spectra of each sample, are in the expected range for TI of 2 - 410^{12} cm^{-2} corresponding to very high average mobilities of about 3000 cm^{2}/Vs, with the maximum value of 4600 cm^{2}/Vs at 14 K, see Fig. 2C.

Recently, we have also been able to establish a procedure by means of which the Bi_{2}Te_{3} films can be capped such that they do not suffer from exposure to air **[Höfer2015].** This capping can be easily removed, still leaving the properties of the Bi_{2}Te_{3} films intact. This will allow us future access to many more investigative tools.

For the topological properties of Bi_{2}Te_{3} to arise, a significant spin-orbit coupling is required while electronic correlations do not play any appreciable role. However, it was recognized recently that also in materials in which strong electronic correlations are present topologically protected surface states may have to form. This applies particularly for the so called Kondo insulators, or strongly correlated semiconductors. In these materials the spin-orbit coupling is typically sizeable, of the order of 0.5 eV. Moreover, the more localized f-states exhibit odd parity while the conduction band is usually of a more d-like character. Some dispersion of the f states is required though in order to open up an energy gap at the Fermi level; in this respect the Kondo insulators are well suited.

The intermediate valence system SmB_{6} is such a material in which hybridization plays a decisive role for its low-temperature properties—in particular an insulating ground state due to the formation of a gap at the Fermi energy—to evolve.

_{6}surface as observed by atomically resolved Scanning Tunneling Microscopy (STM). The surface was obtained by cleaving the sample in situ and at low temperature. The brighter and higher corrugations represent Sm atoms which form chain-like, zig-zag patterns residing on top of an otherwise flat B terrace.

We conducted **scanning tunneling microscopy (STM)** and spectroscopy (STS) on SmB_{6} **[Rößler2014].** Here we made use of the unique capabilities of this technique to relate spectroscopic results unambiguously to specific topographies.

This assignment turned out to be of utmost importance for an interpretation of the different spectroscopic results which can not be achieved by other methods like ARPES. The hybridization gap, inter-multiplet transitions and possible crystal field excitations are observed by STS **[Rößler2014]. **The temperature evolution of these spectra again points toward the Kondo effect being at play. However, a plethora of different surface terminations is found; one example of a disordered, reconstructed surface is presented in Fig. 3. These different surface terminations were found to exhibit profoundly different spectroscopic properties. Here, we take advantage of the unique capability of STM to obtained combined topographic and spectroscopic information. STS conducted on non-reconstructed surfaces gave results in excellent agreement with expectations for a Fano resonance, as presented in Fig. 4 for a Sm and a B terminated surface, respectively. In the simplest description, the tunneling conductance, dI∕dV , is given by

where Γ and q are the resonance width and the asymmetry parameter, respectively. The latter is related to the ratio for tunneling probabilities into the 4f states vs. into the conduction and hence, it is expected to be higher for a Sm terminated surface as indeed observed (fit values of q are given in Fig. 4). In addition, q also depends on the particle-hole asymmetry of the conduction band. Note that additional excitations are observed in case of a Sm-terminated surface, possibly resulting from crystal field or surface states. We also observed electronic inhomogeneities on non-reconstructed surfaces of several tens of nanometers in extent which we interpret as subtle inhomogeneities in the Sm valence at the surface.

As outlined above, the enormous topical interest in SmB_{6} is driven by the theoretical prediction of topologically protected surface states in this material.

_{5∕2}, 4f

_{7∕2}, 5d and 6s orbital character derived bands are represented by red, green, blue and pink colored symbols, respectively. The size of the symbols represent the weight of the various orbital contributions to the underlying bands. The d - f band inversion happens at the X-point, resulting in a non-trivial topology.

Although our present STS results cannot provide clear-cut evidence in favor of, or against, this prediction we note that all types of surfaces, reconstructed and non-reconstructed, displayed a finite zero-bias conductance of considerable magnitude. The single crystals of SmB_{6} and, for comparison EuB_{6}, for our measurements were provided by the group of **Dr. Zachary Fisk** (UC Irvine, USA).

As a next step, we tried to identify other systems with localized f electrons, that not only possess non-trivial topology but also retain many non-polar surfaces such that the Dirac cone properties can be easily investigated. Using density functional theory (DFT) based calculations, **we** have identified an oxide, SmO, a mixed valent correlated compound to possess a non-trivial band structure **[Kasinathan2015].**

_{Bl}(k)) of the first 12 SmO layers of a semi-infinite solid with [001] (left) and [110] (right) surfaces.

The electronic structure calculations are performed using the full-potential non-orthogonal local orbital code (FPLO). Collected in Fig. 5 is the FPLO non-spin polarized, full-relativistic band structure of SmO with the inclusion of the strong Coulomb interaction (LDA+SO+U) for the experimental lattice constant. The 4f states are split into lower-lying and filled 4f_{5∕2} states that can accommodate 6 electrons, and higher-lying (above 5.5 eV), empty 4f_{7∕2} states. The material is semi-metallic with a “warped gap” such that everywhere in the Brillouin zone (at E_{F}), band number x is always below band number x+1. There are small electron and hole pockets at X and Γ, respectively, but there are no band crossings between the highest occupied 4f_{5∕2} bands and the lowest unoccupied bands. In contrast to the localized and not very dispersive 4f bands, the 5d bands of samarium are dispersive, broad and dip below the Fermi level, retaining a 100% weight at the X point.

As a consequence, we observe a 4f-5d band inversion at the X point, resulting in a nontrivial topology, and the ensuing topological indices 1;(000) allows us to classify SmO as a 3D strongly topological semi-metal.

To provide additional confirmation of the topological non-triviality in SmO and to explicitly identify the protected surface states, we have calculated the Bloch spectral density (A_{Bl}(k)) of the 12 topmost surface layers of a semi-infinite solid with [001] and [110] surfaces (Fig. 6). Consistent with the topological indices, we obtain an odd number of Dirac-cones. For both the surfaces, a single Dirac-cone at , which becomes a surface resonance due to the semi-metallic nature of SmO, is observed, along with two cones each at and for the [100] and [110] surfaces, respectively.

Having established the nontrivial topology in SmO for the experimental bulk lattice parameter, we considered the scenario of growing thin films of SmO. In general, the lattice parameter of the substrate plays a decisive role in determining the lattice parameter of a thin film. Then, the relevant question to answer is the robustness of the topological semi-metal state as a function of lattice parameter variation. To this end, we have investigated the topological indices for various lattice parameters (Fig. 7). Treating the strong 4f correlations on a mean-field level, the topological semi-metal state is quite robust and remains so, up to 5.38 Å , a 9% increase in lattice parameter. On an experimental level, this result is very promising, since it provides a large range of lattice constants of the substrate onto which SmO thin films can be grown and reveal topological non-triviality. During the evaluation of the topological indices, we became aware of another interesting feature, a nontrivial topology due to s-f band inversion at Γ for 11% and larger lattice constants (≥ 5.5 Å). Albeit no gap will be opened at Γ due to the degenerate quartet in cubic symmetry, thin films of SmO will acquire a tetragonal symmetry induced by strain which lifts the degeneracy and a gap opens, creating a topological insulator.

Thus, our calculations show that the topological nontrivial ground-state of SmO is stable for a large range of Sm-O distances, including both positive and negative strain. The Sm-f band filling is thus tunable by strain, which opens up the possibility to create correlated topological nontrivial bands at different filling. SmO is therefore an ideal candidate for further investigations in bulk and thin-film form.

[1] K. Höfer, C. Becker, D. Rata, J. Swanson, P. Thalmeier and L. H. Tjeng Intrinsic conduction through topological surface states of insulating Bi_{2}Te_{3} epitaxial thin films. Proc. Natl. Acad. Sci. USA 111, 14979-14984 (2014) http://dx.doi.org/10.1073/pnas.1410591111.

[2] K. Höfer, C. Becker, S. Wirth and L. H. Tjeng Protective capping of topological surface states of intrinsically insulating Bi_{2}Te_{3}. AIP Adv. 5, 097139 (2015) http://dx.doi.org/10.1063/1.4931038.

[3] S. Rößler, T.-H. Jang, D. J. Kim, L. H. Tjeng, Z. Fisk, F. Steglich and S. Wirth Hybridization gap and Fano resonance in SmB_{6}. Proc. Natl. Acad. Sci. USA 111, 4798-4802 (2014) http://dx.doi.org/10.1073/pnas.1402643111.

[4] D. Kasinathan, K. Koepernik, L. H. Tjeng and M. W. Haverkort SmO thin films: A flexible route to correlated flat bands with nontrivial topology. Phys. Rev. B 91, 195127 (2015) http://dx.doi.org/10.1103/PhysRevB.91.195127.