Prediction and observation of the first antiferromagnetic topological insulator
Despite immense advances in the field of topological materials, the antiferromagnetic topological insulator (AFMTI) state, predicted in 2010, has been resisting experimental observation up to now. Here, using density functional theory and Monte Carlo method we predict and by means of structural, transport, magnetic, and angle-resolved photoemission spectroscopy measurements confirm for the first time realization of the AFMTI phase, that is hosted by the van der Waals layered compound MnBiTe. An interlayer AFM ordering makes MnBiTe invariant with respect to the combination of the time-reversal () and primitive-lattice translation () symmetries, , which gives rise to the topological classification of AFM insulators, being equal to 1 for this material. The -breaking (0001) surface of MnBiTe features a giant bandgap in the topological surface state thus representing an ideal platform for the observation of such long-sought phenomena as the quantized magnetoelectric coupling and intrinsic axion insulator state.
A number of remarkable experimental observations and theoretical predictions recently made for antiferromagnetic (AFM) materials indicate clearly that they can be of great practical importance Gomonay and Loktev (2014); Jungwirth et al. (2016); Baltz et al. (2018); Jungwirth et al. (2018); Duine et al. (2018); Gomonay et al. (2018); Železnỳ et al. (2018); Němec et al. (2018); Šmejkal et al. (2018). Indeed, such effects as magnetoresistance, spin torque, and ultrafast dynamics observed in certain AFM materials promise significant advances in novel electronics Baltz et al. (2018). At the crossroad of antiferromagnetism and the emerging field of “topotronics”, i.e. electronics based on the properties of topologically-nontrivial systemsHasan and Kane (2010); Qi and Zhang (2011); Bansil et al. (2016), an AFM topological insulator (AFMTI) emerges Mong et al. (2010), predicted to give rise to both exotic physics Li et al. (2010) and practically relevant phenomena Li et al. (2010); Ghosh and Manchon (2017). An AFMTI can be realized in materials breaking both time-reversal and primitive-lattice translational symmetry , but preserving their combination , that leads to a topological classification Mong et al. (2010). To date, no AFMTI was found.
Here, using state-of-the-art ab initio techniques and Monte Carlo simulations along with the structural, transport, magnetic, and photoemission measurements we discover a first ever AFMTI compound, MnBiTe. Built of septuple layer blocks stacked along the  direction and weakly bound to each other by van der Waals forces, MnBiTe develops an interlayer AFM state, in which ferromagnetic (FM) Mn layers of neighboring blocks are coupled antiparallel to each other, while the easy axis of the staggered magnetization points perpendicular to the blocks. This type of magnetic structure makes MnBiTe -symmetric in bulk, while the strong spin-orbit coupling (SOC) of Bi and Te leads to the appearance of an inverted bandgap of about , converting the material in a three-dimensional (3D) AFMTI. Unlike the bulk, the (0001) surface of MnBiTe, i.e. its natural cleavage plane, is -breaking Mong et al. (2010), which causes the opening of a giant gap in the Dirac point (DP), as evidenced by our photoemission measurements. These results promise to facilitate the eventual observations of such exotic phenomena as quantized magnetoelectric coupling Qi et al. (2008); Essin et al. (2009); Tse and MacDonald (2010); Mong et al. (2010) and the axion insulator state Li et al. (2010); Ooguri and Oshikawa (2012); Sekine and Nomura (2014); Wang et al. (2016), both being intimately related to the AFMTI state of matter.
Ii Theoretical prediction
The -group structure of bulk MnBiTe (Fig. 1a) was reported in Ref. [Lee et al., 2013]. Since the magnetism of MnBiTe was not experimentally investigated, we begin our study by calculating the exchange coupling parameters from first principles. It is seen in Figs. 1b and 1c that among the intralayer interactions the one between first nearest neighbors in the Mn layer is clearly dominant (), while the interactions with more distant neighbors are an order of magnitude weaker. The calculated single-site spin stiffness coefficient (; the sum runs over all ()-pairs of interacting moments), that accounts for the energy required to flip a local magnetic moment, is positive and equal to for the intralayer coupling. Therefore, an FM ordering is expected within each septuple layer (SL) block of MnBiTe. In contrast, the interlayer coupling constants are mostly negative. The corresponding single-site spin stiffness coefficient is also negative and equal to , which means that the overall coupling between neighboring Mn layers is antiferromagnetic. This conclusion is consistent with the results of previous total-energy calculationsEremeev et al. (2017), confirming that the interlayer AFM structure, shown in Fig. 1d, is the magnetic ground state of MnBiTe. We then calculate the magnetic anisotropy energy of MnBiTe that turns out to be positive and equal to per Mn atom, indicating the easy axis with an out-of-plane orientation of the local magnetic moments (). Our Monte Carlo simulations confirm the interlayer AFM structure suggested by ab initio calculations with a Néel temperature () of 3D ordering of (Supplementary Figure 1).
Given the magnetic ground state, the bulk electronic structure was calculated. As shown in Fig. 1e, the system is insulating, the fundamental bandgap value, determined from the GGA calculation of the density of states (DOS), being equal to . The gap magnitude decreases just slightly (to ) if the HSE06 functional is used, which is known to provide accurate semiconductor gaps and a better description of correlations. To determine whether the gap is negative (inverted), we performed the DOS calculations decreasing the SOC constant stepwise from its natural value to . It was found that at the gap is closed, while at other ratios it is nonzero, which points towards a nontrivial topology of MnBiTe.
Mong, Essin, and Moore Mong et al. (2010) introduced a classification of AFMTIs, that is provided by a combination of the time-reversal and primitive-lattice translational symmetry . Since MnBiTe is -symmetric, the invariant can be calculated based on the occupied bands parities Mong et al. (2010); Fang et al. (2013). We find , which unambiguously classifies MnBiTe as AFMTI. Note that, unlike GdBiPt, which was predictedMong et al. (2010); Chadov et al. (2010); Lin et al. (2010) (but still not experimentally confirmed) to be an AFMTI under pressure, MnBiTe is an intrinsic AFMTI.
The implication of the bulk bandgap inversion in a TI is seen at its surface, where the topological phase transition is manifested by the appearance of the topological surface state. In the case of nonmagnetic TIs, this surface state is gapless Hasan and Kane (2010); Qi and Zhang (2011); Bansil et al. (2016), but at the (0001) surface of the AFMTI MnBiTe we find, however, a 88-meV-wide bandgap (Fig. 1f). Indeed, unlike conventional strong nonmagnetic TIs Hasan and Kane (2010); Qi and Zhang (2011); Bansil et al. (2016), not all of the AFMTI surfaces are gapless, but typically only those preserving the symmetry Mong et al. (2010). In the case of an FM layer near the AFMTI surface, this symmetry is broken and an out-of-plane magnetization component opens a surface gap Mong et al. (2010).
Iii Experimental confirmation
For the first time, high-quality MnBiTe single crystals were grown at the TU Dresden (hereinafter D samples, Fig. 2a) and at IP Baku (B samples, Fig. 2b) as described in the Methods section. X-ray single-crystal diffraction experiments confirm the lattice symmetry (space group ) found in Ref. [Lee et al., 2013]. The structure refinement yields some degree of statistical cation disorder in the Mn and Bi positions (Fig. 2c) in contrast to earlier reported ordered modelLee et al. (2013). Mn/Bi antisite defects in two fully occupied cation positions do not, however, lead to a change of translational symmetry or superstructure ordering. Energy-dispersive X-ray spectroscopy reproducibly results in the stoichiometric MnBiTe composition, ruling out a possibility of large compositional variations in our samples. Similar cation intermixing was observed in the GeBiTe TI compoundKarpinskii et al. (1997); Okamoto et al. (2012) and other mixed BiTe ( Sn, Pb) semiconductors isostructural to MnBiTe, and is, therefore, characteristic for these materials.
Temperature- and field-dependent magnetization measurements performed on D samples (Fig. 2d,e) establish a 3D AFM order below , in agreement with the prediction by the Monte Carlo method. Below , a strongly anisotropic magnetic susceptibility is observed, that decreases much steeper for plane. No splitting between zero-field-cooled and field-cooled-warming curves was observed. The paramagnetic regime above was fitted with a modified Curie-Weiss law, , in the to range. Here, stands for a temperature-independent magnetic susceptibility of both diamagnetic closed electron shells and a Pauli paramagnetic contribution due to some degree of metallicity in this material (see below). considers a temperature-dependent Curie-Weiss susceptibility of mostly localized Mn moments. The fitted effective paramagnetic moment of is in rough agreement with the high-spin configuration of Mn (), while the small and positive value of the Curie-Weiss temperature () strongly depends on the fitted contribution. However, given the AFM order below , it indicates competing AFM and FM exchange couplings in MnBiTe. The curve acquired below for shows an indicative spin-flop transition at (Fig. 2e), which is in line with an out-of-plane easy axis of the staggered magnetization.
To gain additional insight into the magnetic ordering in MnBiTe, we performed X-ray magnetic circular dichroism (XMCD) experiments at the Mn absorption edge (Fig. 2g; D samples). The XMCD data were acquired in total electron yield (TEY) mode with a probing depth of typically only a few nm Abbate et al. (1992). The XMCD signal obtained at an external field of and in normal light incidence verifies a magnetic polarization of the Mn ions. After removal of the external field () the XMCD signal collapses as expected for an AFM ordering. Yet, a small residual signal is still observed in remanence, indicating a finite net out-of-plane polarization within the probed volume of the sample. This residual signal appears to be inconsistent with an AFM intralayer coupling where the orientation of the moments within a Mn layer alternates on the atomic scale. For FM intralayer coupling, however, the first SL, which is preferentially probed in the TEY mode, is expected to be composed of mesoscopic domains with the magnetization pointing in or out of the surface plane. We attribute the residual XMCD signal in remanence to a preferential sampling of one domain type with the micron-sized synchrotron beam spot. This supports our first-principles calculations predicting an FM ordering within individual SLs. Performing the same experiment in grazing light incidence, i.e. with sensitivity to in-plane magnetization, we observe no finite polarization in remanence.
Temperature- and field-dependent resistivity measurements were performed on B samples (Fig. 2f). The metallic-like behavior characteristic of the presence of free carriers, is observed at as the resistivity increases with rising temperature. This is consistent with the results of the Hall effect measurements revealing the -type conductivity of these samples (Supplementary Figure 2). A well-defined kink at indicates a magnetic transition in agreement with the magnetization studies and Monte Carlo simulations. In a series of measurements under an external field applied normally to the (0001) plane the kink shifts to lower temperatures as the field increases from to . Above the critical field of to , the curve slope decreases much steeper below , which could be related to the observed spin-flop in the curve (Fig. 2e). We would like to note at this point that the dependence in Fig. 2e for plane shows no saturation even at a maximal field used () as the magnetization is only per formula unit instead of . This is a clear manifestation of the strength of the AFM interlayer exchange coupling acting against the external magnetic field.
Altogether, the experimental scope of evidence (Fig. 2) allows to identify MnBiTe as an interlayer antiferromagnet, in which FM Mn layers are coupled antiparallel to each other and the easy axis of staggered magnetization points perpendicular to the layers (Fig. 1d). The abovementioned statistical disorder does not affect the overall magnetic structure of MnBiTe, which appears to be as predicted by our ab initio calculations and Monte Carlo simulations. Further total-energy calculations, performed for a scenario with the Mn/Bi intermixing, show that the magnetic moments of the Mn atoms at the Bi sites are coupled ferromagnetically to those in the Mn layer.
To study the surface electronic structure of MnBiTe(0001) we performed angle-resolved photoemission spectroscopy (ARPES) experiments. The ARPES spectrum measured near the Brillouin zone center along the direction at a temperature of is shown in Fig. 3a (; B sample). One can clearly see two almost linearly dispersing bands forming a Dirac-cone-like structure with strongly reduced intensity at the crossing point. The energy distribution curves (EDCs) reveal an energy gap of about at the -point (Fig. 3b) that separates the upper and lower parts of the cone. A similar result was obtained for the D samples (Supplementary Figure 3). These results are in a qualitative agreement with those of the MnBiTe surface bandstructure calculations (see Fig. 1f).
In order to identify other theoretically calculated spectral features of MnBiTe(0001) in the ARPES maps, we performed extensive measurements at different photon energies. Careful inspection of the data acquired with (Fig. 3c) allows one to note that other features than at show a pronounced spectral weight: namely, the intense electron- and hole-like bands coming to the -point at the binding energies of about and , respectively. A comparison with the theoretically calculated bulk-projected bandstructure allows to identify these bands as the bulk conduction and valence bands, respectively. The analysis of the -point EDC shows that both valence and conduction bands can be fitted with two peaks (Supplementary Figure 4) in a qualitative agreement with the result of our calculations, showing two bulk bands with a weak dispersion both below and above the Fermi level (Fig. 1f). Based on our photoemission measurements, we estimate the bulk bandgap to be close to 200 meV, again, in agreement with the calculated values. In Fig. 3d, the second derivative representation provides further insight, apart from the bulk bands revealing also the gapped Dirac cone that at appears with lower intensity.
These data confirm the 3D AFMTI phase in MnBiTe, which thus is the first experimentally identified system in the AFMTI class. To check whether the inverted character of the bulk bandgap in MnBiTe is directly related to the AFM order we measured the surface bandstructure above , at (Fig. 3e; ), i.e. in the -preserving paramagnetic state. It can be seen that the topological surface state does not disappear at high temperature, implying that the bulk bandgap inversion persists across the magnetic phase transition. Thus, the -breaking 3D AFMTI phase in MnBiTe is changed by the -symmetric (paramagnetic) 3D TI phase above . Note that increasing the temperature does not lead to the DP gap closing, which is similar to what was reported for the surface states of magnetically-doped TIs Chen et al. (2010); Xu et al. (2012); Sánchez-Barriga et al. (2016) in the paramagnetic phase. In the case of the BiMnSe(0001) surface, resonant scattering processes due to impurity in-gap states were suggested to be a possible reason of such a behavior Sánchez-Barriga et al. (2016). To check whether similar effects take place in our MnBiTe samples, we performed resonant photoemission measurements at the Mn edge. The results, shown in Fig. 3f, reveal no resonant features and, hence, no Mn-3-DOS in the region of the DP gap, whereupon we discard such a mechanism of the DP gap opening in MnBiTe. Thus, together with previous findings Chen et al. (2010); Xu et al. (2012); Sánchez-Barriga et al. (2016); Hirahara et al. (2017), our results stress the importance of acquiring a better insight of how thermal paramagnetic disorder, potentially in combination with structural defects, influences the low-energy excitation spectrum near the DP in the paramagnetic phase of magnetic TIs in general. An additional solid support to the topological nontriviality of MnBiTe comes from the results of the surface bandstructure calculations, in which the material was artificially driven into a topologically-trivial phase by suppressing the SOC constant: these results strongly disagree with those of our ARPES measurements. Namely, in the trivial phase there are neither surface states in the bulk bandgap of MnBiTe at/near the -point nor the resonance states near the bulk bandgap edges (Supplementary Figure 5). In contrast, our ARPES data acquired at different photon energies unambiguously confirm the gapped Dirac cone to be a surface state in agreement with the calculated surface bandstructure of the MnBiTe AFMTI (Fig. 1f).
Iv Outlook and conclusions
Our experimental and theoretical results establish MnBiTe as the first observed AFMTI. Although in our experiment it is -doped (which is typical for the bulk TI crystals), it is -doping that allows to observe the topological surface state with ARPES, which probes only the occupied states. A common strategy in synthesizing truly insulating TI crystals is the admixture of Sb to the Bi sublattice of tetradymite-like compoundsZhang et al. (2011); Arakane et al. (2012); Chang et al. (2013), which is expected to work for MnBiTe as well. Note that such a tuning of composition is not supposed to affect its interlayer AFM ordering Eremeev et al. (2017). On the other hand, recent progress in the molecular beam epitaxy growth of the TI filmsWu et al. (2016) rises hope that a nearly charge neutral MnBiTe can be fabricated.
All this would constitute an important step towards many novel applications. Indeed, according to Ref. [Mong et al., 2010], an AFMTI with the type of antiferromagnetic order established here for MnBiTe represents ideal platform for observing the half-integer quantum Hall effect (), which may aid experimental confirmation of the quantized magnetoelectric coupling. A material showing this effect is known as an axion insulator, which up to now was being sought for in magnetically-doped sandwich-like TI heterostructures Mogi et al. (2017a, b); Xiao et al. (2018). Unfortunately, the latter were found to show superparamagnetic behavior Lachman et al. (2015, 2017); Krieger et al. (2017) – a drawback that MnBiTe does not suffer from, which makes it a promising intrinsic axion insulator candidate. Incidentally, since a possibility of realizing the dynamical axion field in the paramagnetic TI state has been pointed out recently Wang et al. (2016), the gap at the MnBiTe(0001) surface above might facilitate the embodiment of this idea at room temperature. On the other hand, the FM SL blocks of MnBiTe can be utilized for the creation of topologically-nontrivial heterostructures using the recently proposed magnetic extension effect Otrokov et al. (2017a, b); Hirahara et al. (2017). Unlike magnetic doping or proximity effect, it yields a giant bandgap in the topological surface state of a nonmagnetic TI thus providing a promising platform for achieving the quantum anomalous Hall effectChang et al. (2013) at elevated temperatures. Finally, beyond topotronics, another direction of further studies of MnBiTe and related materials, such as MnBiSeEremeev et al. (2017); Hirahara et al. (2017); Hagmann et al. (2017) and MnSbTeEremeev et al. (2017), lies within the rapidly growing field of van der Waals magnets Gong et al. (2017); Huang et al. (2017). Strongly thickness-dependent properties, expected for the van der Waals compounds in the ultrathin film limit, combined with the magnetic degrees of freedom and the strong SOC present in MnBiTe, make it an interesting candidate to couple the emerging fields of antiferromagnetic spintronics and layered van der Waals materials.
In summary, we have theoretically predicted and experimentally confirmed a three-dimensional AFMTI phase in the layered van der Waals compound MnBiTe. These results culminate almost a decade-long search of an AFMTI material, first predicted in 2010. In a broader sense, MnBiTe represents the first intrinsically time-reversal-breaking stoichiometric TI compound realized experimentally. As an outcome of this discovery, a number of fundamental phenomena Qi et al. (2008); Essin et al. (2009); Tse and MacDonald (2010); Mong et al. (2010); Li et al. (2010); Chang et al. (2013); Wang et al. (2016); Sekine and Nomura (2016); He et al. (2017); Šmejkal et al. (2018) are expected to either eventually be observed, being among them quantized magnetoelectric couplingQi et al. (2008); Essin et al. (2009); Tse and MacDonald (2010); Mong et al. (2010) and axion insulator state Li et al. (2010); Wang et al. (2016), or become accessible at elevated temperatures Otrokov et al. (2017b), like the quantum anomalous Hall effect and chiral Majorana fermions Chang et al. (2013); He et al. (2017).
Electronic structure calculations were carried out within the density functional theory using the projector augmented-wave (PAW) method Blöchl (1994) as implemented in the VASP code Kresse and Furthmüller (1996); Kresse and Joubert (1999). The exchange-correlation energy was treated using the generalized gradient approximation Perdew et al. (1996a). The Hamiltonian contained scalar relativistic corrections and the SOC was taken into account by the second variation method Koelling and Harmon (1977). In order to describe the van der Waals interactions we made use of the DFT-D2 Grimme (2006) and the DFT-D3 Grimme et al. (2010, 2011) approaches, which gave similar results. Both GGAAnisimov et al. (1991); Dudarev et al. (1998) and HSE06Becke (1988); Perdew et al. (1996b); Heyd et al. (2003) calculations of the bulk DOS were performed. The magnetic anisotropy energy, , was calculated taking into account both the band contribution, , and the energy of the classical dipole-dipole interaction.
For the equilibrium structures obtained with VASP, we calculated the Heisenberg exchange coupling constants also from first principles, this time using the full-potential linearized augmented plane waves (FLAPW) formalism Wimmer et al. (1981) as implemented in Fleur bib (). The GGA+ approach was employed Anisimov et al. (1997); Shick et al. (1999). The constants were extracted by Fourier inversion of the magnon energy dispersion Sandratskii and Bruno (2002); Ležaić et al. (2006); bib (); Kurz et al. (2004).
The Monte Carlo (MC) simulations were based on a classical Heisenberg Hamiltonian including a magnetic anisotropy energy
where the magnetic moments at site and are described by unit vectors and , respectively, and the magnetic coupling constants are determined by ab initio calculations as described above.
Dresden samples (D samples)
High-quality bulk single-crystals of MnBiTe were grown from the melt by slow cooling of a 1:1 mixture of the binaries BiTe and -MnTe. The binaries were synthesized by mechanical pre-activation and annealing of stoichiometric mixtures of the elements. Crystal size and quality could be controlled via different cooling rates within a narrow temperature region at around and varying annealing times. Further details of crystal-growth optimization will be reported elsewhere Zeugner et al. (2018). Single-crystal X-ray diffraction was measured on a four-circle Kappa APEX II CCD diffractometer (Bruker) with a graphite(002)-monochromator and a CCD-detector at . Mo-K radiation () was used. A numerical absorption correction based on an optimized crystal description X-S () was applied, and the initial structure solution was performed in JANA2006 Petricek et al. (2011). The structure was refined in SHELXL against Fo2Sheldrick (2014, 2008). Energy dispersive X-ray spectra (EDS) were collected using an Oxford Silicon Drift X-MaxN detector at an acceleration voltage of and a accumulation time. The EDX analysis was performed using the standardless method (where = atomic no. correction factor, = absorption correction factor, = fluorescence factor, and = peak to background model).
Baku samples (B samples)
The bulk ingot of the Baku sample was grown from the melt with a non-stoichiometric composition using the vertical Bridgman method. The pre-synthesized polycrystalline sample was evacuated in a conical-bottom quartz ampoule sealed under vacuum better than . In order to avoid the reaction of the manganese content of the sample with the silica container during the melting process, the inside wall of the ampoule was coated with graphite by thermal decomposition of acetone in an oxygen-poor environment. The ampoule was held in the ”hot” zone () of a two-zone tube furnace of the MnBiTe Bridgman crystal growth system for to achieve a complete homogenization of the melts. Then, it moved from the upper (hot) zone to the bottom (cold) zone with the required rate of . Consequently, the bulk ingot with average dimensions of in length and in diameter was obtained. Further details will be reported elsewhere Aliev et al. (2018). The as-grown ingot was checked by X-ray diffraction (XRD) measurements and was found to consist of several single crystalline blocks. With the aid of XRD data high quality single crystalline pieces were isolated from different parts of the as-grown ingot for further measurements.
The magnetic measurements as a function of temperature and magnetic field were performed on a stack of single crystals of MnBiTe (D samples) using a SQUID (Superconducting Quantum Interference Device) VSM (Vibrating Sample Magnetometer) from Quantum Design. The temperature dependent magnetization measurements were acquired in external magnetic fields of and for both zero-field-cooled (zfc) and field-cooled-warming (fcw) conditions. A thorough background subtraction was performed for all the curves.
Part of the magnetic measurements were carried out in the resource center “Center for Diagnostics of Materials for Medicine, Pharmacology and Nanoelectronics” of the SPbU Science Park using a SQUID magnetometer with a helium cryostat manufactured by Quantum Design. The measurements were carried out in a âpullâ mode in terms of temperature and magnetic field. The applied magnetic field was perpendicular to the (0001) sample surface.
The ARPES experiments were carried out at the BaDElPh beamline Petaccia et al. (2010) of the Elettra synchrotron in Trieste (Italy) and BL-1 of the Hiroshima synchrotron radiation center (Japan) using -polarization of the synchrotron radiation and laser Iwasawa et al. (2017a, b). The photoemission spectra were collected on freshly cleaved surfaces. The base pressure during the experiments was better than . Part of the ARPES experiments were also carried out in the resource center “Physical methods of surface investigation” (PMSI) at the Research park of Saint Petersburg State University.
ResPES data were acquired at the HR-ARPES branch of the I05 beamline at the Diamond Light Source. The measurements were conducted at a base temperature of with a beam spot size and resolution of and . The difference of on- and off-resonant spectra for the Mn transition corresponds directly to the Mn density of states. A photon energy series was conducted in order to determine suitable transition energies. The corresponding angle integrated spectra of on-resonant () and off-resonant () conditions can be seen in Fig. 3f.
XMCD measurements were performed at the HECTOR end-station of the BOREAS beamline at the ALBA synchrotron radiation facility Barla et al. (2016). The data were collected in total electron yield mode. The spot size and the resolving power of the supplied photon beam were and , respectively. Measurements were performed at the Mn- edges at a temperature of , i.e. well below .
M.M.O. and E.V.C. thank A. Arnau and J.I. Cerdá for stimulating discussions.
We acknowledge the support by the Basque Departamento de Educacion, UPV/EHU (Grant No.
IT-756-13), Spanish Ministerio de Economia y Competitividad (MINECO Grant No. FIS2016-75862-P),
and Tomsk State University competitiveness improvement programme (project No. 8.1.01.2017).
The supports by the Saint Petersburg State University grant for scientific investigations (Grant No.
18.104.22.1685), Russian Science Foundation (Grant No. 18-02-00062), and the Science Development Foundation under the President of the Republic of Azerbaijan (Grant No. EF-BGM-4-RFTF-1/2017-21/04/1-M-02) are also acknowledged. A.U.B.W., A.I., and B.B. acknowledge support by the DFG within the SFB 1143. A.Z., A.E., and A.I. acknowledge the support by the German Research Foundation (DFG) in the framework of the Special Priority Program (SPP 1666) âTopological Insulatorsâ and by the ERA-Chemistry Program. S.V.E. acknowledges support by the Fundamental Research Program of the State Academies of Sciences for 2013–2020. A.K. was financially supported by KAKENHI number 17H06138 and 18H03683. The calculations were performed in Donostia International Physics Center, in the Research park of St. Petersburg State University Computing Center (http://cc.spbu.ru), and in Tomsk State University.
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