Abstract
Largegap quantum spin Hall insulators are promising materials for roomtemperature applications based on Dirac fermions. Key to engineer the topologically nontrivial band ordering and sizable band gaps is strong spinorbit interaction. Following Kane and Mele’s original suggestion, one approach is to synthesize monolayers of heavy atoms with honeycomb coordination accommodated on templates with hexagonal symmetry. Yet, in the majority of cases, this recipe leads to triangular lattices, typically hosting metals or trivial insulators. Here, we conceive and realize “indenene”, a triangular monolayer of indium on SiC exhibiting nontrivial valley physics driven by local spinorbit coupling, which prevails over inversionsymmetry breaking terms. By means of tunneling microscopy of the 2D bulk we identify the quantum spin Hall phase of this triangular lattice and unveil how a hidden honeycomb connectivity emerges from interference patterns in Bloch p_{x} ± ip_{y}derived wave functions.
Introduction
The electronic wave functions of quantum spin Hall materials wind in momentum space in a topologically distinct way from ordinary insulators, as described by the corresponding \({{\mathbb{Z}}}_{2}\)invariant. The quantized transport via spinpolarized boundary modes is protected by timereversal symmetry, making quantum spin Hall insulator (QSHIs) technologically attractive^{1,2}. Ideal platforms are twodimensional (2D) honeycomb systems, as these naturally host massive Dirac fermions at the K/K′ points in momentum space. As drawn in Fig. 1a, spinorbit coupling (SOC) opens a nontrivial gap, whereas inversionsymmetry breaking (ISB) counteracts SOC and favors the trivial phase^{3}. To achieve roomtemperature operability, a large gap is essential. Graphene is, for example, a poor QSHI because its SOC arises from weak 2ndnearestneighbor hopping processes between outofplane p_{z} orbitals^{4}. To improve on the gap size, 2D materials made of heavier elements relying on local SOC are hence superior^{5,6,7,8,9}. So far, this type of materials design has been achieved only in bismuthene, a honeycomb system featuring purely planar bonding of its 6p orbitals^{10}.
The strategy of replacing carbon with heavier atoms faces two serious challenges. First, in the unavoidable presence of a substrate highZ elements tend to form buckled structures that are hostile to topology^{11}. Second, many elements order preferentially in triangular rather than in honeycomb lattices when deposited on hexagonal templates (Supplementary Note 1), with no experimental realization of a QSHI phase hitherto^{12}. Synthesizing triangular QSHI would therefore potentially accelerate the steps towards the first singlelayer QSHI device concept as, for instance, the growth process would profit from the simplicity of a nonbipartite lattice.
Here, we realize a triangular lattice of indium on SiC(0001) with topological band inversion at the valley momenta K/K′. In “indenene”, SOC arises locally from In 5p orbitals and opens a gap between valence and conduction bands of about 100 meV. A global gap is guaranteed by the presence of the substrate which induces an anticrossing of the bands derived from the indium p_{z} orbital and the two planar p_{±} ∝ p_{x} ± ip_{y} chiral orbitals, respectively^{13}. The valence and conduction bands are further spinsplit at the valleys, as a consequence of the inplane ISB of SiC(0001). The C atom of the surface SiC layer renders the two halves of the indenene unit cell (labeled as “A” and “B” in Fig. 1b and hereafter) inequivalent.
The splitting of the Dirac bands at K/K′ allows to determine the topological nature via a direct energyresolved analysis of the bulk bands: As illustrated in Fig. 1c, the phases of the p_{±}derived Bloch wave functions give rise to constructive and destructive interference in A and B. The resulting charge localization at the voids of the triangular lattice induces an emergent honeycomb connectivity (see cyangreen and orange spots in Fig. 1b). The energy ordering of these valleystates and the A/B character of the corresponding bulk wave function distinguishes the QSHI from the trivial phase in an unambiguous way, as sketched in Fig. 1a. This, therefore, stands out as an example of a topological classification through the spatial symmetry of the electronic bulk wave functions probed by scanning tunneling spectroscopy (STS).
Results and discussion
Theoretical model
For a microscopic understanding of the physics of a triangular model of p orbitals on a substrate we use the three spherical harmonics
In this basis, we stepwise introduce the key interactions relevant to stabilize the QSHI phase with lowenergy Dirac states at K/K′, as illustrated in Fig. 2. The following tightbinding Hamiltonian captures the lowenergy electronic structure of any realistic implementation, as we will see later in the density functional theory (DFT) calculations for indenene on SiC(0001). The latter is the actual material realization that we propose here and our modeling allows us to precisely determine the conditions under which its QSHI phase is realized.
In the freestanding triangular layer (Fig. 2a, d) the D_{6h} point symmetry yields Diraccrossings of the p_{±} inplane orbitals at K/K′ and prohibits the hybridization with the p_{z} subspace resulting in a metallic phase. Local (atomic) SOC (H^{SOC} = λ_{SOC} L ⋅ S) becomes relevant at band touchings and opens a gap at K/K′ between the J = 3/2 and 1/2 states of size λ_{SOC} within the inplane subspace (see inset to Fig. 2a).
In the presence of a homogeneous substrate (Fig. 2b, e), the mirror symmetry along the surface normal direction is broken and a hybridization gap opens between the in and outofplane orbitals. Considering SOC, a nontrivial insulating ground state is realized and the spindegeneracy of the states with mixed p_{±} and p_{z} orbital composition is lifted. This Rashbalike splitting does not involve the Kramers doublet of the lowenergy states at K/K′ that are protected by the C_{6v} symmetry (see inset to Fig. 2b).
Introducing a honeycomb substrate, such as SiC(0001) (Figs. 1b and 3a), the symmetry is further reduced to C_{3v}. As a consequence, ISB renders the A and B halves of the unit cell inequivalent, as schematically illustrated in Fig. 2f. The corresponding nonlocal term, of strength λ_{ISB}, acts within the inplane subspace and opens a gap of size \(3\sqrt{3}{\lambda }_{{{{{{{{\rm{ISB}}}}}}}}}\) at K/K′. H^{ISB} is diagonal in the spherical harmonics basis and promotes orbital angular momentum (OAM) polarization along the surface normal, competing with the topologically nontrivial local SOC gap (see inset to Fig. 2c). The p_{±} valleyHamiltonian reads
Depending on the relative strength of λ_{SOC} and λ_{ISB}, the gap at K/K′ is dominated by either of the two types of interaction, which defines the topological phase as indicated in Fig. 1a. All cases shown in the insets of Fig. 2a–c correspond to \({\lambda }_{{{{{{{{\rm{SOC}}}}}}}}}\, > \, 3\sqrt{3}{\lambda }_{{{{{{{{\rm{ISB}}}}}}}}}\), i.e., to the topologically nontrivial band ordering. More details on the model can be found in Supplementary Note 2.
As mentioned above, the ISB potential, whose strength depends on the substrate and the bonding distance d, distinguishes between A and B (Fig. 1b). Consequently, the charge will tend to localize on the energetically lower triangle. The arrows in Fig. 1c sketch the interference mechanism between the lattice Bloch and the OAM phases determining the chargedensity profile (Supplementary Note 2, 5). In the trivial ISBdriven phase, this interplay leads to both spinvalence (conduction) bands at K and K′ localizing in the A (B)triangle—see band structure without SOC in the main panel of Fig. 2c. The situation changes if the SOCsplitting dominates: the charge associated with the valence band doublet is localized alternatingly in the A and B voids and the same is true for the two unoccupied eigenvalues, as illustrated by the corresponding colors in the inset to Fig. 2c. A crucial observation is that the charge localization pattern is identical at both valley momenta, since the OAM polarization and the Bloch phase are odd under inversion.
An interference pattern of similar nature has been theoretically discussed in twisted bilayer graphene, though with lattice phases originating from the moiré superstructure, i.e., extending over much longer interatomic distances than in this case^{14}. Here, the A/B character represents an emergent honeycomb lattice degree of freedom, intimately linked to the topology of the triangular p model. Its role resembles that of the sublattice index in the graphene Kane–Mele Hamiltonian and it can be associated with the topological gap inversion. Further, it induces the (chiralitydependent) realspace localization of the bulk wave functions that can be measured directly in scanning tunneling microscopy (STM).
Indenene on SiC(0001)
A monolayer of indium atoms deposited on a Siterminated 4HSiC(0001) is an ideal approach to attain a physical realization of our model. Its synthesis is achieved by molecular beam epitaxy, leading to highquality indenene films as characterized by standard surface science tools (see “Methods” section and Supplementary Note 3). Topographic imaging by STM confirms the wellordered triangular lattice formed by the In atoms as shown in Fig. 3a. According to the atomic arrangement obtained by total energy minimization within DFT (see “Methods” section) the In atoms bind directly to the uppermost silicon atoms of the SiC substrate (T1 position), as depicted in Fig. 3a, c. This adsorption geometry translates into a (1 × 1) surface periodicity of the indenene layer with respect to the SiC substrate, with identical inplane lattice constants as confirmed by the STM line profiles in Fig. 3d^{15}. Note in particular the asymmetric height profile along the orange path (Fig. 3d) which reflects the ISB imposed by the C atoms in the first bilayer of the SiC substrate.
Figure 3b shows an STM image of an indenene layer next to the uncovered SiC substrate. Due to the identical triangular lattice, both surfaces appear structurally indistinguishable. Electronically, though, both systems differ significantly in the differential tunneling conductance dI/dV (Fig. 3e), a measure of the local density of states (LDOS). While SiC displays the expected wide energy gap^{16}, we find for indenene finite spectral weight throughout the entire probed energy region.
Before further analyzing the indenene LDOS, we first turn to its momentumresolved electronic structure. The red curves in Fig. 4a represent the DFT band structure of the fully relaxed indenenesubstrate combination. Apart from the substraterelated bands below −1.5 eV at the center of the Brillouin zone, all other bands are of In p character and reproduce the features seen in our tightbinding model. In particular, we observe a Diraclike dispersion around K/K′ with an additional spin splitting resulting in four distinct bands, as expected in the presence of SOC. Correspondingly, a fundamental band gap of size E_{gap} = 70 meV is present (note that the energy scale in Fig. 4 refers to the experimental Fermi level position; DFT per se places E_{F} in the gap).
Figure 4a also shows the experimental band structure determined by angleresolved photoelectron spectroscopy (ARPES). It consists of welldefined band features whose dispersions are in remarkable agreement with the DFT prediction. The only notable deviation concerns the position of the Fermi level, which in the experiment is shifted into the upper Dirac halfcone by ≈250 meV due to electronic charge transfer from the strongly ndoped substrate (see “Methods” section and Supplementary Note 4 for details). This extrinsic population of the conduction band minimum puts us into a position to probe the bandgap directly by ARPES. For this purpose, Fig. 4a shows a zoomin of the gap region at the Kpoint. Clearly the quasilinear dispersions of the upper and lower Dirac cones do not connect to each other (see inset of Fig. 4a). A peakfit of the energy distribution curves (EDCs) at the Kpoint in Fig. 4c decomposes the spectrum into three distinct peaks, namely the two first valence band states (denoted by VB1 and VB) and the lowest conduction band state (CB). The next state (CB+1) is essentially cutoff by the Fermi–Dirac function. Extending this decomposition to selected kvectors around the Kpoint yields the orange markers in Fig. 4b and excellently traces the DFT bands. Their smallest separation is indeed found at K, yielding E_{gap} ≈ 125 meV, in reasonable correspondence with the DFT value, considering that DFT tends to underestimate band gaps.
With this information at hand, the minimum seen in the experimental LDOS around −0.3 eV (arrow in Fig. 3e) is readily identified as the Dirac point, shifted to negative bias voltage due to the finite ndoping. The fact that it appears only as a spectral dip rather than a truly vanishing dI/dV signal, is a characteristic consequence of the partially occupied CB^{17,18,19} (see also Supplementary Note 5). We note in passing that our STS spectra show nmscale spatial fluctuations of the chemical potential of the order of ±40 meV (Supplementary Note 5), presumably induced by inhomogeneities in doping concentration as known from related semiconducting substrates^{20,21}. In our ARPES data, this effect will be spatially smeared out by the large photon spot (diameter ≈ 1 mm) and contribute to the energy distribution curve (EDC) peak widths. Noteworthy, the indenene LDOS exhibits a second spectral depression that is always pinned at zero bias, irrespective of local fluctuations. Various mechanisms have been suggested as the origin of such a zerobias anomaly (ZBA) which, however, depend on the specific probing details^{22,23,24}. It is therefore not considered as an intrinsic feature of the electronic structure.
Having established the existence of a sizable band gap at the valley momenta, we now address the question of its topological character. Our DFT calculation for the fully relaxed structure indicates a nontrivial phase (\({{\mathbb{Z}}}_{2}\,=\,1\)) as derived from the ab initio Wannier charge center movement^{25} (Supplementary Note 2). Interestingly, as we have seen from our model, the topology can be tuned by the relative strength of ISB and SOC (Fig. 1a). For our particular case of In/SiC the InSi bond length d turns out to be the relevant control parameter (Fig. 3c): the smaller the separation to the substrate, the stronger will be the impact of substrateinduced ISB on the indenene layer. For small bond lengths, this implies λ_{ISB} ≫ λ_{SOC} (trivial band gap) whereas in the opposite case the system is in the QSHI phase. This picture is confirmed by DFT for fixed (nonrelaxed) bond lengths d, with the topological transition at d^{crit} = 2.57 Å. The equilibrium bonding distance for our In/SiC system is d^{DFT} = 2.68 Å (see “Methods” section) in excellent agreement with the measured value of \({d}^{\exp }=(2.67\pm 0.04)\) Å obtained by Xray standing wave (XSW) photoemission (Supplementary Note 3). From the distance, we hence get a first, though indirect, hint that indenene is on the nontrivial side of the topological phase diagram. In the following, we present an unambiguous experimental determination of its topology, directly linked to the interference argument anticipated in Fig. 1c.
Topological classification
The chiral symmetry of the Dirac states on the triangular lattice can be exploited to access the topological nature directly from 2D bulk properties. Being composed of p_{±} orbitals with defined OAM the valley states assume an e^{±iϕ} angular dependence around each atomic center. In combination with the Bloch phase picked up from one lattice site to the next the superposition of neighboring atomic orbitals causes characteristic interference effects, namely the localization of the respective wavefunction at either the A or the B voids of the unit cell, visualized in Fig. 1c. As inferred from our model, the actual information on the trivial vs. inverted character of the bandgap is encrypted in the energy sequence of the A/B localization pattern of the four valley states at K and K′ (cf. Fig. 1a and inset of Fig. 2c).
Experimentally, the energydependent charge distribution is best addressed by STS, probing the LDOS with atomic resolution. The upper row of Fig. 5a shows dI/dV maps taken at selected bias voltages and covering several unit cells. For comparison, the lower row shows the corresponding DFT simulations, accounting for the dopinginduced Fermi level shift between experiment and theory. Analogously to the topography map in Fig. 3a, we first calibrate the atomic positions and lattice orientation by probing the indenene p_{z}dominated states at an experimental bias of 300 mV.
By lowering the tunneling voltage into the energy range of interest and with no other contribution from elsewhere in the Brillouin zone, the STS signal becomes exclusively sensitive to the K/K′ valley states. Indeed, at 190 mV the charge maximum has shifted away from the atomic center to the B void of the unit cell. Tuning the bias to smaller and eventually negative values (−150 mV) leads to a switch of the charge localization, now peaking at the A position. Going to even more negative bias voltage repeats the switching pattern, with the charge peak located at B and A at −400 and −550 mV, respectively. Most importantly, the observed alternation of charge localization is in excellent agreement with the corresponding DFT maps of the nontrivial system.
This behavior of the LDOS can be further analyzed by directly comparing the continuous energy dependence of the charge asymmetry between the A and B halves of the unit cell (Fig. 5c, d for DFT and experiment, respectively) to the valley band structure (Fig. 5b). Clearly, the charge difference switches sign each time a new valley state contributes to the LDOS. The noisy behavior of the experimental difference spectrum around the Fermi level is attributed to the ZBA in the total dI/dV curve (Fig. 5d) which tends to amplify small extrinsic fluctuations in the A and B charge signals when taking their difference. Overall, we find remarkable qualitative agreement between experiment and theory. Specifically, an alternating ABAB charge localization sequence is established when following the valley states in energy from VB1 to CB+1, in clear distinction from the AABB sequence predicted for a trivial insulator (Fig. 1a). Our STS data thus confirm that indenene on SiC is a largegap triangular QSHI.
Outlook
From a general perspective, the concept of an emerging honeycomb lattice in the interatomic voids of a triangular atomic arrangement paves the way for the design of novel 2D QSHIs. Our approach promotes inplane chiral Dirac fermions at K/K′ whose mass term is determined by the interplay of local SOC and ISB. These tunable electronic properties, combined with the simple triangular geometry facilitating largescale domain growth, are highly desirable for roomtemperature transport applications based on the utilization of topologically protected edge states.
The reported topological classification is achieved exclusively by means of local observables and it is intimately linked to the nature of the bulk wave functions. For this reason, it represents an interesting complement to the common identification schemes based on the bulkboundary correspondence. Its connection to the orbital angular momentum polarization in kspace can be experimentally unveiled by exploiting the coupling of circularly polarized light to the orbital magnetization and Berry curvature^{26,27}. It also establishes fast earlystage material screening that is complementary to challenging quantum transport experiments and can become relevant to topology beyond solidstate physics, e.g., in optical lattices of ultracold gases^{28}.
Methods
Indenene synthesis, STM, and photoemission measurements
4HSiC(0001) samples (12 mm × 2.5 mm, ntype doped (0.01–0.03) Ωcm) with an atomically flat and wellordered surface were prepared in a gaseous hydrogen dryetching process^{29,30}. Here, 2 slm H_{2} and 2 slm He both with a purity of 7.0 were additionally filtered in gas purifiers and eventually introduced in a dedicated ultrahigh vacuum (UHV) chamber with a pressure of ~950 mbar. The SiC sample was then etched at 1180 °C for 5 min. The smooth hydrogen passivated SiC sample^{29,30} was then transferred in situ to the epitaxy chamber where the surface quality was inspected with lowenergy electron diffraction (LEED) prior to the indium epitaxy. After a heating step that removed the Hsaturation from the substrate, highly pure indium (99.9999%) was evaporated from a standard Knudsen cell. Excessive indium was reduced thermally until only (1 × 1) LEED diffraction spots remained (Supplementary Note 3).
STM data were acquired using a commercial Omicron lowtemperature LTSTM operated at 4.7 K and a base pressure lower than 5 ⋅ 10^{−11} mbar. The chemically etched Wtip was conditioned and inspected on an Ag(111) crystal before and after measuring a sample. dI/dV maps were taken at constant height using a standard lockin technique with a modulation frequency of 971 Hz and modulation voltage of V_{rms} = 10 mV. dI/dV curves were recorded using the same lockin technique. We achieved a semiconstant height mode (CHM) by interrupting the feedback loop at tunneling parameters with featureless topography in constant current mode (CCM) (e.g., at I_{set} = 50 pA and V_{set} = −900 mV) followed by an approach of the tip to the sample surface by Δz = −2.8 Å in order to generate a sufficiently large tunneling signal.
ARPES and Xray photoelectron spectroscopy (XPS) data were recorded in our homelab photoemission setup from Specs equipped with a hemispherical analyzer (PHOIBOS 100), a HeVUV lamp (UVS 300) generating photons of 21.2 eV, and a 6axis LHecooled manipulator (20 K for ARPES, room temperature for XPS). The base pressure of this UHV setup lies below 1 ⋅ 10^{−10} mbar. During LHecooled measurements the He partial pressure of the differential pumped HeVUV lamp did not exceed 1 ⋅ 10^{−9} mbar in the UHV chamber.
Room and lowtemperature XSW measurements were performed at beamline I09 at Diamond Light Source in UHV environment. The samples were prepared and characterized by ARPES in our home lab before shipping them in situ in a UHV suitcase with base pressure below 1 ⋅ 10^{−9} mbar. For more details, see Supplementary Note 3.
DFT calculations
For our theoretical study of indium on SiC(0001) we employed stateoftheart firstprinciples calculations based on the density functional theory as implemented in the Vienna ab initio simulation package (VASP)^{31}, within the projectoraugmented planewave (PAW) method^{32,33}. For the exchangecorrelation potential, the HSE06 functional was used^{34}, by expanding the Kohn–Sham wave functions into planewaves up to an energy cutoff of 500 eV. We sampled the Brillouin zone on an 12 × 12 × 1 regular mesh, and when considered, SOC was selfconsistently included^{35}. The energy decomposed densities are calculated on refined kgrids with a sampling of at least 90 × 90 × 1 and 54 × 54 × 1 for the lowenergy states at K and at M, respectively, by selecting all relevant kpoints with states inside the investigated energy window with the help of a Wannier Hamiltonian. The indenene lowenergy models are extracted by projecting onto In p and SiC sp_{3}like functions (MLWF) by using the WANNIER90 package^{36} to compute the \({{\mathbb{Z}}}_{2}\) topological invariant by following the general method of Soluyanov and Vanderbilt^{25}. We consider a (1 × 1) reconstruction of triangular In on four layers of Siterminated SiC(0001) with an inplane lattice constant of 3.07 Å. The equilibrium structure is obtained by relaxing all atoms until all forces converged below 0.005 eV/Å resulting in an InSiC distance of d_{In–SiC} = 2.68 Å. To disentangle the electronic states of both surfaces a vacuum distance of at least 25 Å between periodic replicas in zdirection is assumed and the dangling bonds of the substrate terminated surface are saturated by hydrogen. Structural models are visualized with VESTA^{38}.
Tightbinding model
We consider a triangular lattice with a (In) p basis with a nearestneighbor interaction given by Slater–Koster parameters^{37}. The onsite energies and transfer integrals are extracted from a Wannier Hamiltonian. Detailed information on the model can be found in Supplementary Note 2.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
References
Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).
Qi, X.L. & Zhang, S.C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).
Kane, C. L. & Mele, E. J. Z_{2} topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).
Kane, C. L. & Mele, E. J. Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005).
Liu, C.C., Feng, W. & Yao, Y. Quantum spin Hall effect in silicene and twodimensional germanium. Phys. Rev. Lett. 107, 076802 (2011).
Xu, Y. et al. Largegap quantum spin Hall insulators in tin films. Phys. Rev. Lett. 111, 136804 (2013).
Li, G. et al. Theoretical paradigm for the quantum spin hall effect at high temperatures. Phys. Rev. B 98, 165146 (2018).
Chen, P. et al. Large quantumspinHall gap in singlelayer 1\({T}^{\prime}\) WSe_{2}. Nat. Commun. 9, 1–7 (2018).
Di Sante, D. et al. Towards topological quasifreestanding stanene via substrate engineering. Phys. Rev. B 99, 035145 (2019).
Reis, F. et al. Bismuthene on a SiC substrate: a candidate for a hightemperature quantum spin Hall material. Science 357, 287–290 (2017).
Rivero, P., Yan, J.A., GarcíaSuárez, V. M., Ferrer, J. & BarrazaLopez, S. Stability and properties of highbuckled twodimensional tin and lead. Phys. Rev. B 90, 241408 (2014).
Wang, Z. F., Jin, K.H. & Liu, F. Quantum spin Hall phase in 2D trigonal lattice. Nat. Commun. 7, 12746 (2016).
Petersen, L. & Hedegård, P. A simple tightbinding model of spinorbit splitting of spderived surface states. Surf. Sci. 459, 49–56 (2000).
Koshino, M. et al. Maximally localized wannier orbitals and the extended hubbard model for twisted bilayer graphene. Phys. Rev. X 8, 031087 (2018).
Stockmeier, M., Müller, R., Sakwe, S. A., Wellmann, P. J. & Magerl, A. On the lattice parameters of silicon carbide. J. Appl. Phys. 105, 033511 (2009).
Seyller, T. Electronic properties of SiC surfaces and interfaces: some fundamental and technological aspects. Appl. Phys. A 85, 371–385 (2006).
Feenstra, R. M. & Stroscio, J. A. Tunneling spectroscopy of the GaAs(110) surface. J. Vac. Sci. Technol. B 5, 923–929 (1987).
Whitman, L. J., Stroscio, J. A., Dragoset, R. A. & Celotta, R. J. Scanningtunnelingmicroscopy study of InSb(110). Phys. Rev. B 42, 7288–7291 (1990).
AlBrithen, H. A., Smith, A. R. & Gall, D. Surface and bulk electronic structure of \({{{{{\rm{ScN}}}}}}(001)\) investigated by scanning tunneling microscopy/spectroscopy and optical absorption spectroscopy. Phys. Rev. B 70, 045303 (2004).
Weidlich, P. H., DuninBorkowski, R. E. & Ebert, P. Quantitative determination of local potential values in inhomogeneously doped semiconductors by scanning tunneling microscopy. Phys. Rev. B 84, 085210 (2011).
Zhang, S., Huang, D., Gu, L., Wang, Y. & Wu, S. Substrate dopant induced electronic inhomogeneity in epitaxial bilayer graphene. 2D Mater. 8, 035001 (2021).
Ming, F., Smith, T. S., Johnston, S., Snijders, P. C. & Weitering, H. H. Zerobias anomaly in nanoscale holedoped Mott insulators on a triangular silicon surface. Phys. Rev. B 97, 075403 (2018).
Zhang, Y. et al. Giant phononinduced conductance in scanning tunnelling spectroscopy of gatetunable graphene. Nat. Phys. 4, 627–630 (2008).
Butko, V. Y., DiTusa, J. F. & Adams, P. W. Coulomb gap: How a metal film becomes an insulator. Phys. Rev. Lett. 84, 1543–1546 (2000).
Soluyanov, A. A. & Vanderbilt, D. Computing topological invariants without inversion symmetry. Phys. Rev. B 83, 235401 (2011).
Schüler, M. et al. Local Berry curvature signatures in dichroic angleresolved photoelectron spectroscopy from twodimensional materials. Sci. Adv. 6, eaay2730 (2020).
Ünzelmann, M. et al. Orbitaldriven rashba effect in a binary honeycomb monolayer AgTe. Phys. Rev. Lett. 124, 176401 (2020).
Becker, C. et al. Ultracold quantum gases in triangular optical lattices. N. J. Phys. 12, 065025 (2010).
Glass, S. et al. Atomicscale mapping of layerbylayer hydrogen etching and passivation of SiC(0001) substrates. J. Phys. Chem. C 120, 10361–10367 (2016).
Seyller, T. Passivation of hexagonal SiC surfaces by hydrogen termination. J. Phys. Condens. Matter 16, 1755–1782 (2004).
Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio totalenergy calculations using a planewave basis set. Phys. Rev. B 54, 11169–11186 (1996).
Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmentedwave method. Phys. Rev. B 59, 1758–1775 (1999).
Blöchl, P. E. Projector augmentedwave method. Phys. Rev. B 50, 17953–17979 (1994).
Krukau, A. V., Vydrov, O. A., Izmaylov, A. F. & Scuseria, G. E. Influence of the exchange screening parameter on the performance of screened hybrid functionals. J. Chem. Phys. 125, 224106 (2006).
Steiner, S., Khmelevskyi, S., Marsmann, M. & Kresse, G. Calculation of the magnetic anisotropy with projectedaugmentedwave methodology and the case study of disordered Fe_{1−x}Co_{x} alloys. Phys. Rev. B 93, 224425 (2016).
Mostofi, A. A. et al. Wannier90: a tool for obtaining maximallylocalised Wannier functions. Comput. Phys. Commun. 178, 685–699 (2008).
Slater, J. C. & Koster, G. F. Simplified LCAO method for the periodic potential problem. Phys. Rev. 94, 1498–1524 (1954).
Koichi, M. & Izumi, F. VESTA 3 for threedimensional visualization of crystal, volumetric and morphology data. J. Appl. Crystallogr. 44, 1272–1276 (2011).
Acknowledgements
We acknowledge Diamond Light Source for time on beamline I09 under proposals NT264191 and SI251514. The research leading to these results has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie SkłodowskaCurie Grant Agreement No. 897276. We gratefully acknowledge the Gauss Centre for Supercomputing e.V. (https://www.gausscentre.eu) for funding this project by providing computing time on the GCS Supercomputer SuperMUCNG at Leibniz Supercomputing Centre (https://www.lrz.de). We are grateful for funding support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy through the WürzburgDresden Cluster of Excellence on Complexity and Topology in Quantum Matter ct.qmat (EXC 2147, Project ID 390858490) as well as through the Collaborative Research Center SFB 1170 ToCoTronics (Project ID 258499086).
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M.B. and J.E. have realized the epitaxial growth and surface characterization and carried out the ARPES and STM experiments and their analysis. P.E. has conceived the theoretical ideas and performed the DFT, Wannier and Berryology calculations. On the experimental side, contributions came from P.K.T., J.G., T.L.L., J.S., S.M., and R.C., while D.D.S. and G.S. gave inputs to the theoretical aspects. R.C. and G.S. supervised this joint project and wrote the manuscript together with all other authors.
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Bauernfeind, M., Erhardt, J., Eck, P. et al. Design and realization of topological Dirac fermions on a triangular lattice. Nat Commun 12, 5396 (2021). https://doi.org/10.1038/s4146702125627y
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DOI: https://doi.org/10.1038/s4146702125627y
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