Abstract
Twodimensional (2D) StiefelWhitney insulator (SWI), which is characterized by the second StiefelWhitney class, is a class of topological phases with zero Berry curvature. As an intriguing topological state, it has been well studied in theory but seldom realized in realistic materials. Here we propose that a large class of liganded Xenes, i.e., hydrogenated and halogenated 2D groupIV honeycomb lattices, are 2D SWIs. The nontrivial topology of liganded Xenes is identified by the bulk topological invariant and the existence of protected corner states. Moreover, the large and tunable bandgap (up to 3.5 eV) of liganded Xenes will facilitate the experimental characterization of the 2D SWI phase. Our findings not only provide abundant realistic material candidates that are experimentally feasible but also draw more fundamental research interest towards the topological physics associated with StiefelWhitney class in the absence of Berry curvature.
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Introduction
With the rapid progress of topological states, the concept of Berry curvature and associated topological invariants, such as Chern numbers^{1,2}, mirror or spin Chern numbers^{3,4,5,6,7}, and FuKane invariants^{8,9}, have been widely applied to condensed matter physics. Recently, a class of topological state with zero Berry curvature, which is characterized by the StiefelWhitney (SW) class, was proposed in spinless systems with spacetime inversion symmetry I_{ST} = PT or C_{2z}T, where P, T, and C_{2z} are spatial inversion, timereversal, and twofold rotation symmetry, respectively^{10,11,12,13,14,15}. This is the socalled SW insulator (SWI), which is topologically distinguished by a different topological invariant, i.e., the second SW number w_{2}^{16}. Different from topological states associated with Chern class which possess topological boundary states due to the bulkboundary correspondence, a 2D SWI features topologically protected corner states in the presence of additional chiral symmetry, indicating it is also a special class of 2D secondorder topological insulators (SOTIs)^{17,18,19,20,21}. So far, SOTIs have been proposed in various systems, including crystalline solids and artificial structures^{22,23,24,25,26,27,28,29,30,31}. In the pioneering work of Lee et al.^{32}, the bulk topology of monolayer graphdiyne, which is a candidate for 2D SOTI, was characterized by a nontrivial second SW number w_{2} = 1, indicating that it is also a candidate material for 2D SWI. However, the recently proposed 2D SWI was mainly studied in theory but seldom in realistic materials, which greatly hinders the experimental study of SWIs. It is thus emergent to search for 2D SWIs in realistic materials.
Meanwhile, in the field of 2D materials, a monoelemental class of 2D honeycomb crystals termed Xenes (X refers to C, Si, Ge, Sn, and so on)^{33,34,35} have attracted tremendous attention as they provide an ideal platform to explore various topological physics. More than a dozen different topological phases, including the quantum spin Hall (QSH)^{36,37,38,39}, quantum anomalous Hall^{40}, quantum valley Hall states^{41,42,43}, and topological superconductors^{44,45,46,47}, are predicted to emerge in Xenes, and these topological states are easily tuned, for example, by chemical functionalization with ligands. In particular, depending on the type of ligands, hydrogenated, or halogenated derivatives of Xene can be largegap QSH or trivial insulators with tunable gaps^{36,37}.
In this work, we extend the theoretical prediction and experimental applicability of the topological physics associated with SW class by recognizing that the liganded Xene family XL (X=C, Si, Ge, Sn, L=H, F, Cl, Br, I), a large, well studied, and readily synthesizable class of materials^{48,49,50,51}, are 2D SWIs. Based on firstprinciples calculation and theoretical analysis, the chemical bonding configuration, bulk topological invariant, and ingap topological corner states are calculated to identify the SW topology. Moreover, the large and tunable bandgaps of liganded Xenes will largely facilitate experimentally observing ingap corner states. Since some liganded Xenes have been experimentally synthesized, we believe our proposal has strong feasibility to be detected by transport and STM measurements, which may draw immediate experimental attention.
Results
SW topology in hydrogenated graphene
We begin our discussion by introducing the atomic structure and associated crystalline symmetry. Since the compounds in the ligandterminated Xenes family have similar crystal structures and electronic structures, we take the singleside and doubleside hydrogenated graphene (also named graphane), denoted as sCH and dCH, as an example hereafter. Figure 1a and b shows crystal structures for dCH and sCH, respectively. The dCH with H alternating on both sides of the nanosheet is in a buckled hexagonal honeycomb structure with space group 164 (\({D}_{3d}^{3}\)) including inversion symmetry P, while the sCH is in a planar configuration with space group 183 (\({C}_{6v}^{1}\)), which contains C_{2z}, the twofold rotation symmetry about the zaxis. As timereversal symmetry T exists in both systems, therefore, the spacetime inversion symmetry required for 2D SWIs are I_{ST} = PT and C_{2z}T for dCH and sCH, respectively. The optimal lattice constants are 2.54 and 2.84 Å for dCH and sCH, respectively, which are consistent with previous reports^{52,53}. We further confirm the stability of d and sCH by phonon calculations that show no imaginary frequency (Supplementary Fig. 4).
For the hydrogenated graphene structures, H atoms directly couple to the halffilled p_{z} orbitals in intrinsic graphene, thereby removing π bonding and forming H–C σ covalent states (Supplementary Fig. 3). The structural buckling in dCH further enhances a sp^{3} hybridization of C atoms. There are in total five covalent bonds within the unit cell, including three C–C and two H–C bonds. Since the valence electron configuration of C and H are 2s^{2}2p^{2} and 1s^{1}, covalent bonding states are fully occupied with two electrons per bonding state. Therefore, the system tends to be an insulator with the Fermi level lying in the gap between bonding and antibonding states. As shown in Fig. 2a and d, dCH (sCH) is indeed an insulator with a direct (indirect) gap. Interestingly, the bandgap of sCH (~1.4 eV) is much smaller than that of dCH (~3.46 eV). Although the lack of structural corrugation in sCH leads to deviations away from the sp^{3} hybridization, singleside hydrogenation brings H atoms closer together, which naturally results in a much larger repulsion among the H–C σ bonding states. As a consequence, the occupied band derived from H–C bonds [marked by “” at Γ in Fig. 2d] shifts upwards in energy that gives rise to a smaller indirect bandgap of sCH.
To get a better understanding of the hybridization, covalency, and ionicity of chemical bonds in d and sCH, we construct maximally localized Wannier functions (MLWFs) from the five occupied valence bands. As shown in Fig. 2b and e, it is clear that these MLWFs display the character of σbonded combinations of mixed sp^{2}–sp^{3} hybrids, providing an intuitive chemical signature of covalent bonds. Moreover, it can be seen that the MLWFs of H–C covalent bonds and their Wannier charge centers are shifted towards H due to the difference of electronegativity between H and C.
The common covalent σ bonding states have the average charge center located around the middle of the bond, implying the unconventional feature of the mismatch between average electronic centers and atomic positions (also known as obstructed atomic limits^{54,55}). This is a direct indicator of nontrivial band topology. We then perform a band representation (BR) analysis based on the theory of topological quantum chemistry^{56,57,58,59}. The BR of trivial atomic insulators is solved to be a sum of atomicorbitalinduced BRs (aBRs), while that of unconventional materials, such as SOTIs, must be a combination of some aBRs and an essential BR from an empty Wyckoff position where no atom exists^{60,61}. The BR decomposition of dCH is A_{1}@2d + A_{g}@3e. As shown in Fig. 2c, the C atoms are located at 2d site of space group 164, while 3e sites of the essential BR are the C–C bond centers, which correspond to \({h}_{3c}^{(\bar{3})}\) primitive generator class of SOTIs in the Benalcazar et al.’s notation^{55}. This implies that there is no net dipole in the plane and the corner charge fractionalization will be e/2 in each π/6 sector. Similarly, we found the BR decomposition of sCH is A_{1}@2b + A_{1}@3c, also indicating its nontrivial bulk topology (see Supplementary Table I and II).
Physically, the nontrivial bulk topology of d and sCH can also be understood from the intuitive picture of double band inversion^{13,26,62,63,64}. For dCH with inversion symmetry P, we consider \({N}_{{{{\rm{occ}}}}}^{}({{\Pi }})\), the number of occupied bands with negative parity at timereversal invariant momenta (TRIM, Π = Γ and three M points). It is found that \({N}_{{{{\rm{occ}}}}}^{}(M){N}_{{{{\rm{occ}}}}}^{}({{\Gamma }})=2\), as shown in Fig. 2a. This indicates a double band inversion, because the system cannot be adiabatically connected to the trivial atomic insulator limit where the parity representations at TRIM must be the same. Importantly, the secondorder band topology of 2D inversionsymmetric spinless systems can be characterized by the second SW number w_{2}^{13,14,16,29,32}, which is determined by
where ⌊⋅⌋ is the floor function. Therefore, the double band inversion gives rise to a nontrivial w_{2} = 1, demonstrating that it belongs to the nontrivial SW class. Alternatively, w_{2} for sCH with C_{2z}T symmetry can be obtained by tracing the Wilson loop spectra (see Fig. 3a). It is proved that w_{2} is given by the parity of the number of spectral crossing at Θ = π, where Θ indicates the phase eigenvalue of the Wilson loop operator^{13,14,15}, as detailed in the “Methods” section and Supplementary Figs. 1 and 2. Due to a similar double band inversion with respect to C_{2z} in sCH, as shown in Fig. 2b, we found a nontrivial w_{2} = 1, confirming it is a SWI (see Fig. 3b).
To explicitly identify the secondorder topology in d and sCH, we calculate the fractional corner charge Q_{corner}, which is a bulk topological index for classifying SOTIs^{55}. In 2D insulators with C_{6} and T symmetries (e.g., sCH), it can be evaluated as^{55}
where \([{{{\Pi }}}_{p}^{(n)}]=\#{{{\Pi }}}_{p}^{(n)}\#{{{\Gamma }}}_{p}^{(n)}\) and \(\#{{{\Pi }}}_{p}^{(n)}\) are defined as the number of bands below the bandgap with C_{n} rotation eigenvalues \({{{\Pi }}}_{p}=\exp [\frac{2{{{\rm{\pi i}}}}(p1)}{n}]\) for p = 1, 2, ⋯ , n. Π stands for high symmetric point M and K. For dCH with S_{6} symmetry, the above formula should be modified by replacing \([{M}_{1}^{(2)}]\) with \([{M}_{\pm }^{(i)}]\), which is the difference in the number of bands with inversion eigenvalue even/odd between M and Γ^{65}. Based on the firstprinciples calculations, we have \([{M}_{1}^{(2)}]=2\), \([{K}_{1}^{(3)}]=0\) for sCH and \([{M}_{\pm }^{(i)}]=\pm 2\), \([{K}_{1}^{(3)}]=0\) for dCH. Therefore, Q_{corner} = e/2, indicating both sCH and dCH to be 2D SOTIs^{66}.
To further reveal their topological nature, the firstprinciples calculations are performed to directly verify the gapped topological edge states and ingap topological corner states in d and sCH, which can be used as a fingerprint to distinguish them from other topological phases. As shown in Fig. 4a and d, an edge band appears throughout the gap and detaches from bulk states of a zigzag nanoribbon. Since the first SW number w_{1}, which is equivalent to the Zak phase, is turned out to be trivial (w_{1} = 0), the existence of the 1D edge state is not due to the bulk polarization, but it is more like a danglingbond state. In fact, different from the topologically protected helical edge states of 2D topological insulators, these edge states are less robust and can be removed by saturating the dangling bonds at edges with hydrogen atoms (see Supplementary Fig. 5). Similar flat edge states also occur in the monolayer βSb, which has been proved to be a 2D SOTI^{65,66}.
Due to the modified bulkboundary correspondence, the most direct indicator of SOTIs is the cornerlocalized ingap states. To identify the corner topology of d and sCH, we calculate the energy spectrum for hexagonal nanodisks with ~500 atoms. In order to remove danglingbond edge states from the bulk gap, we passivate the edges with hydrogen atoms (Supplementary Fig. 6). The energy spectrum for the nanodisk of dCH and sCH are plotted in Fig. 4b and e, respectively. Remarkably, there are six states that are degenerate at the Fermi level inside the bulk gap. Moreover, an emergent chiral (sublattice) symmetry, which derives from the bipartite honeycomb lattice, pins the corner modes in the mid of the gap. As shown in Fig. 4c and f, these states are well localized at six corners of the hexagonal sample, confirming that they are the topological corner states. At exact halffilling, three of the six states are occupied, leading to a fraction corner charge of Q_{corner} = e/2 per corner, which is consistent with the above analysis.
Other liganded Xenes
In addition, we also investigated other liganded Xenes. They are counterparts of hydrogenated graphene, corresponding to the silicene, germanene, and stanene monolayer saturated by hydrogen or halogen. Some of these materials, such as hydrogenated graphene^{67,68,69} and germanene^{70}, or fluorinated graphene (also named fluorographene), have been experimentally synthesized^{70,71,72,73}. Their topological properties have been carefully investigated according to the conventional classification of timereversal Z_{2} topology. Owing to their negligible spinorbit coupling, some liganded Xenes are identified as topologically trivial. However, this argument does not forbid an SW topology with zero Berry curvature.
Based on systematic calculations, we found that those prior trivial liganded Xenes are SWIs actually (see Table 1 and Supplementary Figs. 7–10). Since the π orbitals are saturated by H or F, these compounds become insulators with bandgaps in a wide range. Remarkably, the bandgaps of singleside hydrogenated germanane (sGeH) and stanene (sSnH) are larger than that of doubleside hydrogenated ones, which is different from that of hydrogenated graphene. This is because the HX σ level does not shift to the valence band maximum due to the weaker repulsion between these states at large distances. Furthermore, among halogenated Xenes, more electronegative ligands (e.g., from F to I in halogens) tend to withdraw electron density from the Xene framework, and lower the energy of the ligandX sorbital antibonding levels at the conduction band minimum^{33,74}. Therefore, the trend of bandgap reduction and the topological phase transition from SOTI to QSH are observed from F to I in both single and doubleside halogenated Xenes.
Finally, for experimental detection on 2D SWIs, it is preferred to have the corner states sitting deep in the bulk gap. First, the bandgaps of liganded Xenes are large and tunable by the saturation position (s or d), the ligand type (hydrogen or halogens), and external perturbations such as electric field and strain. Second, the emergent approximate chiral (sublattice) symmetry of the honeycomb lattice structure pins the corner states approximately in the middle of the gap. All these features will facilitate the experimental characterization of the SWI phase in liganded Xenes. Moreover, given that some candidate materials have already been synthesized successfully in experiments^{48,49,50,51}, the SWI phase is highly accessible, and may already be realized in existing materials.
Discussion
In summary, we have demonstrated the 2D SWI phase in a large class of hydrogenated and halogenated Xenes that are experimentally feasible. The nontrivial topological nature of these materials is identified through the nontrivial second SW number w_{2} = 1 and the existence of ingap topological corner states. The candidate materials with bandgaps as large as 3.5 eV will facilitate experimentally detecting ingap corner states by STM measurements. Our results enrich the topological physics associated with SW class, and greatly extend the territory of candidate materials for 2D SWIs. In addition, it is also possible to realize 3D weak and strong SWIs by stacking these 2D SWI candidate materials. These discoveries may draw more fundamental research interests of Xenes, and provide a practical avenue for the realization of SWIs in real materials that are experimentally feasible.
Methods
Firstprinciples calculations
We perform the firstprinciples calculations within the framework of density functional theory using the Vienna ab initio simulation package^{75}. The exchangecorrelation functional is treated using the PerdewBurkeErnzerhof (PBE) generalizedgradient approximation^{76}. The predicted topology is further verified by using the HeydScuseriaErnzerhof (HSE) hybrid functional^{77}. The negligible spinorbit coupling is ignored in our calculations. We also generate MLWFs for the analysis of chemical bonding^{78,79}.
Wilson loop method
The first SW number w_{1} and second SW number w_{2} can be calculated using the Wilson loop. The Wilson loop is defined as a path ordered product of the exponential of Berry connections,
where (k_{1}, k_{2}) parametrizes the 2D Brillouin zone, and F_{i} is the overlap matrix whose elements of F_{i} are given by
The spectrum of Wilson loop is gauge invariant. The topological property of the system is determined by the phase factors θ_{m}(k_{2}) ∈ (−π, π] of the eigenvalues λ_{m}(k_{2}) of Wilson operator, i.e., \({\theta }_{m}({k}_{2})={{{\rm{Im}}}}{{\mathrm{log}}}\,{\lambda }_{m}({k}_{2})\). Computed along the k_{1} direction parallel to the reciprocal lattice vector G_{1} from (k_{1}, k_{2}), the set of Wilson loop eigenvalues \(\{{e}^{{{{\rm{i}}}}{\theta }_{m}({k}_{2})}\}\) indicates the position of Wannier centers at given k_{2}, and the corresponding total charge polarization is given by \({p}_{1}=\frac{1}{2{{{\rm{\pi }}}}}\mathop{\sum }\nolimits_{m = 1}^{{N}_{{{{\rm{occ}}}}}}{\theta }_{m}({k}_{2})\), which is equivalent to the first SW number w_{1} in k_{1} direction. The Wilson loop spectrum of liganded Xenes in the SWI phase shows that w_{1} = 0 in both k_{1} and k_{2}, indicating that the total polarization is zero.
The second SW number w_{2} on a torus of the 2D Brillouin zone indicates whether the Wilson loop operator Θ can be continuously deformed to the identity operator or not, modulo an even number of winding on noncontractible cycles. Accordingly, the parity of the number of crossing points on Θ = π gives the second SW number w_{2}. Specifically, a spectrum corresponds to w_{2} = 0 (w_{2} = 1) if the parity is even (odd).
Note added to proof
After submission, we become aware of an independent work on arXiv recently^{80}, where the results of liganded Xenes are consistent with ours.
Data availability
All data used in this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank Jiaheng Gao and Zhijun Wang for the help in determining the BR decomposition. This work was supported by the National Natural Science Foundation of China (Grant No. 12074006), the National Key Research and Development Program of China (No. 2021YFA1401600), and the startup fund from Peking University. The computational resources were supported by the highperformance computing platform of Peking University.
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H.H. supervised the project. M.P. carried out the DFT calculations. H.H. wrote the manuscript. D.L. and J.F. commented on the manuscript. All the authors contributed to the manuscript.
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Pan, M., Li, D., Fan, J. et al. Twodimensional StiefelWhitney insulators in liganded Xenes. npj Comput Mater 8, 1 (2022). https://doi.org/10.1038/s41524021006952
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DOI: https://doi.org/10.1038/s41524021006952
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