Abstract
Topological insulators are a recently discovered class of materials having insulating bulk electronic states but conducting boundary states distinguished by nontrivial topology. So far, several generations of topological insulators have been theoretically predicted and experimentally confirmed, all based on inorganic materials. Here, based on firstprinciples calculations, we predict a family of twodimensional organic topological insulators made of organometallic lattices. Designed by assembling molecular building blocks of triphenylmetal compounds with strong spinorbit coupling into a hexagonal lattice, this new classes of organic topological insulators are shown to exhibit nontrivial topological edge states that are robust against significant lattice strain. We envision that organic topological insulators will greatly broaden the scientific and technological impact of topological insulators.
Introduction
The concept of topological order in condensed matter physics provides a new perspective for understanding the origin of different quantum phases and has generated intense recent interest in searching for nontrivial topological materials, so called topological insulators (TIs)^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21}. The defining signature of a TI is the existence of robust conducting edge or surface states on the boundary of normal insulators. These characteristic boundary states have a topological origin, which are protected from elastic backscattering and localization, and hence hold potential for applications in spintronics and quantum computation devices.
So far, several generations of TI have been predicted theoretically and confirmed experimentally. The first one is twodimensional (2D) TI (quantum spin Hall system) in an HgTebased quantum well^{5,6}. Subsequently, several binary (Bi_{2}Te_{3}, Bi_{2}Se_{3} and Sb_{2}Te_{3})^{7,8,9,10}, ternary (TlBiTe_{2}, TlBiSe_{2} and PbBi_{2}Te_{4})^{11,12,13} and Heusler compounds^{14,15,16} have been found to be threedimensional (3D) TIs. Most recently, the Bi bilayer^{17,18,19,20,21} and actinidebased binary compounds^{22} have been predicted to be new families of 2D and 3D TIs, respectively. However, these existing families of TIs are all inorganic materials. On the other hand, many conventional inorganic materials and devices have later found their way to organic counterparts, such as organic superconductors^{23}, lightemitting diodes^{24}, solar cells^{25} and fieldeffect transistors^{26}. In general, organic counterparts of inorganic materials have the added advantages of low cost, easy fabrication and mechanical flexibility. Therefore, an interesting question is whether an organic TI (OTI) exists?
TIs are distinguished from ordinary insulators by nontrivial topological invariants associated with bulk electronic structure. In particular, their electronic structures are characterized by a bulk spinorbit coupling (SOC) gap and an odd number of Diraclike edge states for a 2D TI (or surface states for a 3D TI) connecting the conduction and valence edge at certain kpoints; the latter can be guaranteed by band inversion in an inversion symmetric system that gives rise to an odd total parity of all timereversal invariant points^{2}. To design or search for a TI material, two key ingredients are essential: one is the lattice symmetry and the other is the SOC. The first condition is readily met by a singleband model on a hexagonal lattice, such as graphene^{1}, which is predicted to be a 2D TI when a SOC gap is present. However, the SOC in graphene is too small to open a gap at finite temperature^{27}, and proposals have been made to extrinsically enhance SOC of graphene by doping with heavy atoms^{28,29}.
The emerging theme of materials genome inspires exciting opportunities to computationally design novel materials with desired properties. Meanwhile, advances on synthetic chemistry and nanotechnology have shown the potential in producing complex 2D lattices, that is, covalent organic frameworks (COFs). Keeping the above two ingredients in mind, we have computationally designed a new family of OTIs. We have looked into organometallic compounds as molecular building blocks, which consist of CC and Cmetal bonds. To satisfy the first condition, we narrowed down our choices to the triphenylmetal compounds with one metal atom bonded symmetrically to three benzene rings, so that they can be readily selfassembled into a hexagonal lattice, as shown in Fig. 1. To satisfy the second condition, we simply choose heavy metal atoms, such as Pb and Bi, with strong SOC.
Results
2D triphenyllead lattice
We first consider the triphenyllead (TL) [Pb(C_{6}H_{5})_{3}] lattice. The TL molecule consists of a Pb atom bonded with three benzene rings with threefold rotational symmetry (Fig. 1a). When bridged with a Pb atom (Fig. 1b), they naturally form a 2D hexagonal lattice (Fig. 1c). There are two Pb atoms (labelled as 1 and 2 in Fig. 1d) and three benzene rings with a chemical formula of Pb_{2}C_{18}H_{12} in each unit cell and the neighbouring benzene rings are bridgebonded through the paraPb atoms. The 2D lattice is slightly buckled with the paraPb atoms moving alternately up and down out of the plane of benzene rings. This is because the Pb has a 6s^{2}6p^{2} electronic configuration that favours the sp^{3} hybridization. The equilibrium lattice constant is found to be 12.36 Å, with the Pb–Pb distance (l) and height difference (h) being 7.46 and 2.18 Å, respectively, as indicated in Fig. 1d. The comparison of lattice energy between the flat and buckled structure and the stability analysis of the TL lattice are shown in Supplementary Figs S1 and S2.
Without SOC, the electronic band structure of TL lattice displays a Dirac cone at the K point, with the Fermi level located exactly at the Dirac point, as shown in Fig. 2a. Turning on the SOC, a gap of ~8.6 meV opens at the K point, as shown in Fig. 2c. We label this SOCinduced gap as , to be distinguished from the conventional insulator bandgap. For a 2D TI, there should be an odd number of Diraclike edge states connecting the conduction and valence edge at certain kpoints. Thus, we have checked the existence of nontrivial topological edge states of the TL lattice, using the Wannier90 package^{30}. We fit a tightbinding Hamiltonian with maximally localized Wannier functions to the firstprinciples band structures. With these maximally localized Wannier functions, the edge Green’s function^{31} of the semiinfinite TL lattice is constructed and the local density of state of edges is calculated. This method provides a direct connectivity between the edge states and the bulk states. In Fig. 2e, a comparison between the Wannier and firstprinciples bands is given around the Diraccone gap, showing very good agreement. The local density of state of the zigzag edge is shown in Fig. 2f, where one can clearly see the nontrivial topological gapless edge states that connect the band edge states between K and K′ and form a onedimensional (1D) Dirac cone at the boundary of the Brillouin zone (BZ), signifying our designed TL lattice to be a 2D TI.
We note that because the Dirac bands have a relative small bandwidth (~1 eV), there could be other instabilities of gap opening mechanisms, such as those associated with magnetism, dimerization and charge density wave. These can be interesting topics of future studies but beyond the scope of current work within the framework of density functional theory (DFT). Based on our DFT calculations, the system remains always a nonmagnetic ground state.
Effective Hamiltonian
Because the topological nature of the TL lattice can be determined by the physics near the K (K′) point, it is possible to write down an effective Hamiltonian to characterize its lowenergy properties (see Supplementary Figs S3 and S4, and Supplementary Note 1). Similar to the silicene model^{32,33,34}, which is also predicted to be a 2D TI, the effective Hamiltonian for the TL lattice around the K and K′ can be written as
where η=±1 corresponds to K and K′ points, respectively, σ (τ) is the Pauli matrix of spin (sublattice), ν_{F} is the Fermi velocity, ε_{0} is the onsite energy, I is the 4 × 4 identity matrix, λ_{SO} is the effective SOC, λ_{R} is the Rashba SOC and a is lattice constant. A comparison between the firstprinciples and effectiveHamiltonian band structure is shown in Fig. 2b. The corresponding fitting parameters are ε_{0}=0 eV, ħν_{F}=1.13 eV Å, λ_{SO}=0.0043, eV, aλ_{R}=0.316 eV Å. We note that there is a nonzero Rashba SOC in our model due to structural buckling, which is absent in the flat graphene structure. Buckling makes the two sublattices of metal atoms in the different plane to have both inplane and outofplane SOC components^{33}. The inplane component, like in graphene, is responsible for gap opening. The outofplane component, resulting from the potential gradient in zdirection^{33} (absent in graphene), changes slightly the band dispersion (slope) around the Dirac point, but not important to topology. So the Rashba SOC here is intrinsic different from the extrinsic Rashba SOC induced by breaking of the inversion symmetry.
2D triphenylbismuth lattice
The TL lattice represents a ‘true’ TI having the right band filling with the Fermi level located exactly in the middle of the Diraccone gap (or Dirac point without SOC), but the gap is relatively small, ~8.6 meV. One may choose a different metal atom with larger SOC to increase this gap. For example, we have calculated the electronic and topological properties of triphenylbismuth (TB) [Bi(C_{6}H_{5})_{3}] lattice by replacing Pb with Bi, as shown in Fig. 3. The stability and structural properties of TB lattice are similar to the TL lattice, as shown in Supplementary Fig. S1. It has all the required electronic and topological properties for a TI and a Diraccone gap as large as ~43 meV. However, its Fermi level is ~0.31 eV above the Dirac point without SOC (Fig. 3a) because Bi has two extra electrons than Pb. Thus, to truly make the TB lattice a TI material, one must move the Fermi level into the Diraccone gap by doping. This requires doping of two holes per unit cell, because there are two degenerate bands in between the Fermi level and Diraccone gap (Supplementary Fig. S5 and Supplementary Note 2). This can possibly be achieved by gating effect. It is worth noting that the OTIs we demonstrate here represent a real material system that realizes the KaneMele model^{1} of 2D TI with a sizable gap, comparable to the known inorganic 2D TI of HgTe quantum well^{5,6}.
Strain effects
For an inversion symmetric system, its topology can also be identified by band inversion that gives rise to an odd total parity of all timereversal invariant points^{2}, via calculation of topological index Z_{2} number. For the lattices we consider here, there are four timereversal invariant kpoints, one Γ and three M’s in the BZ. We have used this method to also check the strain tolerance of the nontrivial topology of the TL and TB lattices, without repeating the calculation of edge state as a function of strain. As the proposed lattices are likely to be assembled on the substrate, they may not maintain their freestanding lattice constant. We found that the SOC gap () varies only slightly with the inplane biaxial strain, and there is no gap closing even for strains up to ±10%. This can be reasonably understood because the strength of SOC is a local property associated with metal atom (Pb versus Bi) rather insensitive to strain. The calculated Z_{2} number is one within the range of ±10% strain applied for both TL and doped TB lattice. This means that the TL and TB lattices maintain a nontrivial topology (for a trivial topological phase, Z_{2}=0), which is very stable against strain. As the topological invariance is a global property in the whole BZ of the lattice, it is expected to be robust under uniform lattice deformation, making it easier for experimental realization and characterization.
Discussion
Finally, we comment on the experimental feasibility to produce and characterize our proposed 2D OTIs. Organometallic chemistry and substratemediated molecular selfassembly are both wellestablished methods for producing exotic materials. Most importantly, recent developments in this field have already successfully synthesized the oriented 2D COFs involving directional metalC, CC as well as other organic bonds^{35,36,37,38,39,40}, which are exactly what needed for producing our proposed structure. These existing COFs are produced with welldefined lattice symmetry (including the needed hexagonal symmetry) and longrange periodicity, as well as good uniformity in large sample size (see, for example, scanning tunnelling microscope and low energy electron diffraction images in Fig. 2 of ref. 40). It is also worth noting that these COFs can be experimentally made with different metal atoms (such as Fe, Cu and Co in refs 39 and 40) and variety of molecules, while all the metals and molecular precursors needed in our proposed structures are commercially available and have been used in organometallic chemistry and molecular selfassembly before. Therefore, we believe the existing experimental results of COFs have clearly demonstrated the feasibility of producing our proposed OTIs with desired lattice symmetry, good crystalline quality and sufficient size.
Methods
DFT calculations
DFT calculations for the stability, band structure and topological properties of 2D organometallic lattices made of TL and TB were carried out in the framework of the Perdew–Burke–Ernzerhof type generalized gradient approximation functional using the VASP package^{41}. All selfconsistent calculations were performed with a planewave cutoff of 600 eV on a 7 × 7 × 1 MonkhorstPack kpoint mesh on supercells with a vacuum layer more than 15 Å thick to ensure decoupling between neighbouring slabs. For structural relaxation, all the atoms are allowed to relax until atomic forces are smaller than 0.01 eV Å^{−1}.
Additional information
How to cite this article: Wang, Z. F. et al. Organic topological insulators in organometallic lattices. Nat. Commun. 4:1471 doi: 10.1038/ncomms2451 (2013)
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Acknowledgements
We thank Chaoxing Liu for helpful discussion. This work was supported by US DOEBES (grant no. DEFG0204ER46148) and NSFMRSEC (Grant No. DMR1121252). Z.F.W. also acknowledges support from ARL (Cooperative agreement no. W911NF1220023). We thank the CHPC at University of Utah and NERSC for providing the computing resources.
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Z.F.W and Z.L. carried out the theoretical calculations. F.L. was responsible for the overall direction. Z.F.W and F.L. wrote the manuscript.
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Supplementary Figures S1S5 and Supplementary Notes 12 (PDF 661 kb)
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Wang, Z., Liu, Z. & Liu, F. Organic topological insulators in organometallic lattices. Nat Commun 4, 1471 (2013). https://doi.org/10.1038/ncomms2451
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DOI: https://doi.org/10.1038/ncomms2451
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