Electronic Lieb lattice signatures embedded in two-dimensional polymers with a square lattice

Exotic band features, such as Dirac cones and flat bands, arise directly from the lattice symmetry of materials. The Lieb lattice is one of the most intriguing topologies, because it possesses both Dirac cones and flat bands which intersect at the Fermi level. However, the synthesis of Lieb lattice materials remains a challenging task. Here, we explore two-dimensional polymers (2DPs) derived from zinc-phthalocyanine (ZnPc) building blocks with a square lattice (sql) as potential electronic Lieb lattice materials. By systematically varying the linker length (ZnPc-xP), we found that some ZnPc-xP exhibit a characteristic Lieb lattice band structure. Interestingly though, fes bands are also observed in ZnPc-xP. The coexistence of fes and Lieb in sql 2DPs challenges the conventional perception of the structure–electronic structure relationship. In addition, we show that manipulation of the Fermi level, achieved by electron removal or atom substitution, effectively preserves the unique characteristics of Lieb bands. The Lieb Dirac bands of ZnPc-4P shows a non-zero Chern number. Our discoveries provide a fresh perspective on 2DPs and redefine the search for Lieb lattice materials into a well-defined chemical synthesis task.


INTRODUCTION
Exotic electronic structures, exemplified by Dirac cones and flat bands, have emerged as a focal point in contemporary research due to their unique electronic properties, including exotic charge carrier mobilities and the induction of topological effects. 1,2An ample example of these exotic electronic structures are the Dirac cones in honeycomb (hcb) lattice, which were first predicted theoretically and only later gained importance with the discovery of graphene. 3,4Since then, graphene has found many applications in electronic devices, high-speed transistors, spintronics, photonics and optoelectronics. 5All these applications are possible thanks to the Dirac cone, characterized by crossing bands with linear dispersion intersecting at K point of the Brillouin zone.This implies the existence of massless electrons from a non-relativistic perspective, consequently leading to exceedingly high electron mobility and topological effects. 4,6On the other hand, flat (dispersionless) bands are characterized by electrons with extraordinarily large effective masses and energies that are independent of the carrier momentum. 2,7,8Partially filled flat bands can then result in novel phases of matter, such as superconductivity, magnetism, and metal-insulator transitions. 9e relationship between honeycomb lattice and Dirac cones can be generalized to a statement that electronic structure features arise directly from the lattice symmetry of the materials. 101][12] Among these, the Lieb lattice signature electronic structure is a very interesting one because it contains flat bands exactly crossing the Dirac cone (denoted as "Lieb bands" in the remaining manuscript) (Figure 1). 7These bands may be interesting from the viewpoint of electronic topology, since the Dirac bands in an ideal Lieb lattice (the corner and edge sites are in the same energy, dE=0) are topologically non-trivial and the contact between them and the flat bands is protected by the real-space topology.When dE≠0, where the corner and edge sites are in different energies, the flat band becomes topologically non-trivial, while one of the Dirac bands becomes trivial with a band gap between it and the flat band being opened. 10,13,14Theoretical predictions using the Tight-Binding (TB) model show that Lieb bands require ideal lattice symmetry and strict conditions on state energies. 13,15,16Because of these very strict criteria, the electronic Lieb lattice has rarely been achieved experimentally.So far, Lieb lattice has been investigated mainly by using optical lattices, 17 or by surface deposition of small molecules. 18A recently synthesized two-dimensional (2D) sp 2 carbon-conjugated covalent organic framework (COF) 19 was theoretically demonstrated to have Lieb bands. 13,16Soon after, the ZnPc polymer, an analogue to the experimentally achievable FePc polymer, 20 with zinc-phthalocyanine as the lattice center and benzene ring as the linker, was predicted to have Lieb bands, which remain topologically non-trivial after chemical substitution or physical strain engineering. 15However, while ZnPc band structure was identified as a Lieb lattice in the original paper, it much more resembles that of fes lattice, which has been in depth studied in refs 11,21 .Fes characteristic bands (denoted as fes bands in the remaining manuscript, Figure 1c), have two highsymmetry crossing points (Γ and M), with one locally flat band and two conical bands crossing at a Dirac point. 10,11 this study, we have investigated a series of hypothetical two-dimensional polymer (2DP) structures derived from the zinc-phthalocyanine (ZnPc) 2DP as model sql polymers.These derivatives, denoted as ZnPc-xP, feature linkers with varying lengths, where x represents the number of aromatic rings in the linker (ranging from 1 to 5, Figure 2). 22Interestingly, the band structure of the ZnPc-xP 2DPs, while structurally having a simple sql lattice, exhibits an evolution from fes bands to Lieb bands, depending on the linker chain length.In particular, the Lieb bands of ZnPc-4P material are in perfect agreement with the TB model of Lieb lattice, including their non-trivial electronic topology.We have also shown that the features of the Lieb bands are preserved when the Fermi level is shifted by both simple electron removal or atom substitution, thereby transforming the challenge of Lieb lattice search into a well-defined chemical synthesis task.

RESULTS
The basic building units of the 2DPs of interest in this study are the ZnPc molecule, and acenes (benzene, naphthalene, anthracene, tetracene, and pentacene, denoted as 1P, 2P, 3P, 4P, and 5P, respectively).These building blocks assemble into flat 2D sheets with square pores, forming ZnPc-xP 2DPs, where x=1, 2, …, 5 represents the number of aromatic rings in the linker.
The ZnPc-xP 2DPs family represents a very interesting case study into the lattice structure and electronic properties relationship.From the structural point of view, common in experimental/synthetic materials community, they would be considered as sql.However, in a deeper look, Lieb and fes lattices can also be formally projected on the ZnPc-xP geometries (Figure 3).In order to understand how the electronic topology originates from the atomistic structure, we have constructed three simplistic, full-atomic hypothetical carbon allotrope models featuring fes and sql/Lieb lattices similar to the TB lattice models.The first model (Figure 6a), T-graphene, 23 constitutes exactly fes lattice, and produces nearly ideal fes bands.The electronic features of the fes lattice are preserved in T-graphtriyne 24 (Figure 6b), which incorporates an extended linker between the rhombic nodes.If the rhombic center of T-graphtriyne is substituted with a single Zn atom forming Zn-diyne (Figure 6c), which would traditionally be considered a sql lattice, distorted Lieb bands emerge instead of the sql bands. 10These simple models confirm that sql TB model is too simplistic to describe square-pore 2DP systems, since even as simple D4h symmetry structure as in Figure 6c contains Lieb bands.The reason for this is that the electronic structure is defined by topology of the scalar field of electron density rather than by simple geometry, which means that the lattice of the material cannot be simply mapped by its atomistic structure.The primary limitation of the studied 2DPs lies in the positioning of the Lieb bands below the Fermi level.Achieving nontrivial structural properties in these bands thus requires shifting the Fermi level while preserving the Lieb bands.To address this, we first conducted an analysis of the atom contributions to the charge density within the Lieb bands (VB1, VB2, and VB3) using ZnPc-4P 2DP as a model system (Figure 7a, Figure SI-5).The primary contribution to the Lieb bands comes from the linker (tetracene) π-electrons, which contribute to all VB1-3, albeit the carbons in the phthalocyanine center also partially contribute to the Dirac bands VB1 and VB3 (Figure 7).
The contribution of the metal atom to the Lieb bands is only minimal.This is also confirmed using a structure without a metal atom in phthalocyanine center (Pc-4P), which gives almost identical Lieb bands as ZnPc-4P (Figure 7).However, the fes features in conduction bands collapse after the removal of the metal atom.
Additionally, we have designed a hypothetical reference porphyrin polymer characterized by an "ideal atomic Lieb lattice" structure ZnPc-ZnPc 2DP (Figure 7c, Figure SI-6).Both fes features in conduction bands and Lieb bands in valence bands are present with only minor changes.Lieb bands again contain the full π-system of the linker, despite its bigger size.This suggests that the well-ordered conjugated π-system is important for achieving high quality Lieb electronic materials.We have investigated the modulation of the Fermi level position by two different methods: the direct removal of electrons from the system and atomic substation. 25The removal of two electrons per unit cell effectively shifts the Fermi level toward the flat band.This adjustment results in a slight increase in the dispersion of the flat band, but leaves the fundamental Lieb bands intact (Figure 8).The charge density contributions from the Lieb bands show identical features to those observed in the pristine structure.The Chern number of CB1 is 2, showing non-trivial topological properties, while it is 0 for VB1 and VB2.The charge density analysis of VB1-3 (Figure 7) shows that in order to access the flat band, electrons should be removed from the aromatic system of the linkers, especially in the edge atoms of the linker.We have tested replacing individual carbons (2 atoms per UC, one per linker) in the linker with boron atoms, shifting the Fermi level to the flat band; and with nitrogen, shifting the Fermi level to the fes bands (Figure 9).In the B-substituted structure, the Fermi level shifts to the Lieb bands, and most of the band structure features are preserved, although the dispersion of the "flat band" is stronger.The two Dirac bands show non-trivial topological properties with Chern number of (-1 0 1) for the Lieb bands.In the N-substituted structure, the Lieb bands are preserved, with an additional flat band crossing.These substituted model structures, although not easily achieved chemically, could provide a general guide how Lieb bands could be accessed in other square pore 2DPs with more suitable structures for substitution.

Conclusions
We have investigated the ZnPc-xP 2DPs as a model system for electronic structure topology of square pore 2DPs.While these materials are traditionally considered to have a sql topology well known from TB models, our results challenge this oversimplified view.We found that ZnPc-xP 2DPs exhibit both fes bands and Lieb bands while completely lacking the expected sql features.This is because the electronic structure of these materials is governed by the topology of the electron density, rather than by simple atomic geometry.Furthermore, the flat band and bottom Dirac band of the Lieb bands possess non-trivial topological properties, as evidenced by non-zero Chern numbers.Unfortunately, the Lieb bands in the ZnPc-xP 2DPs are lo-cated below the Fermi level, as well as in most other model materials studied here.However, we have shown that shifting the Fermi level by controlling the number of electrons in the system via gating or substitution preserves the Lieb bands including their topological character.The challenge of finding Lieb lattice structures thus turns into a well-defined chemical synthesis task.This opens up possibility of designing materials with unique properties, such as topologically non-trivial phase.We hope that our work will stimulate further experimental exploration of Lieb-lattice-based topological materials.

Methods
The geometries were optimized using the self-consistent-charge density functional based tight binding (SCC-DFTB) 26 method as implemented in the Amsterdam Modelling Suite (AMS) ADF 2019. 27The 3ob-3-1 parameter set 28 was used for systems with X-Y element pair interaction (X, Y = C, H, Zn), and the matsci-0-3 parameter set was applied for systems including boron. 29Band structure calculations were performed employing FIH-aims with TIER1 basis set with 4×4×1 k-mesh grid 30 , using DFT with the Perdew, Burke, and Ernzerhof (PBE) functional. 31The key parameter "tight" regarding computational accuracy was used to control all integration grids, and the accuracy of the Hartree potential.The Berry curvature and the intrinsic anomalous Hall conductivity were performed using the WANNIER90 package. 32Calculations of edge states and the Chern number were carried out using WannierTools package. 33SOC and spin polarization were taken into account in the topological calculations.The geometries and corresponding band structures for all systems investigated in this work are available at the NOMAD repository as a dataset under doi: 10.17172/NOMAD/2023.11.26-1.

Electronic Lieb lattice signatures embedded in two-dimensional polymers with square lattice
Yingying Zhang, a Shuangjie Zhao, a Miroslav Položij, abc Thomas Heine* abcd

Figure 1 .
Figure 1.Schematic models and band structures of (a) sql, (b) Lieb, and (c) fes lattices in the Tight-Biding model, considering only 1 stneighbor interactions.The red dashed lines indicate the unit cells.Yellow areas indicate Lieb, green areas fes bands.

Figure 2 .
Figure 2. Schematic representation of ZnPc-xP 2DPs series structure.The red shaded area indicates the shared atoms between the center and linker molecules.
Thus, we have investigated the relationship between ZnPc-xP 2DP structures and electronic properties, particularly considering them as possible materials possessing Lieb bands.Indeed, the band structures of ZnPc-xP 2DPs shown in Figure 4 include both fes bands and Lieb bands features.The fes bands are prominently present in the band structure ofZnPc-1P 2DP.The Lieb bands can also be observed below the Fermi level (set to 0) with a distorted flat band located at about -2.3 eV.With increasing linker length, the (almost) flat band gradually approaches the Fermi level, as indicated by the yellow arrow in Figure 4.The Lieb bands are slightly distorted because the flat band has a small dispersion.It is noteworthy that the position of the Dirac cone alternates between Γ and M points as the linker length changes in ZnPc-xP 2DPs (Figure 4 which originates from the parity of the symmetry.The parity in 2DPs structures has recently been reported in the same material by Raptakis et al. 22 Upon further investigation, we found that the same behavior can be observed in other classes of square polymers, as shown in the model structures in Figure SI-1.

Figure 4 .
Figure 4. Band structure and the top view of the ZnPc-xP 2DPs of (a) ZnPc-1P, (b) ZnPc-2P, (c) ZnPc-3P, (d) ZnPc-4P, and (e) ZnPc-5P.The red and blue bands indicate the orbital contribution from the center/linker, respectively, which are also highlighted in the crystal structure using the same color scheme.fes bands and Lieb bands features are highlighted with green and yellow backgrounds in the band structures.

Figure 5 .
Figure 5. (a) Lieb lattice with four parameters, the center site EA, the corner site EB, the nearest neighbor hopping t1 and the next-nearest neighbor hopping t2.(b) the TB fitting of DFT calculated band structures.

Figure 6 .
Figure 6.Band structures and the top view of model structures connected by (a) T-graphene (b) T-graphtriyne, and (c) Zn-diyne.Features of fes and Lieb in the band are colored in red and blue, respectively.

Figure 7 .
Figure 7. Structure, band structure, and charge density distribution at Γ point of the upper three VB bands of (a) ZnPc-4P with Zn atom, (b) Pc-4P without Zn atom, (c) ZnPc-ZnPc 2DP.The red and blue bands indicate the orbital contribution from the center/linker, with the centerlinker partitioning highlighted in the crystal structure using the same color scheme.

Figure 8 .
Figure 8. Band structure and charge density distribution at Γ point of the three Lieb bands of ZnPc-4P with two electrons removed (two holes introduced) per unit-cell.The red and blue bands indicate the orbital contribution from the center/linker.

Figure 9 .
Figure 9. Band structures, and top view of the structures of (a) pristine (b) B-substituted, (c) N-substituted ZnPc-4P.Lieb bands are colored in blue.The arrow indicates the Fermi level shift direction.
Figure SI-1.Band structure and the top view of the ZnPc-xP COF, connected with porphyrin and phenyl group.(a) ZnPP-3P COF, (b) ZnPP-1P COF, and (c) ZnPP-2P COF, showing the parity of the Dirac cone position, altered between M and Γ.

Figure SI- 2 .
Figure SI-2.Band structure and the top view of the ZnPc-4P COF in different boarder position between center/linker.The red and blue bands indicate the orbital contribution from center and linker, which are highlighted with the same color in the crystal structure.

Figure SI- 4 .
Figure SI-4.Edge states of the fitted TB model of ZnPc-4P at different degrees of spin-orbit coupling (SOC).

Figure SI- 5 .
Figure SI-5.Band structure, and charge density distribution at Γ point of the top three VB bands of ZnPc-5Bz.