Abstract
Lieb lattice has been predicted to host various exotic electronic properties due to its unusual Diracflat band structure. However, the realization of a Lieb lattice in a real material is still unachievable. Based on tightbinding modeling, we find that the lattice distortion can significantly determine the electronic and topological properties of a Lieb lattice. Importantly, based on firstprinciples calculations, we predict that the two existing covalent organic frameworks (COFs), i.e., sp^{2}CCOF and sp^{2}NCOF, are actually the first two material realizations of organicligandbased Lieb lattice. Interestingly, the sp^{2}CCOF can experience the phase transitions from a paramagnetic state to a ferromagnetic one and then to a Néel antiferromagnetic one, as the carrier doping concentration increases. Our findings not only confirm the first material realization of Lieb lattice in COFs, but also offer a possible way to achieve tunable topology and magnetism in organic lattices.
Introduction
The electronic properties of a crystal are determined by its crystalline lattice symmetry. Lieb lattice, a twodimensional (2D) edgedepleted square lattice (Fig. 1a), can be regarded as the reduced 3D perovskite lattice^{1}. As one of the most important frustrated lattices, the ideal Lieb lattice can host exotic electronic structures, which is featured by the Dirac cone intersected by a flat band (Diracflat bands)^{2}, as shown in Fig. 1c. Interestingly, various physical phenomena, e.g., topological Mott insulator^{3}, superconductivity^{4}, ferromagnetism^{5,6}, and fractional quantum Hall (FQH)^{7,8,9,10} effects, have been predicted to exist in the ideal Lieb lattice systems. However, until now only a few Lieb lattices have been realized in artificial lattice systems, e.g., molecular patterning lattices on metal substrates^{11,12}, photonic and coldatom lattices^{13,14,15,16}, rather than a real material system, which significantly prevent the realization of these unusual physical properties of Lieb lattice for practice applications.
Although it is found that the Lieb lattice is extremely difficult to be realized in an inorganic material, it is still possible to realize such a lattice in an organic one, as the structural flexibility allows an organic system exhibiting diverse crystalline symmetries^{17,18,19,20,21,22,23,24,25}. Especially, it is noticed that the covalentorganic frameworks (COFs) can host various 2D lattices^{18,21,24,26}, including square lattice^{21,26}, which makes the discovery of a Lieb lattice in a COF possible. Usually, the molecular orbitals (MOs) play an essential role in determining the electronic properties of a ligandbased organic semiconductor, e.g., a COF. Since the MOs in an organic system can be more easily modulated than the atomic orbitals (AOs) in an inorganic system by the lattice distortions, it is expected that the tunable electronic and magnetic states could be achieved in an organic Lieb lattice.
In this article, using tightbinding (TB) modeling and firstprinciples densityfunctional theory (DFT) calculations, we find that the electronic and topological properties of a Lieb lattice can be effectively modulated by its lattice distortions. Interestingly, we discover that sp^{2}CCOF and sp^{2}NCOF, which are synthesized in recent experiments^{27}, are the first two material realizations of organicligandbased Lieb lattices. The electronic structures of these two COFs around the band edges can be well characterized by the MOsbased Diracflatband models. Furthermore, we find that the lattice distortions can also dramatically affect the bandwidth of the Diracflat bands, which in turn determines its electronic instability against spontaneous spinpolarization during carrier doping. Remarkably, it is found that both ferromagnetic (FM) and Néel antiferromagnetic (AFM) phases can be realized in these distorted Lieb lattices under certain hole doping concentrations (n_{h}). Our discovery not only confirms the first material realization of Lieb lattice in COFs, but also suggests an interesting routine to realize tunable topological and magnetic states in d(f) orbitalfree organic lattices.
Results
Threeband Lieb lattice models
We begin with the threeband tightbinding Hamiltonian of Lieb lattice [without spinorbit coupling (SOC) effects], including one corner and two edge sites in a 2D linecentered square lattice unitcell:
where c_{i} (\(c_i^\dagger\)) annihilates (creates) an electron with the energy of E_{i} on the ith site (i = 1, 2, or 3) in the unitcell, and t_{1} and t_{2,3} are nearestneighbor (NN, \(\left\langle {i,j} \right\rangle\)) and next nearestneighbor (NNN, \(\left\langle {\left\langle {i,j} \right\rangle } \right\rangle\)) hopping integrals, respectively, as shown in Fig. 1a. The details of TB modeling can be found in the Supplementary Notes 1–3.
For an ideal Lieb lattice, the onsite energy of the three sites are identical, i.e., E_{1} = dE/2 = 0 and E_{2} = E_{3} = −dE/2 = 0. Then, we can calculate the (threeband) electronic structure of an ideal Lieb lattice. When the NNN hopping is not included, as shown in Fig. 1c, the calculated band structure is featured by the Dirac cone intersected by a flat band. The wavefunction of flat band is mostly localized at the edge sites, while the wavefunctions of Dirac bands are equally contributed by both corner and edge sites. When NNN hopping is included, e.g., t_{2} = t_{3} = 0.2t_{1}, the features of the Diracflat bands still maintain besides that the flat band becomes more dispersive around the Γ point in the Brillouin zone (BZ), as shown in Fig. 1d.
Since the ideal Lieb lattice is very rare in solids^{3,4,5,6,7,8,9,10}, it is curious for us to further understand the electronic structures of distorted Lieb lattices. Firstly, we consider the situation of θ ≠ 90°, e.g., θ < 90°, corresponding to the geometrical lattice distortions. In this case, the fourfold rotation symmetry of the system is no longer valid, then t_{2} ≠ t_{3}. As shown in Fig. 1e, the Diracflat bands decompose into two sets of Dirac cones that deviate away from M (M1) to Γ point along the ΓM (ΓM1) in the BZ. Secondly, we consider the situation of dE ≠ 0, which corresponding to different atom (ligand) occupations at corner and edge sites in an inorganic (organic) lattice. In this case, the band degeneracy of Diracflat bands is broken and a bandgap Δ_{1} (hereafter, Δ_{1} is defined as the bandgap between the top Dirac and middle flat bands) is induced, as shown in Fig. 1f. Meanwhile, the flat band and the bottom Dirac band still keep in touch with each other. Interestingly, it is found the wavefunction components of these Diracflat bands are dramatically different from the ideal Lieb lattice, i.e., the top Dirac band is contributed by the corner sites while the flat and bottom Dirac bands are contributed by the edge sites. Finally, we consider the distortions with both θ ≠ 90° and dE ≠ 0, as shown in Fig. 1g, which correspond to the realistic situations in many materials, including the COFs focused in our study. In this case, the changes of Diracflat bands could be considered as a combined effect of θ ≠ 90° (Fig. 1e) and dE ≠ 0 (Fig. 1f). Interestingly, only one Dirac cone can survive around M1 point along the Γ − M1 line.
Distorted Lieb lattice model with SOC effects
We further consider the SOC effects on the distorted Lieb lattice model, with the total Hamiltonian H = H_{0} + H_{SO}. H_{SO} is considered as an imaginary hopping between the NNN sites, similar to Kane and Mele’s SOC term^{9,28}:
where λ is the SOC constant and d_{ik} (d_{kj}) denotes the unit vector from edge (corner) site i (k) to corner (edge) site k (j) (see Supplementary Fig. 1). s is the Pauli matrix representing the electron spin. Generally, we find that SOC effects can lift the degeneracy of the Dirac cone, and the topological properties of these Diracflat bands strongly depend on the value of dE (the values of other parameters are the same as those in Fig. 1g).
For dE = 0 (Fig. 2a), the bandgaps of \({\mathrm{\Delta }}_1\)and \({\mathrm{\Delta }}_2\) (hereafter, \({\mathrm{\Delta }}_2\) is defined as bandgap between middle flat and bottom Dirac bands) can be induced around M and M1 points. It is noted that \({\mathrm{\Delta }}_2\) can only be induced by SOC effects. For the case of \(dE \ne 0\), we find that there is a competing mechanism between dE and SOC for the changes of \({\mathrm{\Delta }}_1\). When 0 < dE < dE_{c} (\(dE_{\mathrm{c}} = 2\sqrt {\left( {t_2  t_3} \right)^2 \, + \, 4{\uplambda}^2}\), see Supplementary Notes 3–5), \({\mathrm{\Delta }}_1\) decreases as dE increases. When dE = dE_{c}, \({\mathrm{\Delta }}_1 = 0\), as shown in Fig. 2b, two Dirac cones can form around M and M1 points. When dE > dE_{c} (Fig. 2c), \({\mathrm{\Delta }}_1\) opens again and increases as dE increases, which is opposite to the case of dE < dE_{c}. Meanwhile, when dE changes through dE_{c}, the wavefunction compositions of top Dirac band and middle flat band are switched, as shown in Fig. 2a, c.
The topological properties of each band can be characterized by its spin Chern number, due to the spin degeneracy and conservation of S_{z}. The spin Chern number for the nth band is defined as \(C_n^{\mathrm{s}} = \left( {C_{n \uparrow }  C_{n \downarrow }} \right)/2\), and \(C_{n{\upsigma}}\) is Chern number for the spin\({\upsigma}\) (\({\upsigma} = \, \uparrow , \downarrow\)) of the nth band, which can be calculated from the integral of the Berry curvature over the whole BZ^{29,30} (see “Method” section and supplementary Fig. 2). As shown in Fig. 2, when dE = 0, the calculated \(C_1^{\mathrm{s}} =  1\) and \(C_3^{\mathrm{s}} = + 1\) for the top and bottom Dirac bands, respectively, and the calculated \(C_2^{\mathrm{s}} = 0\) for middle flat band. Interestingly, when dE > dE_{c}, the top Dirac and middle flat bands can switch their topologies, as shown in Fig. 2c, i.e., \(C_1^{\mathrm{s}} = 0\) (\(C_2^{\mathrm{s}} =  1\)) for the top Dirac (middle flat) band.
The nontrivial topology of these Diracflat bands in Fig. 2a–c can be reflected by edge states calculations, as shown in the according Fig. 2d–f. It can be seen that the helical edge states with opposite spin channels connect the two bands with nonzero \(C_n^{\mathrm{s}}\), indicating the quantum spin Hall effect.
It is emphasized that in a realistic material, only the total spin Chern number C_{s}, the sum of \(C_n^{\mathrm{s}}\) for all the occupied bands, is associated with the observable quantum conductance of the system. Therefore, the topological properties of a realistic Lieb lattice material sensitively depend on the position of Fermi level, and the charge doping may be needed to achieve a nontrivial state.
Realization of distorted Lieb lattices in COFs
Taking advantage of the structural flexibility of organic materials, numerous COFs with various crystalline symmetries have been synthesized in recent years. Remarkably, after extensive structural analysis, we notice that two COFs with sp^{2} hybridized backbones fabricated in the very recent experiments, named as sp^{2}CCOF and sp^{2}NCOF, can realize slightly distorted square Lieb lattices^{27} (θ < 90°), as shown in Fig. 3a, b, respectively. Meanwhile, the ligands at the corner and edge sites are different in monolayer sp^{2}CCOF and sp^{2}NCOF, corresponding to the case of dE ≠ 0. In both COFs, a pyrene surrounded by four phenyl rings is located at the corner site, named as 1,3,6,8tetrakis(2phenyl)pyrene (TPPy), while the ligands at the edge sites in these two COFs are slightly different, i.e., 1,4bis(2crylonitrile)benzene (BCNB) in sp^{2}CCOF and 1,4bis(2aldimine)benzene (BAIB) in sp^{2}NCOF. Meanwhile, crylonitriles (CN) and aldimine (AI) bond to the bis positions on the phenyl ring of edge sites in sp^{2}CCOF and sp^{2}NCOF, respectively. The enlarged view of connection parts between corner and edge sites are shown in Supplementary Fig. 3.
We have calculated the electronic properties of monolayer sp^{2}CCOF and sp^{2}NCOF using DFTPBE calculations, as shown in Fig. 3c, d, while the results for bulk sp^{2}CCOF and sp^{2}NCOF can be found in Supplementary Fig. 4. Generally, it is found that monolayer and its corresponding bulk COFs have similar electronic properties due to the relatively weak interlayer van der Waals (vdW) interactions. The bandgaps of monolayer sp^{2}CCOF and sp^{2}NCOF are 1.45 and 1.30 eV, respectively, which is smaller than the experimental values (e.g., 1.9 eV for sp^{2}CCOF)^{27} due to the underestimation of bandgaps in DFT calculations.
Here, we mainly pay our attention to the band edges, especially for the top three valence bands, i.e., VB1–VB3 marked in Fig. 3c, d, in both COFs. In the band structure of sp^{2}CCOF (Fig. 3c), VB1 separates from VB2 and VB3 in energy, opening an energy gap of \(\Delta _1\)~0.4 eV. Notably, VB2 and VB3 touch with each other forming a single Dirac cone near the M1 point on the \({\mathrm{\Gamma }}  {\mathrm{M}}1\) line. The wavefunction distributions of energy bands can be better understood by the ligandbased MOs than AOs in a COF. Generally, the band edges in a ligand lattice are formed by the HOMOs and LUMOs of the ligands, respectively^{31}, and the interaction between the HOMOs and LUMOs are negligible. Therefore, the VB1–VB3 can be considered separately from the CB1–CB3 in a TB modeling based on the Lowdin perturbation theory. As shown in Fig. 3c, the charge densities of VB1–VB3 are projected into the ligands of sp^{2}CCOF. Interestingly, it is found that the top VB1 is solely contributed by the HOMO of TPPy (corner ligands), while the less dispersive VB2 and the bottom VB3 are contributed by the HOMOs of both BCNBs (edge ligands), respectively. Interestingly, it is found that the band dispersions and wavefunction distributions of VB1–VB3 (Fig. 3c) indeed agree well with the threeband distorted Lieb lattice model under the situation of dE ≠ 0 and θ ≠ 90° (Fig. 1g). The band structure of sp^{2}NCOF (Fig. 3d) is similar to that of sp^{2}CCOF. The \(\Delta _1\) in sp^{2}NCOF is ~0.2 eV, which is smaller than that in sp^{2}CCOF. Meanwhile, it is also noted that VB1 in sp^{2}CCOF is much less dispersive than that in sp^{2}NCOF. The calculated SOC effects (\({\mathrm{\Delta }}_2\)~0.03 meV) are negligible in these two COFs, indicating that it could be challenging to observe the nontrivial topological properties (Fig. 2) in these two COFs.
Since VB1–VB3 in these two COFs are similar to that of threeband distorted Lieb lattice model, we can further understand the band dispersions of VB1–VB3 using TB band fitting of DFT calculations. Overall, we find that the differences of VB1–VB3 band dispersions in these two COFs are mainly determined by the difference of dE between the HOMOs of corner and edge sites in these two systems. It is found that dE determines the size of \(\Delta _1\), i.e., the larger the dE, the larger the \(\Delta _1\) will be (without SOC effects). Meanwhile, \(\Delta _1\) can in turn determine the band dispersion of VB1, i.e., the larger the \(\Delta _1\), the flatter the VB1 will be. In addition, the interfacial torsion angle between the corner and edge ligands in the sp^{2}CCOF is larger than that in the sp^{2}NCOF (see Supplementary Fig. 3b, e). The larger torsion angle can reduce the conjugation in a larger degree in the COF and lead to a smaller t_{1}. Therefore, the larger dE and torsion angles in sp^{2}CCOF than in sp^{2}NCOF can give rise to a flatter VB1 in sp^{2}CCOF, agreeing well with the DFT results. On the other hand, our understandings suggest a possible way to design flat bands in organic lattices, as the flat bands could have great importance for realizing many novel physical phenomena.
Similarly, the bottom three conduction bands (CB1–CB3), formed by the LUMOs of the edge and corner ligands, can also be understood by the MOsbased Lieb lattice model. The TB band fitting of CB1–CB3 within Lieb lattice model is shown in Supplementary Fig. 5. Due to the small dE between the LUMOs of corner and edge ligands, the CB1–CB3 formed Diracflat bands (Fig. 3c, d) are close to the ideal ones (Fig. 1c).
Ferromagnetism in sp ^{2}CCOF and sp ^{2}NCOF
In the experiments, the FM phase, with an easy axis perpendicular to the COF plane, i.e., along the z direction in Fig. 3a, b, is found in sp^{2}CCOF after iodine oxidization at a low temperature (8.1 K)^{27}. This is unusual as the sp^{2}CCOF not containing any transitionmetal atoms can evoke FM ordering, which inspires us to further understand the magnetic properties of these COFs after iodine doping. Our extensive test calculations indicate that the most important role of iodine doping is to introduce extra holes in sp^{2}CCOF and sp^{2}NCOF. The n_{h} depends on the iodine concentration, but it is insensitive to the adsorption sites of iodine atoms. Besides of the role of hole doping, ionized iodine dopants can introduce some localized defect levels around E_{F}. In the following, we will discuss the mechanism of n_{h}dependent magnetic phases in the main text and leave the results of iodine doping in Supplementary Fig. 6. According to the Mermin–Wagner theorem^{32}, for a 2D system there is no realistic longrange order at finite temperature. Hence our discussion should be constrained on finitesize 2D lattices.
Figure 4 shows the band structures of COFs under a typical doping concentration of n_{h }= 0.5 holes per unitcell (u.c.). The VB1 in both sp^{2}CCOF and sp^{2}NCOF can become partially occupied. According to the Stoner model^{33}, the FM of iterative carriers can arise from partially occupied (except for the case of halffilling^{34}) band with a sufficient narrow bandwidth, i.e., the smaller the bandwidth is, the larger the effective electronic interactions will be. Since VB1 in sp^{2}CCOF has a rather narrower bandwidth than that in sp^{2}NCOF, the Stoner criterion for realizing a FM ordering can be more easily satisfied in sp^{2}CCOF than in sp^{2}NCOF. It is found that the spontaneous spin polarization exists in both COFs under n_{h} = 0.5 holes per u.c., and the COFs becomes metallic. The calculated exchange energy splitting of VB1 in sp^{2}CCOF (sp^{2}NCOF) is about 85 (40) meV. The spin density distributions of these two systems are mainly localized around the corner sites (see Supplementary Fig. 7). For the sp^{2}CCOF, the magnetic moment is 0.5 μ_{B} per u.c. at n_{h} = 0.5 holes per u.c., i.e., 1.0 μ_{B} per hole. The FM state is energetically more favorable than the AFM one by \(\Delta E = E_{{\mathrm{FM}}}  E_{{\mathrm{AFM}}} =  5.0\) meV [or than the nonmagnetic (NM) one by −6.9 meV] (see Supplementary Table 1). For sp^{2}NCOF, the magnetic moment is ~0.378 μ_{B} per unitcell and the FM ordering is slightly more stable than the AFM one (\(\Delta E =  1.5\;{\mathrm{meV}}\)). Therefore, the stronger FM stability in sp^{2}CCOF is induced by its narrower VB1 bandwidth, which originates from the larger dE and torsion angles in sp^{2}CCOF than in sp^{2}NCOF.
Although the magnetism originates from the manybody effects, which are usually underestimated by the meanfield approximation (MFA) within the DFT methods, the spinpolarized ground states of similar weaklycorrelated systems have been correctly evaluated^{35,36,37,38,39,40,41} and confirmed in the experiments^{42,43}. The onsite electronelectron (e–e) interactions (U) in the sp^{2}CCOF and sp^{2}NCOF (sp^{2}hybridized C and N systems) are very weak, compared to the traditional magnetic (3d or 4f) systems. On the other hand, the widely acceptable remedy within the DFT formalism to include the electron correlations is DFT + U method. We have performed a GGA + U calculation with U = 1.5 eV on the 2p orbitals of C and N atoms, and our calculations show that the enhanced U can slightly increase the exchange splitting E_{s} (see Supplementary Fig. 8) and hence further enhance the stability of FM groundstates. Therefore, the conclusion of FM ordering in holedoped COFs obtained from DFT calculations can be enhanced under DFT + U calculations.
To estimate the FM transition temperature (T_{c}), we have performed Monte Carlo (MC) simulations within the 2D Heisenberg model with uniaxial anisotropy (See “Methods” section for details), which is successfully applied to simulate the T_{c} of many 2D systems^{44,45}. The reduced exchange integral J and magnetic moment m are 2.5 (1.31) \({\mathrm{meV}} \cdot {\upmu}_{\mathrm{B}}^{  2}\) and 0.5 (0.378) μ_{B} for sp^{2}CCOF (sp^{2}NCOF), respectively. The magnetic anisotropic energy (MAE = E_{outofplane} − E_{inplane}), determined by the energy difference between the outofplane and inplane spin configurations, is ~−0.1 meV for both COFs, which indicates that the magnetic easy axis is perpendicular to the COF planes, agreeing with the experimental observations. Although the value of MAE is small, it is very important for these two COFs to overcome the spin fluctuation and maintain a longrange magnetic ordering at finite temperature. The calculated Tdependent magnetic moment (per TPPy) is shown in Fig. 4c. Here, both sp^{2}CCOF and sp^{2}NCOF are under n_{h} = 0.5 holes per u.c. It can be seen that the inplane components of magnetization are zero at arbitrary temperature for both COFs, but their outofplane components are nonzero at low temperature. The calculated T_{c} is 5.3 K (close to 8.1 K of experiments) and 1.7 K for sp^{2}CCOF and sp^{2}NCOF, respectively.
Magnetic phase transitions in sp ^{2}CCOF
Besides of n_{h} = 0.5 holes per u.c., we have further explored the possible magnetic phase transitions as a function of n_{h} at the DFTlevel calculations. As shown in Fig. 3, the VB1 of sp^{2}CCOF is contributed by the MO of corner ligands, which has a small effective intersite (ligand) hopping t_{1} = 0.1 eV. Meanwhile, the effective onsite (ligand) U is usually in the range of 0.5~1.5 eV for organic systems^{46,47}, which is significantly larger than t_{1}. The calculated \(\Delta E\) as a function of n_{h} is shown in Fig. 5a. When n_{h} < 0.3 holes per u.c. (yellow region), the magnetic moment on each ligand is too small to evoke a preferred magnetic ordering, giving rise to a paramagnetic (PM) phase. When 0.3 < n_{h} < 0.75 holes per u.c., the FM state becomes to be more favorable (\(\Delta E < 0\), green region) according to the Stoner’s criterion. Interestingly, when \(\Delta E\) reaches a maximum value of −5.7 meV at n_{h} = 0.55 hole per u.c., it becomes to be reduced gradually and finally reaches \(\Delta E = 0\) at n_{h} = 0.75 holes per u.c. When 0.75 < n_{h} < 1.0 holes per u.c., the effect of Fermi surface nesting plays a dominated role in determining its magnetic groundstate, as shown in Supplementary Fig. 9, giving rise to a Néel AFM state (blue region).
Employing the Monte Carlo simulations within the 2D Heisenberg model with the uniaxial anisotropy, we further show n_{h} dependent magnetic phase diagram of sp^{2}CCOF in Fig. 5b. It can be seen that the FM or AFM phase can survive at finite temperatures under different n_{h}. Interestingly, our calculated magnetic phase diagram of sp^{2}CCOF system agrees well with that by J. E. Hirsch based on a similar model system (at the MFAlevel calculations) with 6t_{1} < U < 10t_{1}^{34}. Remarkably, the T_{c} of FM ordering can reach a maximum value of ~5.7 K for the sp^{2}CCOF at n_{h}~0.55 holes per u.c. Finally, we predict that at an extremely heavy hole doping concentration (n_{h} = 3.0 holes per u.c.), the system can reach a halfmetallic Dirac semimetal state, as shown in Supplementary Fig. 10.
Discussion
t is emphasized that a Heisenberg model is adopted to simulate the T_{c} of various magnetic phases. Due to finitesize effect, a series of phase transition temperatures can be obtained. Taking such temperature as a proxy to a real T_{c}, we can conclude that the critical temperature of sp^{2}CCOF is higher than that of sp^{2}NCOF. Moreover, the more accurate phase diagram may need more complex methods beyond MFA, e.g., quantum Monte Carlo^{48,49} or dynamical meanfield theory^{50,51,52}, which is out of the scope of the current study.
In conclusion, based on TB modeling and firstprinciples calculations, we predict that the sp^{2}CCOF and sp^{2}NCOF, which have been synthesized in the recent experiments, are actually the first two material realizations of organicligandbased Lieb lattices. The lattice distortion and inequivalent corner and edge sites make the electronic structures of these two systems deviate from the ideal Diracflat bands in a Lieb lattice. Interestingly, these two COF systems can be converted to the FM and AFM states at certain n_{h} due to the less dispersive top valence bands. Our discovery not only can extend our understanding on the electronic and topological properties of distorted Lieb lattices, but also offers a possible way to realize exotic magnetic states in frustrated organic lattices, which opens the possibility to design the novel organic spintronic devices.
Methods
DFT calculations
Our ab initio calculations for the electronic structures of periodic structures were carried out within the framework of the PerdewBurkeErnzerhof generalized gradient approximation (PBEGGA)^{53}, as embedded in the Vienna ab initio simulation package code^{54,55}. All the calculations were performed with a planewave cutoff energy of 450 eV. For the sp^{2}CCOF, we adopted the experimental lattice constants as starting structure to perform geometric optimization. sp^{2}NCOF holds a very similar structure as sp^{2}CCOF based on our calculation. Both COFs were fully relaxed without any constraint until the force on each atom was <0.01 eV Å^{−1}. The final lattice constants are a = b = 24.634 Å, c = 3.569 Å, α = 79.407°, β = 100.593° and γ = 88.299° for the sp^{2}CCOF and a = b = 24.282 Å, c = 3.465 Å, α = 78.492°, β = 101.508° and γ = 88.490° for sp^{2}NCOF. To eliminate the interlayer interaction, we introduced a vacuum layer of 18 Å thickness for monolayer calculations. The Brillouin zone kpoint sampling was set with a 3 × 3 × 13 mesh for bulk unitcell, and a 3 × 3 × 1 one for monolayer calculations, respectively.
Chern number and Berry curvature calculations
The Chern number for one energy band can be calculated by the integral of the Berry curvature over the whole BZ
The Berry curvature Ω_{n}(k) of the nth band can be determined as^{29,56,57}
where \(\psi _n\left( {\mathbf{k}} \right)\) and ε_{n}k are the spinor Bloch wave function and eigenvalue of the nth band at k point, and v_{x(y)} is the velocity operator. The total Chern number^{29} of a system is the sum of C_{n} over all the occupied bands (n):
2D Monte Carlo simulations
We have performed the Monte Carlo simulations based on the 2D Heisenberg model with uniaxial anisotropy, which is written as
where m_{i} and \(m_i^z\) represent the magnetic moment and its z component on the ith moment, and i, j confines the NN neighboring m_{i} and m_{j}. The J is the exchange integral (\(J =  \Delta E/8m^2,{\mathrm{where}}\Delta E = E_{{\mathrm{FM}}}  E_{{\mathrm{AFM}}}\)) with the assumption of \(\left {{\mathbf{m}}_i} \right = m\). D is the reduced onsite MAE, and D = −MAE/m^{2}. A 100 × 100 supercell containing 10,000 local magnetic moments are adopted, and each simulation lasts 10^{9} loops to relaxation and another 10^{9} loops to collect the physical quantities. In each loop, one moment is rotated to a random direction.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The MC code used for simulations is available by written request to the corresponding authors.
Change history
29 January 2020
An amendment to this paper has been published and can be accessed via a link at the top of the paper.
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Acknowledgements
B.C., X.Z., and D.L. acknowledge the support from NSFC (Nos. 11574118). J.W. and B.H. acknowledge the support from Science Challenge Project (No. TZ2016003), NSFC (No. 11574024) and NSAF (No. U1930402). B.C. also acknowledges the support from Shandong Provincial Natural Science Foundation (No. ZR2019MA64) and the NSFC (No. 11404188). Computations were performed at Tianhe2JK at CSRC. The authors thank Dr. Z. Liu (Tsinghua University), Dr. S. Yin and Dr. M. Zhao (Shandong University) for helpful discussions.
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B.C. and B.H. directed the project. B.C. and X.Z. calculated the results. B.C., J.W., D. L., S. X. and B.H. analyzed the results. B.C., J.W. and B.H. wrote the manuscript. All authors discussed the results and commented.
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Cui, B., Zheng, X., Wang, J. et al. Realization of Lieb lattice in covalentorganic frameworks with tunable topology and magnetism. Nat Commun 11, 66 (2020). https://doi.org/10.1038/s4146701913794y
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DOI: https://doi.org/10.1038/s4146701913794y
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