Lieb lattice with digital inhomogeneity
We begin from the elemental TB mannequin evaluation of the Lieb lattice. There are 4 key parameters figuring out the band construction of a Lieb lattice, i.e., the on-site vitality distinction ΔE, the dimerization interplay δ between the nook and the edge-center websites, the nearest-neighbor (NN) hopping t, and the next-NN (NNN) hopping t′, as indicated in Fig. 1a. For an excellent Lieb lattice with ΔE, δ, and t′ all equal to zero, the band construction is characterised with Dirac cones shaped on the M level, intersected by a flat band, as proven in Fig. 1b. The NNN interplay t′ is thought to have an effect on the band dispersion alongside Γ-X/Y direction6, which is normally relatively small and negligible. Then again, the ΔE and δ are identified to suppress the dispersion of the Dirac bands considerably and elevate the band degeneracy on the M level (Fig. 1c, d)6. It is very important word that nonvanishing ΔE and δ trigger a finite variation of native digital potentials at completely different lattice websites, specifically a finite diploma of EI. The bigger the ΔE and/or δ, the bigger the EI, which may drastically alter the digital, and particularly the magnetic properties of the Lieb lattice, as we elaborate under.
Particularly, a optimistic/destructive ΔE isolates the higher/decrease Dirac band (see e.g., Fig. 1c for optimistic ΔE) from the opposite bands, and a non-zero δ time period totally separates the three bands (see Fig. 1d). Though the band dispersion and the degeneracy at sure okay factors change with these two parameters, principal options of the Lieb lattice stay the same6. The Dirac band and the flat band are primarily contributed by the nook and edge-center states, respectively10, as displayed by the projected density of states (PDOS) within the inset of Fig. 1b. We word that there could be completely different quantity (N) of websites on edges, and all result in related band buildings, i.e., Dirac bands intersected by flat bands (see Supplementary Fig. 1). These lattices are named as Lieb-(2N + 1) lattice, the place 2N + 1 is the variety of websites per unit cell. Apparently, Fig. 1a exhibits a Lieb-Three lattice with N = 1.
Materials realization of the Lieb lattice: sp
We first analyze the molecular construction of the experimentally synthesized 2D sp2c-COF (Fig. 2a), having a characteristic of full π-conjugation alongside each x and y instructions28. The nook and edge-center websites of sp2c-COF are occupied by pyrene (Py) and 1,Four-bis(cyanostyryl)benzene (BCSB) ligands, as highlighted by blue and purple ellipses, respectively. Due to this fact, from the structural standpoint, one might contemplate the sp2c-COF, Py(BCSB)2, to be described by a barely distorted Lieb-Three lattice. To substantiate this, we then perform an digital band matching evaluation. In Fig. 2b, we present the DFT calculated digital band construction of the Py(BCSB)2 within the kz = zero airplane. Clearly, the intrinsic Py(BCSB)2 is a nonmagnetic insulator, and the band hole is discovered to be round 1.zero eV (see Suplementary Fig. 2), which is smaller than the experimentally measured 1.9 eV28. That is cheap as hole dimension is thought to be underestimated by the usual DFT technique, which, alternatively, can fairly seize the dispersions and orbital compositions of each valence and conduction bands. Due to this fact, to avoid wasting time we’ll use the usual DFT technique, because the hole dimension is not going to have an effect on our principal conclusions about magnetic properties. We word that there’s additionally a noticeable dispersion alongside the kz path, indicating robust interactions between the layers (see Supplementary Fig. 2). Moreover, from the orbital-resolved projected band construction in Fig. 2b, one sees that the bands across the Fermi degree are primarily contributed by the pz orbitals, displaying the characteristic of full π-conjugation shaped by the sp2 hybridized C and N atoms. These agree completely with the experimental outcomes28.
2D COF Py(BCSB)2. a High and facet view of crystal construction of Py(BCSB)2 with Py (blue ellipse) and BCSB (purple ellipse) ligands sitting on the nook and edge-center websites of the distorted Lieb lattice, respectively. b DFT calculated digital band construction for kz = zero airplane and the orbital-resolved projected band for pz (black circle) and px,y (inexperienced circle) orbitals. The purple and blue bands spotlight the Lieb-Three and Lieb-5 band construction, respectively
We now give attention to the highest three valence bands (VBs) under the Fermi degree, which are literally in keeping with Lieb-Three bands, i.e., one practically flat band in between two Dirac bands, albeit extremely perturbed by EI. To disclose this, we calculate the band-resolved cost distribution for these three bands (see Supplementary Fig. Three). We discover that the cost distribution for the second band is usually localized on the BCSB ligands on the edge heart, whereas these for the opposite two bands are primarily distributed on the Py ligands on the nook. We additional carried out scanning tunneling spectroscopy simulations by calculating native density of states of edge heart and nook websites (see Supplementary Fig. Three), which present a transparent characteristic of Lieb-Three lattice10. The big EI exhibited by the Py(BCSB)2 is comprehensible, as a result of a big ΔE is predicted between the molecular orbitals (MOs) of Py and BCSB ligands and a non-zero δ is predicted from the structural distortion of the Py(BCSB)2 in contrast with the best Lieb-Three lattice. These outcomes strongly recommend that the valence bands are derived from a non-ideal Lieb-Three lattice. As well as, based mostly on the identical evaluation, we discovered that the underside 5 conduction bands (CBs) above the Fermi degree are the Lieb-5 bands (see Supplementary Fig. Four), having two flat bands in between three Dirac bands.
Moreover, to higher perceive the mechanism of band formation, it’s instructive to review the molecular info of the 2 constructing models, i.e., Py and BCSB. As proven in Fig. Three, we discovered that the MOs across the Fermi degree for each Py and BCSB are all π-conjugated, confirming once more the complete π-conjugation of the crystalline Py(BCSB)228. The wavefunctions of the best occupied MOs (HOMOs) of Py (Fig. 3a) and BCSB (Fig. 3b) present precisely the identical form because the band-resolved cost distribution for the Lieb-Three VBs (see Supplementary Fig. Three), indicating the digital Lieb-Three VBs are constructed by the HOMOs of the nook Py and edge-center BCSB ligands. Equally, the one lowest unoccupied MO (LUMO) of Py and the 2 LUMOs of BCSB with two localized cost facilities every (highlighted by purple ellipses in Fig. 3b) kind the Lieb-5 band (see Supplementary Fig. Four). This exhibits a uncommon coexistence of each Lieb-Three and Lieb-5 bands in a single single lattice, which is attributable to a coincidence of the energetically completely aligned MOs between Py and BCSB ligands. We have now additionally calculated molecular info utilizing Gaussian bundle displaying constant outcomes (see Supplementary Fig. 5). To extra concretely affirm the Lieb-lattice-like nature of the Py(BCSB)2, we carried out the maximally localized Wannier features becoming utilizing the Wannier90 bundle44. The fitted band construction and the corresponding maximally localized Wannier features present good consistency with the above DFT calculation outcomes and TB analyses under (see Supplementary Fig. 6).
Molecular info. MO vitality ranges and the related wavefunctions of a a Py and b a BCSB molecule. The purple and blue ellipses spotlight the cost localization of LUMOs and HOMO, respectively
Subsequent, we quantify the diploma of EI in Py(BCSB)2 by becoming the TB mannequin to the DFT band to estimate ΔE and δ. As a result of the three VBs and 5 CBs of Py(BCSB)2 are shaped by completely different MOs and separated distinctly within the vitality house, they are often independently reconstructed by a three- and five-band TB mannequin on the Lieb-Three and Lieb-5 lattice, respectively. We then modify the ΔE based mostly on the vitality diagram calculated for MO evaluation, and match the DFT calculated bands with completely different t and δ (see Supplementary Fig. 2 and Supplementary Be aware 1). Particularly for the Lieb-Three band, we discovered the hopping integral (t ≈ zero.13 eV) is relatively small, which is attributable to a big separation between the HOMOs of Py and BCSB ligands. Extra importantly, we discovered comparatively a big on-site vitality distinction (ΔE ≈ t ≈ zero.14 eV) and a robust dimerization interplay (δ ≈ zero.3t ≈ zero.04 eV) between the 2 HOMOs, indicating a excessive diploma of EI, giving rise to a really slim bandwidth (W ≈ zero.36 eV). Consequently, the higher Dirac band turns into extremely localized and is totally remoted from the 2 bands under. We then in contrast our TB becoming parameters with these of the Wannier fitted Hamiltonian, which exhibits an excellent settlement, additional confirming the robust EI impact to induce a localization of the VB (see Supplementary Be aware 2).
Ferromagnetism in sp
Understanding the sp2c-COF to be a Lieb-lattice-like system, we now proceed to elucidate the bodily mechanisms underlying the experimentally noticed magnetic behaviors of the sp2c-COF upon iodine doping. It’s well-known Lieb lattice can spawn intriguing magnetic properties related to the unique flat band1,2. Nevertheless, the flat band in Py(BCSB)2 is the second VB, which is past the attain of typical doping degree. Due to this fact, one should search for a definite mechanism. We understand that based mostly on Stoner criterion45, the ferromagnetism can come up from any localized band below partial filling. A more in-depth take a look at the primary VB of Py(BCSB)2 reveals that this typical dispersive Dirac band has develop into extremely localized as a result of robust EI. As a result of the VB is fabricated from 2p state, the Stoner parameter is in actual fact even bigger than the 3d state46. Consequently, one might anticipate that this band will now be subjecting to an instability towards spin polarization upon gap doping. To substantiate this, we stock out a computational experiment to dope the system with one gap per unit cell, and calculate the corresponding digital and magnetic properties.
From the band construction and PDOS plot proven in Fig. 4a, one can clearly see that the insulating intrinsic Py(BCSB)2 turns into metallic after doping (see Supplementary Fig. 7). The half-filled spin-degenerate bands spontaneously break up (spin splitting J ≈ 20 meV), giving rise to an itinerant FM state. From the distinction of cost distribution earlier than and after doping (see Supplementary Fig. eight), the doped holes are discovered to be primarily situated on the corner-site Py ligands with the identical form as HOMO of Py, which is the one to kind the Dirac band of the Lieb-Three lattice (inset of Fig. 1b and Supplementary Fig. Three). Persistently, the FM aligned spins are primarily localized on the Py ligands as properly, as proven by the spin distribution plot in Fig. 4b. We additional calculated the vitality of anti-ferromagnetic state to verify the FM floor state. After extracting the vitality distinction and becoming to the simplified Heisenberg mannequin (see Supplementary Fig. 9 and Supplementary Be aware Three), the Curie temperature is estimated to be ~9.Three Okay, which agrees very properly with the experimental outcomes (~eight.1 Okay). Utilizing non-collinear spin calculations contemplating spin–orbit coupling, we discovered the magnetic moments favor to align within the path perpendicular to the airplane (see Supplementary Be aware Four).
Magnetism in Py(BCSB)2. a Spin-polarized digital band construction and PDOS of the hole-doped Py(BCSB)2. Spin distribution of b the hole-doped and c the iodine-doped Py(BCSB)2 with iodine (highlighted by blue arrow) sitting on the off-corner web site. Inset exhibits the HOMO of Py
In experiments, the Py(BCSB)2 is chemically oxidized (hole-doped) utilizing iodine vapor28. Due to this fact, moreover the computational experiment of gap doping, we carry out calculations of the Py(BCSB)2 with iodine doping utilizing a number of consultant structural configurations. To decouple the iodine ions between the layers, we assemble a two-layer supercell with one iodine. This provides a comparable doping focus (zero.5 gap per unit cell) with the experiments (zero–zero.7 gap per unit cell)28. The off-corner web site is discovered to be essentially the most secure place (Fig. 4c) after inspecting a number of attainable positions for the iodine (see Supplementary Fig. 10 and Supplementary Be aware 5), which is used to calculate the digital and magnetic properties. From the cost distinction earlier than and after the iodine doping (see Supplementary Fig. eight), we discover a cost switch from the Py(BCSB)2 to the iodine. The generated holes are primarily distributed on the nook Py ligands. Once more, the bottom state of the iodine-doped Py(BCSB)2 is discovered to be FM and the corresponding spin distribution plot is proven in Fig. 4c. We word that moreover the magnetization contributed by Py ligands, the iodine additionally carries certain quantity of magnetization as a result of a partial filling of its p orbitals. We have now additionally examined one other two structural configurations with the iodine sitting on the off-edge-center and nook web site that has ~zero.04 and ~zero.36 eV per supercell greater vitality, respectively, displaying related magnetic conduct (see Supplementary Fig. 11).
Basically, the narrower the bandwidth is, the stronger the magnetization can be45. Recall that there’s a noticeable dispersion alongside kz path within the band construction of bulk Py(BCSB)2 (see Supplementary Figs. 2 and seven) attributable to interlayer interactions, which will increase considerably the width of the Dirac band. Interlayer interplay could be eradicated in a monolayer Py(BCSB)2, and therefore one might anticipate enhanced magnetization. To check this concept, we calculate the digital and magnetic properties of the monolayer Py(BCSB)2, and its three key properties, i.e., bandwidth (W), spin splitting (J), and the best magnetization are listed in Supplementary Desk 1 compared with the majority. We discover that W of the Dirac band is way smaller than that of bulk Py(BCSB)2 and the J is considerably enlarged, which lead to a a lot greater magnetization upon gap doping (see Supplementary Fig. 12).
One other fascinating level is that the magnetization of a flat or extremely localized band relies upon critically on band filling. Due to this fact, it’s pure to anticipate the magnetization to alter with completely different doping degree. We have now calculated the magnetization for each bulk and monolayer Py(BCSB)2 as a operate of doping degree (see Supplementary Fig. 12). One can clearly see the magnetization will increase first at low doping degree of holes when just one spin channel turns into partially stuffed till reaching a most, after which decreases at excessive doping degree when holes begin to fill each spin channels. Because the experiments solely noticed a rise of magnetization with none lower28, we consider the induced magnetization has not reached the utmost for the iodine-doped Py(BCSB)2. That is probably as a result of a restricted quantity of iodine could be doped into the majority pattern with comparatively a small floor space. To extend the doping focus, one might use a thinner or monolayer Py(BCSB)2 with greater surface-to-volume ratios, resulting in enhanced magnetization. To check these concepts, we propose experiments to additional examine the magnetization of sp2c-COF as a operate of doping degree and movie thickness.
It’s price mentioning that one other COF with C=N conjugation (C=N-COF), synthesized by the identical group, additionally exhibits related however weaker magnetic response28. Our calculations present related band construction and magnetic conduct upon gap doping for C=N-COF, however apparently, a stronger magnetic response than the monolayer Py(BCSB)2 (see Supplementary Fig. 13), due to each a narrower W and a bigger J (see Supplementary Desk 1). This distinction may very well be attributable to completely different pattern qualities of those two COFs. We consider a greater pattern high quality might yield an excellent greater spin density within the C=N-COF. One other fascinating level to say is that if the system may very well be doped additional, the unique flat band triggered ferromagnetism would evolve, which may very well be probably used to review the flat-band-related topological state in Lieb lattice.
In conclusion, we have now found that the experimentally synthesized 2D natural system of sp2c-COF is a fabric realization of a Lieb-like lattice. Moreover, we display that it’s a non-ideal Lieb lattice with robust EI, in order that it displays an insulator-to-metal transition and a nonconventional magnetic instability, in keeping with experimental remark. Our findings open the door to exploiting the extremely tunable COFs as a flexible materials platform to discover unique digital and magnetic properties hosted by Lieb lattices. Furthermore, the Stoner mechanism for ferromagnetism supplies a helpful theoretical steering seeking new COF-based natural magnets. The development in elementary data enabled by our theoretical examine may additionally be transferred to technological functions for COFs as digital and spintronic supplies, past the already-known conventional functions as structural supplies. Our work may additionally foster new analysis instructions to review different unique physics of Lieb lattices past magnetism in COFs, similar to topological properties.