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Wenbin
Li
^{a},
Lei
Sun
^{b},
Jingshan
Qi
^{c},
Pablo
Jarillo-Herrero
^{d},
Mircea
Dincă
*^{b} and
Ju
Li
*^{c}
^{a}Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
^{b}Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. E-mail: mdinca@mit.edu
^{c}Department of Nuclear Science and Engineering, Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. E-mail: liju@mit.edu
^{d}Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

Received
17th November 2016
, Accepted 20th January 2017

First published on 8th February 2017

We use first-principles calculations to show that the square symmetry of two-dimensional (2D) metal–organic frameworks (MOFs) made from octaamino-substituted phthalocyanines and square planar Ni^{2+} ions, which enable strong conjugation of π electrons, has a critical impact on the magnetic properties of the lattice. In particular, we predict the unexpected emergence of a rare high-temperature ferromagnetic half-metallic ground state in one case. Among charge neutral MOFs made from (2,3,9,10,16,17,23,24)-octaiminophthalocyanine (OIPc) metallated with divalent first-row transition metal ions (M-OIPc; M = Cr^{2+}, Mn^{2+}, Fe^{2+}, Co^{2+}, Ni^{2+}, Cu^{2+}, Zn^{2+}) and connected through square planar Ni-bisphenylenediimine moieties, NiMn-OIPc exhibits a half-metallic and ferromagnetic ground state with a large exchange energy resulting from the unique strong hybridization between the d/π orbitals of Mn, the Pc ring, and the Ni-bisphenylenediimine nodes. Notably, we show that for NiMn-OIPc there is a considerable difference between the ferromagnetic ordering temperature (T_{c}) predicted by a 2D Ising model, which exceeds 600 K, and a T_{c} of 170 K predicted by our more realistic Monte Carlo simulation that includes magnetic anisotropy. Critically, our simulations adopt two spin models that incorporate magnetic anisotropy in the form of exchange anisotropy and single-ion anisotropy. We further show that in the bulk, 2D layers of NiMn-OIPc adopt a slipped-parallel stacking configuration, and exhibit interlayer magnetic coupling that is sensitive to the relative in-plane displacement between adjacent layers. These results highlight the critical role of magnetic anisotropy in modeling the properties of 2D magnetic systems. More generally, it demonstrates that strong hybridization between open-shell ions and delocalized aromatic π systems with appropriate symmetry, combined with large magnetic anisotropy, will be an effective design strategy to realize ferromagnetic 2D MOFs with high T_{c}.

The 2D MOFs synthesized so far^{8–15} are made from benzene- or triphenylene-derived ligands with imine, phenol, or thiophenol functionalities that bind a variety of late-transition-metal ions in a square planar environment. These 2D MOFs exhibit honeycomb lattices that stack in the third dimension, in a manner analogous to graphene and graphite. A schematic of such 2D honeycomb lattices, contrasted with 2D square lattices, is shown in Fig. 1. Although the presence of metals and the greater compositional variety of 2D MOFs allows significant modulation of band dispersion, absolute band energy, Fermi energy, and density of states, all 2D MOFs studied thus far have hexagonal symmetry and consequently exhibit electronic structures that mimic that of graphene symmetry-wise. Deviations from hexagonal symmetry should give rise to band structures that deviate entirely from those of hexagonal lattices, giving rise to electronic properties that diverge fundamentally from those expected from hexagonal 2D symmetry. It is thus highly desirable to explore 2D MOFs made from ligands and metal nodes that enforce non-hexagonal lattice geometries. In this article, we use first-principles calculations to computationally design metal-phthalocyanine (MPc)-based 2D MOFs with square lattices. Our study reveals rich magnetic behavior in this new class of 2D MOFs. In particular, we find that a charge-neutral MOF made from octaaminophthalocyanines metallated by Mn^{2+} ions and bound by Ni^{2+} ions through the phenylenediimine linkages should exhibit high-temperature ferromagnetic half-metallicity in monolayer form. In the bulk phase, MnPc based 2D MOFs exhibit strong intralayer but relatively weak interlayer magnetic coupling that is sensitive to the relative interlayer in-plane displacement. These results indicate the possibility of engineering the magnetic behavior of bulk or few-layer systems by controlling the interlayer stacking geometries.

Fig. 2 Molecular models of octaamino-metal phthalocyanine (MPc) and NiMPc two-dimensional (2D) metal–organic frameworks (MOFs). M = Cr, Mn, Fe, Co, Ni, Cu and Zn. |

Our first-principles calculations are based on density functional theory (DFT),^{30,31} as implemented in the Vienna ab initio simulation package (VASP).^{32,33} Exchange–correlation functional of the Perdew–Burke–Ernzerhof (PBE) form within the generalized gradient approximation (GGA)^{34,35} was used. Coulomb and exchange interactions of the localized d orbitals in TM elements were treated in the framework of the DFT+U method,^{36} using the Dudarev approach.^{37} The effective Coulomb (U) and exchange (J) parameters in DFT+U are U = 4 eV and J = 1 eV. This set of (U, J) parameters for TM d orbitals have been tested extensively in MPc-based systems.^{28,38–40} We have also tested a slightly different set of (U, J) parameters for MPc recently proposed by Brumboiu et al.^{41} and obtained results with quantitative similarity. Other details of the computational methods can be found in the ESI.†

To determine the structural and magnetic ground states of the 2D MOFs, structural models of the 2D MOFs in unit cells were first fully relaxed in DFT+U with spin-polarized conditions. After relaxation all the systems adopt structures that can be identified as 2D square lattices in the p4m plane group. The CrPc, MnPc, FePc, CoPc and CuPc 2D MOFs exhibit unit cell magnetization of 4, 3, 2, 1, and 1 Bohr magneton (μ_{B}) respectively. The NiPc and ZnPc-based materials are non-magnetic because the TM atoms in these MPc have closed shells.^{26} In the systems that exhibit magnetization, the magnetic moments are predominantly localized on the TM atoms in the MPc moieties. The square-planar Ni ions connecting the MPc moieties do not carry magnetic moments that can be attributed to localized d electrons, since the Ni ions do not possess unpaired electrons in the square-planar coordination mode.

The magnetic ground states of the 2D MOFs with unit-cell magnetization are determined by comparing the energies of the systems with ferromagnetic (FM) or antiferromagnetic (AFM) coupling between the TM atoms in the MPc moieties. This is carried out in a structural supercell, which would be the magnetic unit cell if the system assumed AFM coupling, as illustrated in Fig. 3. Denoting the system energy per supercell by E_{AFM} and E_{FM} for AFM and FM coupling, respectively, the magnetic exchange energy per supercell is defined as E_{ex} ≡ E_{AFM} − E_{FM}. Accordingly, the zero-temperature magnetic ground state of the system would be FM if E_{ex} > 0, and AFM if E_{ex} < 0. In Table 1 we list the computed values of E_{ex} and the energy band gap corresponding to the magnetic ground states. Among the various systems investigated here, the CrPc, FePc, CoPc and CuPc-based systems are all semiconductors with weak AFM ground states. The calculated magnetic ground states of these 2D MOFs are in line with those obtained by Zhou et al. for MPc-based organometallic monolayers in the absence of the square-planar Ni linkages.^{28} Surprisingly, the MnPc-based system, NiMnPc, exhibits a FM metallic ground state with substantial exchange energy (E_{ex} = 183 meV). In contrast to the rest of the NiMPc materials considered here, the exchange energy of NiMnPc, 183 meV, is significantly larger than the MnPc monolayer system studied by Zhou et al., 62 meV (40 meV using our calculation setups), when evaluated in the superlattice. In the following, we investigate the origin of the unusually strong FM coupling in the NiMnPc monolayer system.

Cr | Mn | Fe | Co | Ni | Cu | Zn | |
---|---|---|---|---|---|---|---|

M (μ_{B}) |
4 | 3 | 2 | 1 | 0 | 1 | 0 |

E
_{ex} (meV) |
−36 | 183 | −20 | −19 | — | −0.4 | — |

E
_{g} (eV) |
0.35 | Half-metal | 0.28 | 0.35 | 0.35 | 0.30 | 0.30 |

The electronic band structure of NiMnPc near the Fermi level and the corresponding partial density of states (PDOS) are shown in Fig. 4 (the band structures of the other NiMPc 2D MOFs are shown in Fig. S1†). Notably, the electronic states of NiMnPc in the vicinity of Fermi level are completely spin-polarized. The system is therefore a half-metal whose low-energy electronic phenomena would be dominated by one type of spin-polarized electrons, which could be relevant for spintronic applications. Analysis of the PDOS of bands near the Fermi level reveals that nearly all the bands are derived from the hybridization of π orbitals, namely the symmetry-equivalent d_{xz} and d_{yz} orbitals of Mn and Ni, as well as the p_{z} orbitals of N and C. In combination with the metallic nature of the system, it can be concluded that the π orbitals of the system are delocalized. We note that because the electronic bands near the Fermi level are relatively dispersive, the MnPc based 2D MOFs should also have good electrical conductivity, a highly sought-after property for this class of materials.^{17}

The square-planar Ni–N moieties in the MnPc-based 2D MOFs also play a key role in the strong FM exchange between the Mn atoms by effectively mediating the π electron conjugation in different octaamino-MPc molecules. This is consistent with the fact that 2D MOFs with some of the highest measured electrical conductivity were realized by π-conjugation through square-planar Ni–N linkage.^{9,11} We also find that replacing the nitrogen linker atoms in the square-planar Ni–N moieties by sulfur or oxygen does not significantly increase or decrease the magnetic exchange energy, suggesting that catechol and thiocatechol moieties may also mediate effective π electron conjugation (see also Fig. S2†).

We attempt to predict more realistic T_{c} values for 2D FM systems by using two models that consider finite magnetic anisotropy. The first one (hereafter referred as Model 1) is the anisotropic Heisenberg model, which incorporates magnetic anisotropy by exchange anisotropy:

(1) |

In Model 2, we assume isotropic exchange interactions, but take into consideration the presence of uniaxial single-ion anisotropy. This model enforces an energy penalty on magnetic moments pointing to directions other than the magnetic easy axis (chosen to be the z axis), irrespective of the spins at neighboring sites:

(2) |

In this model, the strength of magnetic anisotropy, represented by γ, can take any positive values. A prefactor 2 is added before γ to facilitate comparison between Model 1 and Model 2, based on the energy cost of rotating the spins. The case γ = 0 in this model again corresponds to the 2D isotropic Heisenberg model, whereas the case γ = +∞ represents the classical 2D Ising model, since spin components in directions other than the magnetic easy axis are now fully penalized.

We use classical Monte Carlo simulations to determine T_{c} with Model 1 and Model 2 as a function of the magnetic anisotropy parameter γ. T_{c} is expressed as a function of J/k_{B}, where k_{B} is the Boltzmann constant. The Monte Carlo simulations were performed using the Metropolis algorithm, and T_{c} was determined by finding the intersection of Binder cumulants^{58} for two system sizes (40 × 40 and 80 × 80, consisting of 1600 and 6400 spins respectively). The calculated T_{c} as a function of γ for Model 1 and Model 2 are plotted in Fig. 7. Our simulations confirm previous studies^{52–55} that even a small amount of magnetic anisotropy can stabilize a FM ground state at finite temperature in the 2D Heisenberg model. When the values of γ are small (i.e. 0.001 < γ < 0.1), the T_{c} values determined by Model 1 and Model 2 were found to be close, as shown in the inset of Fig. 7. However, when γ > 0.1, the T_{c} predicted by Model 2 becomes noticeably larger. Because the value of γ in Model 1 is restricted in the range between 0 and 1, the maximum T_{c} of this model is reached at γ = 1.0, which gives T_{c} ≈ 0.9J/k_{B}. In comparison, Model 2 predicts T_{c} ≈ 1.2J/k_{B} at γ = 1.0. Importantly, in the range of γ from 0 to 1, the T_{c} predicted by both models are significantly lower than that of the 2D Ising model, which has an analytical solution, .^{59} For Model 2, the Ising limit is only asymptotically reached at large values of γ (γ > 100, see Fig. S3†). Hence, caution must be exercised when using the classical Ising model to predict the T_{c} of 2D magnetic systems. Indeed, our results here show that in the absence of particularly strong magnetic anisotropy, the Ising model significantly overestimates T_{c}. Although quite dramatic, this overestimation is not surprising because in the Ising model the size of the accessible phase space for thermal excitation is significantly smaller compared to continuous spin models, low-energy excitations such as spin-waves being absent in the Ising model.

With a model that more comprehensively considers magnetic anisotropy in hand for magnetic phase transitions in the 2D space, we calculated the T_{c} of NiMnPc on the basis of the model parameters obtained from first-principles calculations. The exchange constant J in this system is related to the exchange energy E_{ex} evaluated in the supercell by J = (1/8)E_{ex}, which gives J = 22.9 meV. The magnetic anisotropy constant γ can be evaluated by relating it to the exchange constant J and the magnetic anisotropy energy (MAE), the latter of which can calculated from DFT+U with spin–orbit coupling.^{60} The value of MAE is determined as MAE ≡ E_{∥} − E_{⊥}, where E_{∥} and E_{⊥} represent the energies of the system when the spin moment is parallel or perpendicular to the 2D plane of the MOF monolayer, respectively. A positive MAE implies that out-of-plane spin alignment is energetically more favorable than in-plane alignment. Our calculation gives MAE = 0.74 meV for a unit cell of NiMnPc monolayer. Although it is possible to specifically calculate the contribution of both exchange anisotropy and single-ion anisotropy to MAE from first principles, we have shown in Fig. 7 that the T_{c} values predicted by Models 1 and 2, corresponding to exchange anisotropy and single-ion anisotropy, respectively, are similar for small γ. In both models, the calculated MAE is related to γ through MAE = 2γJ. This gives γ = 0.017, which corresponds to T_{c} ≈ 0.66J/k_{B}, or 170 K. In contrast, the frequently used 2D Ising model, where T_{c} ≈ 2.269J/k_{B}, gives a transition temperature value higher than 600 K. Notably, when the square-planar NiN_{4} moiety in NiMnPc is changed to NiO_{4} or NiS_{4} (i.e., the phenylenediamine units are changed to catechols or thiocatechols) the predicted T_{c} are approximately 230 K and 150 K, respectively. Although the T_{c} values for the NiN_{4}, NiO_{4}, and NiS_{4} analogues of MnPc-based MOFs predicted using our realistic anisotropy model are still below room temperature, they sit above the liquid nitrogen temperature and thus present significant potential for practical applications (we note that we have ignored the phonon excitation and quantum fluctuation effects in calculating the T_{c}). Growth of these 2D MOFs as single layers on appropriate substrates, as might be expected under experimental conditions, could act as additional symmetry-breaking measures to enhance the magnetic anisotropy,^{61–63} further increasing T_{c}.

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## Footnote |

† Electronic supplementary information (ESI) available: (1) Computational methods; (2) electronic band structures of NiMPc MOF monolayers (Fig. S1); (3) calculation results for NiMnPc MOF monolayers with NiO_{4} and NiS_{4} moieties (Fig. S2); (4) ferromagnetic transition temperatures of 2D Heisenberg model with single-ion anisotropy (Fig. S3); (5) structure and energetics of bulk NiMnPc (Fig. S4 and S5); (6) atomistic coordinate data of NiMPc MOF monolayers. See DOI: 10.1039/c6sc05080h |

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