I. Introduction

Metal-organic frameworks (MOFS) consist of clustered metallic atoms connected to a larger organic framework, often referred to as organic ligands [1–20]. Importantly, it has been shown that MOFS can support the desirable electronic properties of many inorganic systems but with the advantage that, as organic systems, they can often be synthesized more rapidly and at lower cost. This situation has motivated expanded experimental and theoretical studies of MOFS.

While such studies have made significant progress, high-throughput screenings for desirable properties such as magnetism and superconductivity have fallen behind those performed for inorganic systems [21, 22]. This is due in large part to the relatively large number of atoms, often between 20 and 200, contained within the unit cell of a typical MOF. This large number of atoms translates to a soaring computational expense in ab initio simulations [23].

The computational demands of these systems are perhaps greatest when studying the phonon spectra, in particular the electron-phonon coupling [24]. The evaluation of electron-phonon coupling strengths poses a tremendous computational challenge in simple inorganic systems, let alone large organic molecules. Nevertheless, as one of the most common mechanisms for superconductivity its evaluation is of vital importance to determining optimal superconducting candidates [24].

Modern machine learning techniques appear perfectly positioned to overcome the enormous computational expense associated with studying organic compounds [25–27]. However such methods require large datasets to be trained for making accurate predictions which do not currently exist for MOFS.

In this work, we attempt to make progress towards the screening of MOFS for superconductivity by quantifying the electron-phonon coupling, developing a strategy, outlined in Fig 1, to make such a search feasible for the first time. In order to carry out this screening while limiting the computational expense, we utilize a recently proposed proxy for the electron-phonon coupling strength [28], λ. The fundamental idea behind this approach is that, (1) where λ_Γ is the electron-phonon coupling strength at the Γ location and it is shown that f can reasonably be considered constant among a given material family. In this way promising superconducting candidates can be isolated by evaluating the electron-phonon coupling strength at the Γ location alone. We remark that by focusing solely on the Γ location it is possible that superconducting candidates with maximal electron-phonon coupling strength at alternative high-symmetry locations will escape detection. However, due to the low-symmetry of many MOFs focusing on the Γ location is advantageous as it offers a uniform criteria among a diverse dataset. Furthermore, we demonstrate, utilizing a known MOF superconductor studied in Ref. [29], that even when the Γ location does not support the maximal value of electron-phonon coupling, this proxy can continue to be successful.

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Fig 1. Methodology for isolating superconducting metal organic frameworks within the organic materials database presented in this work.

https://doi.org/10.1371/journal.pone.0310211.g001

It is important to emphasize that, despite use of this proxy a large computational expense remains present due to the fact that the MOFS analyzed in this work have between 60 and 100 atoms in the primitive unit cell. To obtain suitable accuracy in ab initio computation of the electron-phonon coupling strength for each phonon mode thus requires ≈ × 10⁴ CPU hours per compound.

In section II we outline the process by which we have filtered candidate MOFS from the organic materials database (OMDB) [30–34]. The OMDB is among the largest existing databases of organic compounds, containing approximately 40000 compounds as well as the corresponding band-structure, magnetization, and other relevant properties as computed via high-throughput density functional theory. We further describe the computational strategy employed to identify isolate the compound most likely to support superconductivity.

In section III, direct computation of λ and the superconducting critical temperature, T_c, for this optimal compound, C₉H₈Mn₂O₁₁ is detailed.

II. Candidate selection and evaluation

We select MOFS from the organic materials database (OMDB). This database is extremely useful as it allows for searches to be performed based on keywords and band gap. These functionalities allow for immediate isolation of 150 MOFS, of which we select 30 with a vanishing band gap at the Fermi energy and less than 120 atoms in the primitive unit cell. The band-gap criteria is set as the goal of the project is to identify superconducting candidates which require a Fermi surface. The criteria for number of atoms is imposed due to considerations surrounding the computational expense of relaxation and computing phonon modes.

Once the initial 30 candidates have been selected, the atomic positions are relaxed. In this work, all first principles calculations based on density-functional theory (DFT) are carried out using the Quantum Espresso software package [35–37]. Exchange-correlation potentials use the Perdew-Burke-Ernzerhof (PBE) parameterization of the generalized gradient approximation (GGA) [38]. In each case the initial atomic positions and lattice parameters are taken from the OMDB. The atomic positions are relaxed until the maximum force on each atom is less than 1 × 10⁻⁴Ry/Bohr. A 6 x 6 x 6 Monkhorst-Pack grid of k-points is utilized as well as a plane wave cutoff of 520 eV. We have neglected inclusion of a Hubbard U in these computations, which can impact the computed values and should be a subject of future work.

The formation energy of each compound, computed as E_compound − ∑_AtomsE_A, is plotted in Fig 2. After relaxation we proceed with computation of the electron-phonon coupling at the Γ location. This computation is performed utilizing density functional perturbation theory (DFPT) as offered through the Quantum Espresso package. It is important to utilize DFPT in computing the phonon modes as the size of the primitive unit cell makes computation of the phonon modes via packages such as Phonopy [39] which utilize supercells, computationally prohibitive.

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Fig 2. Formation energy of initial candidate MOFs as labelled by the ID of the compound in the organic materials database.

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From the initial 30 compounds, the resulting phonon modes at the Γ location are converged for 15 candidates. This result does not necessarily indicate that the remaining compounds are unstable, rather it is likely that the relaxation criteria for these compounds must be made more strict to achieve convergence. A common feature among those systems which do not show convergence is a large number of hydrogen atoms. Hydrogen atoms can impede the relaxation process due to their small atomic weight as incremental changes in their positions can push the system into local energetic minima.

Due to the computational expense associated with a subsequent round of relaxation and computation of phonon modes, we discard these compounds. The resulting electron-phonon coupling strength at the Γ location for the 15 converged compounds and Cu-BHT is shown in Fig 3. We include Cu-BHT as it is a known MOF superconductor. The results thus demonstrate that the method utilized in this work would successfully identify this material as a superconducting candidate. In addition, in Ref. [29] it is shown that for Cu-BHT, λ is not maximal at Γ, underscoring the effectiveness of this proxy among a variety of materials despite individual microscopic details.

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Fig 3. Electron-phonon coupling at the Γ location for 15 candidate MOFs in the Organic Materials database as labelled by the compound ID number in the database as well as Cu-BHT, studied in Ref. [29].

Compounds with non-zero values are potential superconducting candidates.

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We observe that the majority of screened compounds yield a near-vanishing electron-phonon coupling strength at the Γ location, indicating that they not likely superconductor candidates. Three compounds appear to support λ_Γ > 0.05, with one compound, C₉H₈Mn₂O₁₁ yielding λ_Γ ≈ 0.28. This value is in-line with the electron-phonon coupling strength of other three-dimensional organic superconductors [29] and is thus a promising candidate warranting further investigation in the following section.

III. Analysis of C₉H₈Mn₂O₁₁

Due to the large electron-phonon coupling strength at the Γ location relative to other candidate MOFs, we isolate C₉H₈Mn₂O₁₁ for further investigation. This compound belongs to symmetry class P₁ and has been experimentally synthesized with the details given in Ref. [40]. As stated previously, we utilize the atomic positions as listed in the OMDB and perform a subsequent relaxation. In order to directly investigate whether this compound is a superconducting candidate we compute the dynamical matrices within density-functional perturbation theory on an irreducible 2 × 2 × 2 grid off q-points in the Brillouin zone using DFPT as implemented in Quantum Espresso.

To subsequently extract the superconducting properties of the system, we implement Migdal-Eliashberg theory via the Electron-Phonon Wannier (EPW) code [41–43]. This is an advantageous approach as the EPW code uses Wannier interpolation to access significantly denser k and q point grids in computation of the electron-phonon matrix elements. The Wannier interpolation is performed using a 8 × 8 × 8 grid of k points. To achieve optimal selection of atomic orbitals, we implement the SCDM method [44, 45]. We subsequently utilize 10 × 10 × 10 k and q point grids for computation of the electron-phonon matrix elements.

The resulting Eliashberg spectral function is shown in Fig 4b as well as the resulting electron-phonon coupling strength, (2)

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Fig 4. Details of C₉H₈Mn₂O₁₁.

(a) Relaxed crystal structure of C₉H₈Mn₂O₁₁ for a single primitive unit cell. (b) Spectral function and electron-phonon coupling, λ = ∫α²F(ω)/(2ω)dω. (c) Electron-phonon coupling on the Fermi surface of C₉H₈Mn₂O₁₁. The low-symmetry of the crystal leads to a complex Fermi surface structure. (d) Superconducting critical temperature as a function of the effective coulomb repulsion strength, μ*.

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We find that λ saturates to maximum value of λ ≈ 0.19. We further demonstrate a plot of λ on the Fermi surface in Fig 4c. This plot underscores the low-symmetry which is common among organic systems. We further estimate the superconducting critical temperature (T_c) by self-consistently solving the isotropic Migdal-Eliashberg equations for multiple effective coulomb repulsion values with the results shown in Fig 4d.

The estimated values of T_c imply that, while posing a non-trivial challenge, the superconducting state can be accessed experimentally. It should be emphasized that we have neglected to include both spin-orbit coupling and a Hubbard U in this computation. Due to the presence of Mn both should effect the results of the realistic system and magnetic pairing mechanism cannot be ruled out [46]. We further point out that recent studies have identified significant changes in the critical temperature when approaching the two-dimensional limit in MOFs [29]. It is possible that such a dependence can be identified in C₉H₈Mn₂O₁₁. Investigation of these effects to future works.

IV. Summary and outlook

Our work validates the utility of the method proposed in Ref. [28] in organic compounds and demonstrates an effective route to screening organic compounds for superconducting candidates. With the necessary computing resources this approach can be scaled up to screen all known MOFs and identify optimal candidates. In future works it will be important to pursue the inclusion of spin-orbit coupling and correlation effects, however accounting for these details significantly increases the computational expense. For this the reason they have been reserved for future studies. Furthermore, inclusion of correlation effects in phonon computations is a topic requiring further independent software development. To complement the data presented in this study, analysis of λ using the rigid-muffin-tin (RMT) method [47] should also be pursued. While the RMT method has been shown to be computationally efficient for many systems, it has been reserved for future study as it requires phonon modes to be recomputed using alternative first-principles software. Furthermore, RMT generally requires phonon-modes computed at all high-symmetry locations in Brillouin zone, increasing the computational expense for studying MOFs admitting a large number of atoms in the unit cell. Despite such approximations and limitations, this work represents an important step forward given the current environmental and financial cost of known superconductors composed of rare earth metals. Our work further puts forward a novel, stable organic superconductor, C₉H₈Mn₂O₁₁ motivating future experimental studies.

Acknowledgments

We thank Sinéad M. Griffin for stimulating discussions and drawing our attention to Ref. [28].