Background & Summary

The recent emergence of atomically-thin materials within the last two decades has precipitated a multitude of families of 2D materials, each offering complementary properties, including remarkable mechanical, optical, and electronic properties1,2. One class of 2D materials with a recent rise in interest is MXenes, a family of layered transition metal carbides and nitrides first discovered in 20113.

The synthesis of MXenes is accomplished by preferential etching of layered MAX phases. These MAX phases consist of M, A, and X-type atoms, where M denotes an early transition metal, A denotes a group 13–16 element, and X denotes carbon or nitrogen. The M, A, and X atoms crystallize into a layered structure of composition Mn+1AXn, where n = 1, 2, 3 and n corresponds to the number of X layers in the MXene flake. This layered structure forms the basis for 2D MXene sheets. Figure 1 shows the three crystal structures observed in MAX phases. From the MAX phase precursor, the comparatively weaker M-A bonds can be etched away, resulting in single- or multi-layered MXenes. This unique top-down synthesis approach gives MXenes more scalability than other families of 2D materials. Beyond their synthesis route, MXenes are desirable for applications in energy storage, catalysis, and electronic sensors due to their metallic conductivity and 2D structure4. For these reasons, MXenes have a large potential in both existing and future applications.

Fig. 1
figure 1

Conventional DTM MAX unit cells, for n = 1, 2, 3 layered structures. These unit cells demonstrate the three crystal structures studied in this investigation and provide the basis for DFT calculations.

A recently emerging sub-class of MXenes known as double transition metal (DTM) MXenes contain two different transitional metal elements5. These DTM MXenes retain the layered structure of mono-M MXenes, with M′ and M″ atoms occupying the M sites. These MAX phases are further subdivided into in-plane ordered (i-MAX), out-of-plane ordered (o-MAX), and solid-solution (ss-MAX) phases, depending on the distribution of the M′ and M″ atoms6. The o-MAX phases are defined by compositions of M′M″AX, M′2M″AX2, and M′2M″2AX3 for n = 1, 2, 3, respectively.

Out of the 8712 possible o-DTM-MAX materials, only a few have been etched into MXenes. Thus, to guide future experimental efforts, we introduce a comprehensive dataset of density functional theory (DFT) results for all possible o-MAX phases, including energetics and structural information.

The multivariate composition of DTM MXenes and their precursor MAX phases offer a high-dimensional domain of exploration well-suited for high-throughput density functional theory. Existing databases of DFT calculations, such as Materials Project7, only contain a small fraction of all possible MAX phases. For example, querying Materials Project’s API for all MAX-formulae with space group P63/mmc reveals only 75 MAX phases, none of which are DTM MAX structures.

Several other studies have reported DFT investigations of DTM MAX structures, but none present a complete view of the entire DTM MAX domain of compositions. Exhaustive computational investigations of mono-M MAX phases and their MXenes have been performed8, but this has not been extended to DTM MAX phases. Several studies have characterized DTM MXenes using DFT, but each was limited in domain to <100 compositions9,10,11. A database of 23,870 DTM MXene phases has previously been established; however, this database contains no precursor MAX phases, and 70% of the MXene properties in the database are machine learning generated12.

This paper aims to fill this gap by establishing a complete database of bulk DTM o-MAX structures, which can accelerate the development of new DTM MXenes. Electronic structure calculations using DFT predict several of the properties of interest in MXene applications and help guide the design of new 2D materials of this family. The composition domain of this database is defined below in Fig. 2, where M′, M″ = Hf, Mn, Cr, Mo, Nb, Sc, Ti, V, Zr, and Ta, and W; A = Cd, Pb, P, S, Tl, As, Al, Si, Ge, Ga, In, and S; and X = C or N, with n = 1, 2, and 3 layers. This database includes 8712 potential MAX phases.

Fig. 2
figure 2

DTM MAX domain. This graphic shows the composition space of the elements being studied in the MAX system, with separate colors for M′, M″, A and X.


Workflow overview

The data and simulations workflow used to characterize and obtain features for all 8712 candidate MAX phases consists of three main steps, see Fig. 3: i) initial structure generation, ii) DFT calculations, and postprocessing/indexing of results into a queryable database. The following subsections describe each step in detail.

Fig. 3
figure 3

High-throughput DFT workflow. This flowchart shows the process by which data is passed from an initial starting structure to the final ResultsDB, hosted on nanoHUB.

Initial structure generation

The unit cell for each MAX structure is first estimated starting with the Ti3 AlC2 MAX structure obtained from the Materials Project7. From each MAX composition, the selected elements are mapped using pymatgen, and the lattice vectors are adjusted based on the atomic radii of the elements involved, using geometric insight on MAX phases to serve as a starting estimate for the DFT calculations13. The expressions used to calculate the a and c lattice parameters for each of the three crystal structures are listed in Table 1.

Table 1 Geometric estimates for lattice parameters for n = 1,2,3 layers.

Density functional theory calculations were performed on each structure using the Vienna Ab initio Simulation Package (VASP)14,15. These simulations utilized Perdew−Burke−Ernzerhof (PBE) exchange-correlation functional and projector-augmented wave (PAW) pseudopotentials16,17. The calculations include collinear spin-polarization with magnetic moments for each ion initialized to one. Starting with the initial structures described above, a fixed-cell ionic relaxation via energy minimization is performed. This is followed by a cell relaxation to relax the stress tensor and a single-point energy calculation on this optimized structure to obtain energetic properties. Ionic relaxations use the conjugate gradient algorithm with 10−5 eV stopping criteria and the cell relaxation uses conjugate gradient algorithm and 10−6 eV threshold. To ensure numerical convergence of the runs, a kinetic energy cutoff of 550 eV and a k-point grid of 18x18x3 was chosen. This was determined to be a sufficient cutoff by ensuring a variability of less than 1% of the total a and c lattice parameter values.

Simulation results postprocessing and indexing into a queryable database

The raw VASP outputs are processed to extract the structural and energetics of interest using the following procedure. The first step is to parse the VASP output using the VaspDrone function of atomate218 to convert it into a structured JSON document. Atomate2 is an open-source set of computational materials science workflows designed for high-throughput automation. To extract the desired quantities of interest and make the data Findable, Accessible, Interoperable, and Reusable (FAIR)19, we use nanoHUB’s20 Sim2Ls21. Sim2Ls are online, end-to-end, and queryable workflows with formally declared inputs and outputs. They are implemented as Jupyter notebooks and can be executed from nanoHUB or using an API. The open source VASPINGESTOR Sim2L ingests JSON files produced by atomate2 and extracts the desired structural and energetic information22. Importantly, all the inputs and outputs, including the JSON file and auxiliary quantities of interest, are automatically indexed into nanoHUB’s results database (resultsDB) where they can be accessed through a web interface or queried using an API. The Section Data Records describes the properties extracted from the JSON file with the raw results and stored in the ResultsDB. We note that the VASPINGESTOR tool is open and available online, it can be used by any researcher using VASP to make their data FAIR.

Data Records

The dataset is available on Zenodo, an open data repository23. In addition, the dataset is available on as both a database24 and an interactive tool22 with a GNU GPL license. As described above, we use nanoHUB’s Sim2Ls to make both the postprocessing workflow and the data FAIR21. All the inputs and the outputs of the vaspingestor Sim2L are indexed in the ResultsDB, they are listed in Tables 2, 3 with an (I) or (O) following their names, respectively. The features extracted from the 8712 DFT simulations can be grouped into two categories, the first being those directly extracted from the VASP output, such as energies, stresses, and structures. These results are listed in Table 2.

Table 2 Features stored in Sim2L.
Table 3 Features derived from VASP results.

From the DFT results, additional quantities of interest useful as descriptors for machine learning applications are extracted, these are provided in Table 3. These quantities are categorized separately as they either require additional DFT calculations, such as formation and cohesive energy, or they are MAX-specific, such as the selected bond lengths and interlayer distances.

The bond lengths and interlayer distances listed in Table 3 are illustrated in Fig. 4.

Fig. 4
figure 4

Interlayer distances and bond length. This figure illustrates the bond lengths and interlayer distances provided in the feature set, to avoid ambiguity in their definition.

Formation energies are calculated using:

$${E}_{{\rm{form}}}^{MAX}={E}_{{\rm{bulk}}}^{MAX}-\sum _{i=\left\{M,A,X\right\}}{N}_{i}^{* }\left({{\rm{e}}}_{{\rm{i}}}^{0}\right)$$

This equation takes the difference of the obtained MAX phase energy \({E}_{bulk}^{MAX}\) and the constituent elements i in their respective standard states, \({e}_{i}^{0}\), which represent the energy per atom in the standard state and Ni the number of atoms of type i in the MAX structure. These equilibrium states are the lowest energy configurations of each element; for example, the equilibrium state of carbon would be the graphite structure. This provides a metric of the relative stability of the MAX phase over the stability of the individual components.

Finally, cohesive energies were calculated using Eq. 2.

$${E}_{{\rm{coh}}}^{MAX}={E}_{{\rm{bulk}}}^{MAX}-\sum _{i=\left\{M,A,X\right\}}{N}_{i}* {E}_{i}^{{\rm{vacuum}}}$$

This equation takes the difference of the obtained MAX phase energy \({E}_{bulk}^{MAX}\) and the constituent elements i as single atoms in vacuum. This provides an additional metric of stability as the difference between a bonded and dissociated structure.

The interactive tool is implemented as a Jupyter notebook, it queries the database and provides several ways of visualizing the data. These include scatter and violin plots of formation energy as a function of M′, A, X, and n, in addition to heatmaps of each M′/M″/A combination for both carbides and nitrides. It also shows an example of the use of the Sim2L inputs to reproduce a VASP calculation, and a visualization of the ternary and quaternary convex hulls.

Technical Validation

Comparison to Frey et al

The largest existing database of MAX phases, published by Frey et al.8, contains 792 MAX phases with single-M structures. This work reported lattice a lattice parameters, formation energies, and cohesive energies for all calculated MAX phases. A comparison between their MAX phase results and this work is provided in Fig. 5 as parity plots of formation and cohesive energies as well as a lattice parameters. Values are generally in agreement for the formation and cohesive energies, with RMSE values of 0.13 and 0.25 eV/atom, respectively. As shown in Fig. 5, the major outliers are limited to Sc-A-C type MAX phases, where A is any of the 12 possible A-group elements. We independently confirmed our calculations and we compared them with the Sc-containing MAX phases in the Materials Project. Both these tests indicate our results to be correct. A possible origin of this discrepancy is differences in the spin state of the systems. Comparisons of lattice parameters indicate an RMSE of 0.12 Å with 76% of the structures showing discrepancies less than 2%. 10% of the structures show differences of over 5% which could not be explained by differences in the methods used. This comparison and analysis highlight the importance of making research data FAIR, as Frey et al. did.

Fig. 5
figure 5

Comparison of lattice parameter, formation and cohesive energies with Frey et al. This plot shows the deviations in formation and cohesive energies are mainly limited to Sc-A-C-type MAX phases with n = 1, while the lattice parameters show much more variation.

Comparison to Materials Project

To further validate our data, a comparison was made to all existing MAX phases in the Materials Project7. The formation energies and a lattice parameters are shown in Fig. 6. In the plot of formation energy, we provided reference lines to account for the disparity in nitrogen reference between datasets. We use nitrogen in its gaseous state as its equilibrium structure, while the Materials Project uses a crystalline form of nitrogen as its reference. When accounting for this difference the formation energies and lattice parameter RMSE values were 0.06 eV/atom and 0.07 Å. The Sc-A-C type MAX phase formation energies have an RMSE of 0.004 eV/atom, showing strong agreement with Materials Project and exhibiting the same deviation from Frey et al. as shown in Fig. 5.

Fig. 6
figure 6

Comparison of formation energies and ‘a’ lattice parameters with Materials Project. This plot shows that our calculated lattice parameters agree with Materials Project, and that correcting for the different nitrogen reference state demonstrates agreement in formation energy as well.