MOenes family materials with Dirac nodal loop, strong light-harvesting ability, long carrier lifetime and conduction-band valley spin splitting

The crystal structures of M${}_{\textit{n}+1}$O${}_{\textit{n}}$T₂ MOenes are shown in Fig. 2. In general, MOenes have two phases: 2H trigonal prismatic and 1T octahedral phases. As shown in Fig. 2a, the 2H phase consists of edge-sharing trigonal prisms belonging to the space group $P\overline{6}m2$ (No. 184). On the other hand, 1T phase has a space group of $P\overline{3}m1$ (No. 164), and is composed of [MO₆] octahedra. Due the unsaturated orbitals of outer M metals, bare MOenes are apt to be terminated by functional groups. In this respect, the most two possible functionalisation sites are considered, as exhibited in Fig. 2. The most energetically stable crystal structures of M${}_{\textit{n}+1}$O${}_{\textit{n}}$T₂ MOenes can be determined by comparing their energies. In the following, we fully studied the bare M${}_{\textit{n}+1}$O${}_{\textit{n}}$ MOenes and terminated M${}_{\textit{n}+1}$O${}_{\textit{n}}$T₂ MOenes from dynamical, mechanical and electronic features by high-throughput calculations.

To give a quick understanding for this emerging MOenes family, we firstly have built an online materials library (http://moenes.online). The workflow for investigating MOenes is shown in Fig. S1. Based on the most energetically stable M${}_{\textit{n}+1}$O${}_{\textit{n}}$ and M${}_{\textit{n}+1}$O${}_{\textit{n}}$T₂ MOenes, we firstly confirm the ground sates of MOenes by comparing energies between nonmagnetic (NM) and ferromagnetic (FM) spin configurations. Phonon spectra of MOenes are next used to confirm their dynamical stability. Besides, their mechanical stability are determined by Born criteria for hexagonal crystal, namely, $C_{11}$ > |$C_{12}$| and $C_{66}$ = ($C_{11}$$-$$C_{12}$)/2 > 0 Mouhat and Coudert (2014). When MOenes have reliable dynamical and mechanical stability, the electronic properties of MOenes are explored, and then their band characters can be determined by the locations of the VBM and CBM).

In brief, the structures of 880 MOenes have been presented in this online materials library, from basic summaries to mechanical properties, phonon spectra and band structures. One can quickly search for the characteristics of the desired MOenes by selecting stoichiometric ratios (n), early-transition metals (M), oxygen group elements (O) and functional groups (T) step by step. Taking Ti₂SF₂ MOenes as illustrations, the search results are shown in Fig. S2. In basic summaries part, we can obtain the optimized lattice constant, favorable absorption site, the most favourable dynamical stability, magnetic moments of early-transition metal, locations of VBM and CBM, band character, band gap in semiconductors/insulators and work functions, etc. In mechanical properties part, the elastic constants, mechanical stability, Young’s modulus (Y) and possion’s ratio ($\nu$) are available. In addition, we can acquire their phonon dispersions within the phonon spectra, and projected band structures (Pbands) are provided as well.

The stability, exotic electronic features including semimetals, direct/indirect semiconductors and insulators are also summarized. The summaries for M₂OT₂ MOenes are shown in Fig. 3. There are 70 M₂OT₂ MOenes are dynamically stable. Obviously, most of them have rich metallic properties, along with several semimetals, such as 1T-Y₂OF₂, 2H-M₂O(O)₂ (M=V, Nb, Ta) and 1T-M₂O(O)₂ (M=V, Ta), in which the Weyl fermion locates at the Fermi level. Besides, there are 10 semiconductors and one insulators. Among them, 2H/1T-Ti₂OF₂, 2H-M₂O(O)₂ (M = Zr, Hf) and 1T-Nb₂O(O)₂ are direct semiconductor with a band gap of 0.14$-$0.50 eV(PBE). 2H/1T-M₂OF₂ (M = Zr, Hf) and 2H-Ti₂O(O)₂ are indirect semiconductors. Besides, 1T-Y₂O(O)₂ is an insulator with a band gap of 4.67 eV (PBE), and futher corrected to 6.20 eV (HSE06). It is worth emphasizing that the summaries for M₂ST₂, M₂SeT₂, M₂TeT₂, M₃O₂T₂, M₃S₂T₂, M₃Se₂T₂ and M₃Te₂T₂ MOenes are available in Supporting Informations. As listed in Table 1 and shown in Fig. S11, there are 14 direct semiconductors in the MOenes family, and their band gap values are 0.32$-$1.28 eV (HSE06), reflecting their promising applications in semiconducting fields. Furthermore, they have a small Y (< 260 N/m) and isotropic $\nu$, meaning their soft properties, and thus can applied in flexible devices Lee et al. (2008).

As presented above, the bare MOenes are metals and usually lack a good stability, but they can transform into dynamically stable phases when their outer orbitals of M are saturated by functional groups. On the other hand, the electronic traits of MOenes are strongly related to the M. In short, the majority of fluorinated and hydroxidized Ti-, Zr-, Hf-based MOenes show semiconducting features, and other fluorinated and hydroxidized MOenes are mostly metallic. In addition, oxidized Sc- and Y-based M₂O MOenes have insulating and semiconducting characters, but oxidized V-, Nb-, Ta-based M₂O MOenes possess semimetal and metallic features. All in all, due to the structural diversity of MOenes, it brings the diversity of its electronic structures, including highly conductive metallic, semimetallic, semiconducting, (topological) insulating, which can bring various applications for MOenes as traditional MXenes.

Since early transition metal elements bring rich electronic structures in MOenes, we have selected group IVB metal elements (Ti, Zr, Hr) and fifth-row transition metal elements (Y, Zr, Nb, Mo) under the chemical formula 1T-M₂OF₂ to fully characterize the influence of "M" sites in MOenes (Fig. 4a). In all fluorinated 1T-M₂O MOenes, because of the lower electronegativity of the transition metals than the O atom and the attached fluorinated functional groups, the transition metals become positively charged by donating their electrons to both O atoms and the fluorinated functional groups. Firstly, Ti, Zr, and Hf are in the same IVB group in the periodic table with the same number of electrons in their outer shells ([Ar]3d²4s², [Kr]4d²5s², and [Xe]4f¹⁴5d²6s²). As expect, fluorinated 1T-Zr₂O and Hf₂O exhibit similar semiconducting properties to those of 1T-Ti₂OF₂Yan et al. (2022). Moreover, their energy states around the Fermi energy are mainly made of the M-d${}_{z^{2}}$ and M-d${}_{x^{2}-y^{2}}$ (M = Ti, Zr and Hf) orbitals. As shown in Fig. 4b, 1T-Ti₂OF₂ is a direct semiconductor together with VBM and CBM located at the $\Gamma$ point. However, when M changes from Ti, to Zr, and even to Hf, the M-d${}_{z^{2}}$ orbitals move down, while the M-d${}_{x^{2}-y^{2}}$ orbitals move up. As a result, the M-d${}_{x^{2}-y^{2}}$ at the K point is higher than the M-d${}_{z^{2}}$ at the $\Gamma$ point in. Thus, 1T-Zr₂OF₂ and 1T-Hf₂OF₂ become indirect semiconductors (Fig. 4c and 4d).

Around the Zr metal element, the fifth-row transition metal elements (Y, Zr, Nb) are further systematically scrutinized to draw the roles of the different number of electrons in their outer shells ([Kr]4d¹5s², [Kr]4d²5s², [Kr]4d⁴5s²). Due to the dynamical instability in 1T-Mo₂OF₂, we don’t consider it further. Clearly shown in Pbands, as the number of electrons in Y, Zr and Nb outer shells progressively increases, the Fermi energy gradually shifts upwards and the d${}_{x^{2}-y^{2}}$ orbitals near the Fermi energy move down largely. As Zr changes to Nb metal, the increase of Fermi energy and the movement of Nb-d${}_{x^{2}-y^{2}}$ orbitals result in the metal characters for 1T-Nb₂OF₂ (Fig. 4f). When the Zr turns to Y, the Fermi energy shifts downwards and cuts directly through the middle of the most highest valance bands in 1T-Zr₂OF₂, leading to a semimetal feature for 1T-Y₂OF₂ (Fig. 4e). Clearly, two salient features can be observed in 1T-Y₂OF₂ without SOC (Figs. 4e and S12a). Firstly, 1T-Y₂OF₂ has two linear band crossings in the Fermi level (P₁ and P₂). Secondly, these linear band crossings are observed for an electron and hole band near the Fermi level. A careful scan of the band structure shows that the intersection P₁ is not isolated, but is actually part of an approximate nodal loop centred on the $\Gamma$ point, as indicated in Figs. S12c and S12e. However, , the approximate nodal ring is not complete, and the exact crossing points are located only on the high symmetry path, such as P₁. Remarkably, the crossing points (labeled N₁ and N₂) still persist when SOC is turned on (Fig. S12b), which is similar to that without SOC due to the weak SOC effect in 1T-Y₂OF₂. The difference is that each band, when considering SOC, is doubly degenerate. Moreover, a complete nodal ring (labeled L₁) exists around the $\Gamma$ point at this time, as shown by the calculated energy gap between the electron-like and hole-like bands in Fig. S12f. Since each band is doubly degenerate, the nodal ring is fourfold degenerate Dirac nodal ring, as illustrated in Fig. S12d. In addition, the Dirac nodal ring of this system lies exactly on the Fermi surface and there are no non-topological bands near the Fermi energy, reflecting the properties of an ideal 2D Dirac nodal loop state.

In order to draw the influence of oxygen group elements, the properties of Ti₂OF₂ (O = O, S, Se, Te) are systematically scrutinized and compared. We first focused on the 1T phase. Firstly, 1T-Ti₂SF₂, 1T-Ti₂SeF₂ and 1T-Ti₂TeF₂ can maintain structural integrity at 900, 600 and 300 K, respectively (Fig. S13), lower than that of 1T-Ti₂OF₂ (1200 K)Yan et al. (2022). Clearly, as the oxygen group elements change from O to S, Se and Te, the thermal stability is weakened due to the increase in chemical bonds. (Table II). Similar to 1T-Ti₂OF₂ shown in Fig. 5a Yan et al. (2022), 1T-Ti₂SF₂ and 1T-Ti₂SeF₂ are direct semiconductors with VBM and CBM located at the $\Gamma$ point, and have a band gap of 0.67 and 0.97 eV (HSE06), respectively (Figs. 5c and 5e). The sum of square of the transition dipole moment (P²) also confirm the allowed electronic transition between VBM and CBM at the $\Gamma$ point. Their VBM states are composed of Ti-d${}_{z^{2}}$ orbitals mostly, but CBM states have hybrid orbitals from Ti-d_xy, Ti-d${}_{z^{2}}$ and Ti-d${}_{x^{2}-y^{2}}$. The same as 1T-Ti₂OF₂ monolayer Yan et al. (2022), three low-energy bands are involved in the band edges at $\Gamma$ point, including one valence band and two degenerate conduction bands composed by one heavy electron-band and one light electron-band. Besides, 1T-Ti₂TeF₂ monolayer has an indirect band gap of 0.40 eV (HSE06), which is also reflected by the results of P² (Fig. 5g). As expected, their band gaps at the $\Gamma$ point first decrease and then increase when the oxygen group elements change from O to S, Se and then to Te. This is due to the decrease in electronegativity of the oxygen group elements, which weakens the bond strength in the 1T-Ti₂OF₂ (O= O, S, Se, Te) MOenes (Table II). The clearer Pbands are shown in Fig. S14. As the oxygen group elements change from O to S, the light electron-band contributed by Ti-d${}_{z^{2}}$ orbitals moves down and gradually becomes a heavy electron-band, leading to a decrease of the band gap firstly. When the oxygen group elements change from S to Se, this band continuously moves down and the orders of the original two degenerate conduction bands are changed, bringing an increase of the band gap. In this progress, the band dispersions of two degenerate conduction bands and valence band around the $\Gamma$ point have been significantly weighted, leading to a larger effective mass and a smaller carrier mobility. Finally, 1T-Ti₂TeF₂ monolayer becomes an indirect semiconductor with VBM and CBM located at the $\Gamma$ and M point, respectively.

As the oxygen group elements change progressively, the optical absorption behaviour of 1T-Ti₂OF₂ (O = O, S, Se, Te) are further compared within a GW method simulated with the random phase approximation (GW+RPA) (without e-h interactions) and the Bethe-Salpeter equation (GW+BSE) (with e-h interactions). The imaginary parts of the frequency-dependent 2D polarizability for 1T-Ti₂SF₂, 1T-Ti₂SeF₂ and 1T-Ti₂TeF₂ are shown in Figs. 5d, 5f and 5h, respectively, and compared with that for 1T-Ti₂OF₂ (Fig. 5a)Yan et al. (2022). Here, the optical band gap $E_{g}^{o}$ is defined as the first dominated peak corresponding to the first optically allowed (bright) exciton. The $E_{g}^{o}$ values for 1T-Ti₂SF₂, 1T-Ti₂SeF₂ and 1T-Ti₂TeF₂ are 0.21 and 0.45 eV, respectively. In term of the spectral edge obtained at the GW+RPA level, the quasi-particle energy gap ($G_{0}W_{0}$ gap) are extracted to 0.74 (1.05) eV for 1T-Ti₂S(Se)F₂. Here, GW correction to the PBE band gap is about 0.59 (0.70) eV. Then, the exciton binding energy $E_{b}$ values for 1T-Ti₂SF₂ and 1T-Ti₂SeF₂ are evaluated to 0.53 and 0.59 eV, respectively. As listed in Table II, the numerical values of the $E_{g}^{o}$ and $G_{0}W_{0}$ gap show the same trend as the band gap values. Generally, the larger the band gap, the weaker the screening, resulting in a higher $E_{b}$. Therefore, the $E_{b}$ values of 1T-Ti₂OF₂ (O = O, S, Se) decrease firstly, and then increase, on going from O to Se. Similar to 1-Ti₂OF₂ monolayerYan et al. (2022), there are several significant peaks in the infrared and visible light regions, making 1T-Ti₂OF₂ (O=S, Se, Te) promising for infrared detectors and solar cells with strong light absorption ability in the desired spectral range.

On the other hand, as the oxygen group elements gradually change from O to Te, the 2H-Ti₂OF₂ (O = O, S, Se, Te) changes from a direct semiconductor to a Weyl semimetal, and finally to an indirect semiconductor (Fig. S15) at the PBE level, undergoing a similar evolution of the Ti-d${}_{z^{2}}$ orbitals discussed above. When spin-orbit coupling (SOC) is not considered, the Weyl fermions in 2H-Ti₂SF₂ and 2H-Ti₂SeF₂ are purely located at the $\Gamma$ point at the Fermi level (Figs. S16a and 16d). At this time, the VBM and CBM touch quadraticall and form a two-component 2D double Weyl fermion with an $E^{\prime}$ irreducible representation of the $D_{3h}$ point group. When SOC is considered, these degenerate Weyl fermions contributed by Ti-d_xy and Ti-d${}_{x^{2}-y^{2}}$ orbitals are split, resulting in a SOC gap of $\sim$19 and 9 meV (PBE) for 2H-Ti₂SF₂ and 2H-Ti₂SeF₂, respectively (Figs. S16b and S16e). Moreover, in Figs. S16c and 16f, the edge states depicted 2H-Ti₂SF₂ and 2H-Ti₂SeF₂ connect the bulk valence and conduction bands. Besides, the $Z_{2}$ invariant values are determined to be 1 via Wannier charge center calculations. Thus, 2H-Ti₂SF₂ and 2H-Ti₂SeF₂ are determined to be 2D topological insulators, whose nontrivial gaps are further corrected to $\sim$24 and 55 meV (HSE06). Although thses nontrivial gaps are smaller than that in Ti₃O₂F₂ and Ti₄O₃F₂ MOenes, they are adequate for practical applications at room temperature Yan et al. (2024).

As summarized before, the most significant advantage identified thus far is that MOenes markedly enhance the scope of direct semiconductors and light absorption within the MXenes family. Therefore, we put our attention on these direct semiconducting MOenes with fascinating properties in the following.