Since the discovery of graphene in 2004^[1], two-dimensional (2D) materials have undergone tremendous development in these two decades^[2]. Numerous families of 2D materials have been discovered experimentally or theoretically, such as group-III/IV/V/VI single-element 2D monolayers^[3–7], transition metal dichalcogenides (TMDCs)^[8,9], MXenes^[10], MA₂Z₄ (M = elements of transition-metal groups IVB/VB/VIB; A = Si/Ge; and Z = N/P/As)^[11], α-III₂VI₃ (III = Al/Ga/In; VI = S/Se/Te)^[12], and so forth. Meanwhile, 2D materials hold great potential in the fields of electronics, photoelectronics, catalysis and sensing^[13–16]. Thus, the exploration of novel 2D materials is a popular topic of research.

Generally, there are three main ways to predict new 2D materials. The first is based on the existing bulk materials, which may obtain stable 2D monolayers by mechanical or liquid exfoliation, such as graphene^[1], MoS₂^[17] and phosphorene^[18]. The second is based on crystal prediction software, such as CALYPSO^[19] and UPSEX^[20]. It sets appropriate structural prediction conditions and then obtains the stable structure in specified conditions. The last one is based on the existing 2D structures. The new structure is predicted by the principle of element replacement or valence electron matching, such as binary V-V compound^[21].

Recently, a new family of 2D materials, i.e., M₂X₂Y₆ (M = metal elements; X = Si/Ge/Sn; Y = O/S/Se/Te)^[22–24] or M_IM_IIGe₂Y₆ (M_I and M_II = metal elements, Y = S/Se/Te)^[25], has been widely studied due to their rich properties. For instance, among these compounds, many materials with intrinsic ferroelectric, ferroelasticity and ferromagnetic properties have been confirmed, showing great potential in 2D memory and spintronic devices. However, we found that in the current research, the metal element M mainly focus on transition metals or group IIIA/VA elements, while the alkaline earth metals are not involved. To explore whether alkaline earth metals can form stable M_IM_IIX₂Y₆-like 2D materials, we designed a series of new materials that are based on the valence electron balance strategy. Take the ion compound In₂Ge₂Te₆^[23] as an example. In the In₂Ge₂Te₆, each In atom loses its three valence electrons to form an In³⁺ ion, so when one of the In³⁺ ions is replaced by a M_I²⁺ ion, a M_II⁴⁺ ion is needed to replace the remaining In³⁺ ions to achieve a valence electron balance.

Based on this idea, in this report, we selected alkaline earth metal Mg as one of the metal elements. The other was selected as the group-IVB elements (Ti/Zr/Hf) to balance the valence electrons. Consequently, we predicted a new family of 2D MgMX₂Y₆ (M = Ti/Zr/Hf; X = Si/Ge; Y = S/Se/Te) monolayers (MLs for short) as free standing 2D semiconductors. The structural stability, electronic structures, mechanical properties, and optical properties of 2D MgMX₂Y₆ have been investigated through the first-principles calculations.

All of the calculations in this work are based on density functional theory (DFT)^[26]. The projector augmented-wave (PAW) method^[27,28] in the Vienna Ab initio Simulation Package (VASP 5.4.4) was adopted^[29,30]. The valence electron configurations were: 3s for Mg, 3s and 3p for S and Si, 4s and 4p for Se and Ge, 5s and 5p for Te and In, 4s and 4d for Ti, 4s, 4p, 5s and 4d for Zr, 6s and 5d for Hf. An energy cutoff of 500 eV with a 9 × 9 × 1 Γ-centered k-mesh was used for the plane wave basis. For the exchange-correlation energy, the Perdew-Burke-Ernzerhof (PBE) functional^[31] formal within generalized gradient approximation (GGA)^[31] was implemented. Due to the band gap underestimation problem in GGA^[32,33], the Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional was further employed in electronic band structure calculations^[34]. In addition, the spin-orbit coupling (SOC) effect has been further considered in the electronic band calculations. In structural optimization, a strict 1 meV/Å force criterion and 10⁻⁷ eV energy criterion were used, respectively. Furthermore, a 20 Å vacuum layer was introduced in all 2D monolayers to avoid interactions with adjacent layers. Phonon dispersions of 2D MgMX₂Y₆ were calculated by the density functional perturbation theory (DFPT) using PHONOPY code^[35]. The ab initio molecular dynamics (AIMD) simulations were further performed with a 3 × 3 × 1 (90 atoms in total) supercell for 5 ps at 300 K.

We start with the crystal structure of 2D MgMX₂Y₆ (M = Ti/Zr/Hf; X = Si/Ge; Y = S/Se/Te), as shown inFig. 1. The ML MgMX₂Y₆ possess a hexagonal symmetry with the space group of P31₂ (No. 149). In ML MgMX₂Y₆, there are five atomic layers, in the order of Y-X-Mg/M-X-Y, while the Mg and M atoms are strictly in the same plane. In addition, there is a X-X dimer in the 2D MgMX₂Y₆, which is similar to that of P-P dimer in 2D ABP₂Y₆ (A = Cu/Zn/Ge/Ag/Cd, B = Ga/In/Bi, Y = S/Se)^[36]. We have summarized the optimized lattice constant a/b, bond length Mg-Y/M-Y/X-X/X-Y and buckling height d of MgMX₂Y₆ MLs inTable 1. As shown inTable 1, the lattice constants, bond lengths and buckling heights of MgMX₂Y₆ MLs all vary periodically. For example, in MgTiSi₂Y₆ MLs, the lattice constants and buckling heights are 6.147/6.461/7.015 Å and 3.16/3.37/3.63 Å for MgTiSi₂S₆/MgTiSi₂Se₆/MgTiSi₂Te₆, respectively. In addition, for comparison, we have also given the calculated results of In₂X₂Y₆ MLs, as summarized inTable 1. The lattice constants, bond lengths and buckling heights of In₂X₂Y₆ MLs are vary periodically and are close to that of MgHfX₂Y₆ MLs.

The structural stability is a vital criterion to evaluate whether the newly predicted 2D materials can be applied in electronic devices. For the newly predicted 2D MgMX₂Y₆ MLs, the first concern is their kinetic stability. To evaluate the kinetic stability, we calculated the phonon dispersions of these 18 monolayers. As shown inFig. 2, all the predicted MgMX₂Y₆ MLs exhibit superior kinetic stability due to the free of imaginary frequency in phonon dispersions. In addition, although some phonon dispersions of In₂X₂Y₆ MLs exhibit tiny imaginary frequency near the Γ point which is caused by the computational error, the kinetic stability of In₂X₂Y₆ MLs are also been confirmed. Subsequently, we examined the thermodynamic stability of MgMX₂Y₆ MLs based on the AIMD simulations. During the simulations, the temperature is kept at 300 K for a time scale of 5 ps. The total energy fluctuations of MgMX₂Y₆ MLs are presented inFig. 3. The corresponding crystal structures of MgMX₂Y₆ MLs after the simulation are also inserted inFig. 3. No obvious structural reconstruction has been found after the simulation and the total energy fluctuations are tiny, indicating the high thermodynamic stability of these monolayers at room temperature.

Furthermore, cohesive energy (E_coh) is an important factor to judge the thermal stability. The E_coh of MgMX₂Y₆ MLs is defined as Ecoh=(EMg+EM+2EX+6EY−EMgMX2Y6)/10 , where the EMg / EM / EX / EY and EMgMX2Y6 represent the energy of single Mg/M/X/Y atom and the total energy of MgMX₂Y₆ MLs, respectively. According to the definition, larger positive value of E_coh indicate higher thermal stability. The calculated cohesive energies of MgMX₂Y₆ MLs are shown inFig. 4. In addition, for comparison, the cohesive energies of In₂X₂Y₆ MLs and three typical 2D materials (buckled arsenene, phosphorene and silicene) have also been given. The calculated cohesive energies of the predicted 18 monolayers are ranging from 3.310 eV/atom (MgTiGe₂Te₆) to 4.670 eV/atom (MgHfSi₂S₆). For the In₂X₂Y₆ MLs, the calculated cohesive energies (seeFig. 4) MLs are ranging from 4.033 eV/atom (In₂Ge₂Te₆) to 5.478 eV/atom (In₂Si₂S₆), which is superior to that of most of the MgMX₂Y₆ MLs. Obviously, the calculated cohesive energies of MgMX₂Y₆ are comparable or superior to that of buckled arsenene (2.989 eV/atom)^[37], phosphorene (3.44 eV/atom)^[38] and silicene (3.94 eV/atom)^[39]. Since these three 2D materials have been realized experimentally, the predicted 2D MgMX₂Y₆ in this work may also possess high experimental feasibility.

We also checked the mechanical stability of MgMX₂Y₆ and In₂X₂Y₆ MLs. To verify the mechanical stability, the independent elastic constants of MgMX₂Y₆ and In₂X₂Y₆ MLs have been calculated and the results are summarized inTable 2. For a stable 2D material, it is required to meet the Born-Huang criterion^[40], i.e., C11C22−C1220 , C660 . Since ML MgMX₂Y₆ and In₂X₂Y₆ possess a hexagonal symmetry, there are C₁₁ = C₂₂. Therefore, for 2D MgMX₂Y₆ and In₂X₂Y₆, it needs to be satisfied C112−C1220 , C660 . As shown inTable 2, all of the monolayers satisfy the Born-Huang criteria and exhibit high mechanical stability. These results indicate that MgMX₂Y₆ MLs possess high kinetic, thermal and mechanical stabilities, and might be realized experimentally in the future.

To further confirm our strategy of achieving valence electron balance based on element substitution, we present the results of Bader charge analysis^[41] for MgMX₂Y₆ and In₂X₂Y₆, as summarized inTable 3. The results show that in MgMX₂Y₆ and In₂X₂Y₆ MLs, both Mg and In atoms lose all of their valence electrons and exhibit strong metallic properties. The Bader charge distribution of similar or identical elements in MgMX₂Y₆ and In₂X₂Y₆ is almost the same. Taking the ML In₂Si₂S₆ and MgHfSi₂S₆ as examples, the Bader charge analysis shows that Si loses 2.98 e and 3.00 e in MgHfSi₂S₆ and In₂Si₂S₆, respectively. While for that of S, it gains 1.99 e and 2.00 e for MgHfSi₂S₆ and In₂Si₂S₆, respectively. Meanwhile, the Mg and Hf lose 2 e and 3.98 e respectively, which is very close to the total amount of electrons lose by two In atoms (6 e). Thus, the corresponding elements in MgHfSi₂S₆ and In₂Si₂S₆ lose or gain almost the same amount of charge. The Bader charge analysis results well proved the valence electron balance strategy. For MgTiX₂Y₆ and MgZrX₂Y₆ MLs, although the Bader charge analysis results are somewhat different from those of MgHfX₂Y₆, they are mainly caused by the differences between Ti, Zr and Hf elements, which does not affect the reasonable prediction of the family of MgMX₂Y₆.

Young's modulus and Poisson's ratio are important indicators to characterize the mechanical properties of materials. According to the crystal symmetry, the Young's modulus and Poisson's ratio of MgMX₂Y₆ and In₂X₂Y₆ MLs should be isotropic. Therefore, the corresponding axial Young's modulus (Y₁₁/Y₂₂, the θ is set to 0°/180° and 90°/270°) and Poisson's ratio (v₁₁/v₂₂) can be calculated by simplified formulas, as follow:

1Y11=Y22=C112−C122C11,

2v11=v22=C12C11.

We have given the independent elastic constants in Table 2. The calculated axial Young’s modulus and Poisson’s ratio of MgMX₂Y₆ MLs are also listed inTable 2. The Young’s moduli of MgMX₂Y₆ MLs range from 37.52 N/m (MgZrGe₂Te₆ and MgHfGe₂Te₆) to 67.19 N/m (MgHfSi₂S₆). For In₂X₂Y₆ MLs, the Young’s moduli range from 42.80 N/m (In₂Ge₂Te₆) to 66.55 N/m (In₂Si₂S₆), which is comparable to that of MgMX₂Y₆ MLs. Compared to some other 2D materials, such as graphene (342.2 N/m)^[42], BN (275.8 N/m)^[43] and MoS₂ (197.9 N/m)^[3], the MgMX₂Y₆ MLs possess much lower stiffness and hold great potential in flexible electronic devices. In addition, the Young's moduli of MgMX₂Y₆ MLs also show a cyclical trend. For instance, the Young’s moduli of MgZrSi₂S₆, MgZrSi₂Se₆ and MgZrSi₂Te₆ are 60.23, 51.52 and 40.09 N/m, respectively, showing a decreasing trend when Y goes from S to Te. This can be explained by the electronegativity. Since 2D MgMX₂Y₆ is dominated by the ionic bond, its bond strength mainly depends on the number of electrons gained and lost in the bonding process. The electronegativity values of S, Se and Te are 2.59, 2.42 and 2.16, respectively. Therefore, in the bonding process, the sulfur-containing system has a stronger ionic bond, and the material possesses larger stiffness. Compared with the periodic change of Young's modulus, the Poisson's ratio of MgMX₂Y₆ and In₂X₂Y₆ MLs is relatively stable. The Poisson’s ratios of MgMX₂Y₆ and In₂X₂Y₆ MLs are around 0.27–0.30 and 0.30–0.31, respectively, larger than that of graphene (0.173)^[42] and BN (0.22)^[43]. The relatively close Poisson's ratio indicates that 2D MgMX₂Y₆ and In₂X₂Y₆ possess similar response to tensile or compressive stresses in the plane.

To further evaluate the mechanical properties, the strain-stress relationships of MgMX₂Y₆ and In₂X₂Y₆ MLs have been plotted inFig. 5 and the details are summarized inTable 2. As shown inFig. 5 andTable 2, the ultimate strains (ε^*) of MgMX₂Y₆ MLs along the x-direction are ranging from 0.11 (MgHfSi₂Se₆) to 0.25 (MgZrGe₂Te₆), much superior to that of In₂X₂Y₆ MLs (around 0.10 to 0.12) and lower than that of ML MX₂ (M = Mo, W; X = Se, Se, Te) (0.28–0.32)^[44]. The corresponding ultimate strengths (σ^*) of MgMX₂Y₆ MLs are about 3.78 N/m (MgTiGe₂Te₆)–6.42 N/m (MgHfSi₂S₆), still larger than that of most of In₂X₂Y₆ MLs (3.03–4.35 N/m). For the tensile strain along the y-direction, the ultimate strains of MgMX₂Y₆ MLs are much smaller than that of x-direction, just about 0.08 to 0.12, which is comparable to that of In₂X₂Y₆ MLs (0.09–0.12). In addition, both the ultimate strength of MgMX₂Y₆ (2.14–6.53 N/m) and In₂X₂Y₆ (2.10–4.17 N/m) MLs along the y-direction are smaller than that of x-direction under the same strain. When undergoing biaxial strain, the ultimate strain of In₂X₂Y₆ MLs is very close to each other (0.13–0.15), while MgMX₂Y₆ MLs shows a relatively large difference (0.10–0.19). The ultimate strength of MgMX₂Y₆ MLs with biaxial strain is around 4.38–8.95 N/m, while for that of In₂X₂Y₆ MLs is about 4.61–6.97 N/m. Generally, the mechanical properties of the MgMX₂Y₆ MLs are comparable to or better than the In₂X₂Y₆ MLs.

The electronic properties of 2D materials are important for their application in nanoelectronics. On the one hand, the GGA-PBE always underestimate the band gaps of semiconductors; while on the other hand, spin-orbit coupling (SOC) effect will strongly affect the band structures of compounds with heavier element, such as Si, Ge, Se and Te. Therefore, we have given the band gaps and band structures of MgMX₂Y₆ and In₂X₂Y₆ MLs based on hybrid functional HSE06 within SOC effect involved. The band gaps ( EgHSE+SOC ) are summarized inTable 1, while the projected electronic band structures of MgMX₂Y₆ and In₂X₂Y₆ MLs are shown inFig. 6 andFig. 7, respectively. All the MgMX₂Y₆ MLs are indirect band gap semiconductors with the band gap values ranging from 0.870 eV (MgTiGe₂Te₆) to 2.500 eV (MgHfSi₂S₆). Such a wide range of band gaps in MgMX₂Y₆ MLs enable the selection of suitable monolayers for different requirements. For In₂X₂Y₆ MLs, the band gaps are ranging from 1.220 (In₂Ge₂Te₆) to 2.704 eV (In₂Si₂S₆). In contrast from the MgMX₂Y₆ MLs, which all exhibit indirect band gap features, the In₂Ge₂Se₆ and In₂Ge₂Te₆ of In₂X₂Y₆ MLs are direct band gap semiconductors with the VBM and CBM both located at Γ point. The direct band gap features of In₂Ge₂Se₆ and In₂Ge₂Te₆ may more favorable for photoelectric device applications^[23,45].

Although the MgMX₂Y₆ MLs are all indirect band gap semiconductors, their morphology of band structures still shows many differences. The morphology of band structures can be divided into three types according to the Y element. For the MgMX₂S₆ MLs, the VBMs are located along the Γ–K direction and closer to K. While for the MgMX₂Se₆ MLs, the VBMs are either located along the Γ–K direction and closer to K point or at Γ point, and the energy differences between these two points are merely several or dozens of meV. For example, the VBM of ML MgZrGe₂Se₆ is located along the Γ–K direction and closer to K point and the energy difference between the VBM and Γ point is only 28 meV. However, the VBM of ML MgHfGe₂Se₆ is just located at Γ point and the energy difference for these two points is merely 6 meV. The last one is MgMX₂Te₆, whose VBMs are all located at the Γ point. For the conduction bands, their morphological change trend is roughly the same, and their CBMs are mainly located at the M point or close to the K point. Furthermore, the band gap values of MgMX₂Y₆ MLs also show a periodic change trend as expected. For instance, in MgHfGe₂Y₆ MLs, the band gaps decreased from 2.232 to 1.189 eV when Y goes from S to Te. Turn to the In₂X₂Y₆ MLs, as shown inFig. 7, the CBMs of In₂Si₂S₆, In₂Si₂Se₆ and In₂Ge₂S₆ are located at Γ point. The CBMs of In₂Si₂S₆ and In₂Ge₂S₆ are located at M point, in contrast from the In₂Si₂Se₆ with VBM located at K point. For In₂Si₂Te₆, the VBM is located at Γ point, while the CBM is located along the K–M direction and close to the K point. Similar to MgMX₂Y₆ MLs, the band gap values of In₂X₂Y₆ MLs also show a periodic change trend.

For the orbital contribution, as shown inFig. 6, the conduction band of ML MgMX₂Y₆ is mainly contributed by the d-orbit of Ti/Zr/Hf atom, while the valence band is dominated by the p-orbit of S/Se/Te element. Similarly, for In₂X₂Y₆ MLs (seeFig. 7), the conduction band is mainly contributed by the p-orbit of In atom, while the valence band is dominated by the p-orbit of non-metal element. In addition, the Si/Ge-p orbit contributes to both the valence band and conduction band in MgMX₂Y₆ and In₂X₂Y₆ MLs. No contribution from Mg-s orbit is found in the range of energy scales shown inFig. 6, mainly due to the electron loss in the bonding process of Mg, so its orbital contribution will be located at deeper energy levels.

Next, we focus on the band edges (VBM and CBM) of MgMX₂Y₆ and In₂X₂Y₆ MLs. At present, the application of 2D semiconductors can be divided into two aspects: the first is to make use of its intrinsic properties, the second is to make use of its heterojunction properties with other materials. Regardless of which aspect, it is necessary to study its band edge characteristics. We have shown the band edges of MgMX₂Y₆ and In₂X₂Y₆ MLs at HSE06+SOC level inFig. 8, and the details are summarized inTable 1. The energy levels of CBMs for MgMX₂Y₆ MLs range from –4.96 to –4.18 eV, while for VBMs, the energy levels are around –6.92 to –5.47 eV. For In₂X₂Y₆ MLs, the energy levels of CBMs for In₂X₂Y₆ MLs range from –4.49 to –4.21 eV, while for VBMs, the energy levels are around –6.92 to –5.48 eV, both are comparable to that of MgHfX₂Y₆ MLs. The work functions of Ag, Ti, Cu, Au and Pt are 4.26, 4.33, 4.65, 5.1 and 5.65 eV, respectively. It can be seen that Schottky contact or Ohmic contact can be achieved in metal/MgMX₂Y₆ and metal/In₂X₂Y₆ heterostructure by selecting appropriate electrode or 2D materials. Meanwhile, as shown inFig. 8, type-I (such as MgTiGe₂Te₆/MgZrSi₂Te₆, In₂Si₂S₆/In₂Si₂Te₆) and type-II (such as MgTiSi₂Se₆/MgTiSi₂Te­₆, In₂Ge₂Se₆/In₂Ge₂Te₆) band alignment can also be realized by the combination of different materials in MgMX₂Y₆ and In₂X₂Y₆ MLs, which is beneficial for the realization of different types of optoelectronic devices. The wide band gaps and band edges make MgMX₂Y₆ and In₂X₂Y₆ MLs potential candidates for realizing multifunctional nanoelectronic devices.

Finally, the band gaps of MgMX₂Y₆ MLs cover a wide range (0.870–2.500 eV), so it is necessary to evaluate its light harvesting capability. We mainly focus on the absorption coefficient, which can be expressed as follows^[46]: α(ω)=2ωc{ε12(ω)+ε22(ω)−ε1(ω)}1/2 , where ω represents the photon frequency, ε1(ω) and ε2(ω) are the real and imaginary part of the dielectric function, respectively. And the dielectric function was obtained by ε(ω)=ε1(ω)+iε2(ω) . The absorption coefficients of ML MgMX₂Y₆ and In₂X₂Y₆ were calculated using the independent particle approximation (IPA) at HSE+SOC level.Fig. 9 shows the absorption coefficients of MgMX₂Y₆ MLs. For MgTiX₂Y₆ and MgZrX₂Y₆ MLs (Figs. 9(a) and9(b)), the absorption coefficients in near-infrared (NIR) and visible light (VIS) region are much higher than that of MgHfX₂Y₆ MLs (Fig. 9(c)), and can reach the order of 10⁵ cm^-1, especially for MgTiGe₂Te₆ and MgZrGe₂Te₆. Furthermore, from the distribution of absorption peaks, it can be seen that the optical band gaps of MgMX₂Y₆ MLs are larger than the electronic band gap. For example, the absorption peaks of MgHfX₂Y₆ MLs are all located in the deep ultraviolet (UV) region (seeFig. 9(c)), while the electronic band gaps of MgHfX₂Y₆ MLs range from 1.189 to 2.500 eV. This is mainly due to the fact that in indirect band gap semiconductors, light absorption is accompanied by absorption or emission of phonons, which will consume a large quantity of energy. In addition, the light absorptions of MgHfX₂Y₆, MgTiSi₂Se₆ and MgTiSi₂Te₆ are mainly concentrated in the UV region, which indicate potential applications of UV optoelectronic devices. Meanwhile, strong optical absorption coefficients in NIR, VIS and UV light have been confirmed for MgTiGe₂Te₆, MgZrSi₂Te₆ and MgZrGe₂Te₆, which suggests a better potential in multifunctional optoelectronic devices. The remaining seven monolayers possess considerable optical absorption in the VIS and UV regions, and are also potential candidates for optoelectronic devices. Finally, for comparison, the optical characteristics of In₂X₂Y₆ MLs are also investigated, as shown inFig. 9(d). For In₂X₂Y₆ MLs, the optical absorption coefficient is mainly concentrated in VIS and UV region, and can reach the order of 10⁵ cm⁻¹, comparable to that of MgZrX₂Y₆ MLs. The suitable band gap value and direct band gap feature make In₂Ge₂Te₆ exhibit stronger visible light absorption than the rest of In₂X₂Y₆ MLs. The considerable optical absorption of In₂X₂Y₆ MLs in the VIS and UV regions suggest its potential for optoelectronic devices.

In summary, based on the valence electron balance, we have predicted a new family of 2D semiconductors; that is, MgMX₂Y₆ (M = Ti/Zr/Hf; X = Si/Ge; Y = S/Se/Te) MLs. The 18 newly predicted 2D monolayers have good kinetic, thermodynamic and mechanical stability. Small Young's modulus and moderate Poisson's ratio indicate that 2D MgMX₂Y₆ is suitable for flexible electronic devices. Electronic structure analysis shows that these monolayers are all indirect band gap semiconductors with band gap values ranging from 0.870 to 2.500 eV. In addition, its band-edge properties are suitable for constructing various types of electronic devices. Optical property analysis shows that 2D MgMX₂Y₆ possesses good absorption in the near-infrared, visible and ultraviolet regions, and holds great potential in multifunctional optoelectronic devices. The discussion in this report would be of great meaningful value for 2D MgMX₂Y₆.