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J. Semicond. > 2023, Volume 44 > Issue 4 > 042101

ARTICLES

A family of flexible two-dimensional semiconductors: MgMX2Y6 (M = Ti/Zr/Hf; X = Si/Ge; Y = S/Se/Te)

Junhui Yuan1, Kanhao Xue1, 2, , Xiangshui Miao1, 2 and Lei Ye1, 2,

+ Author Affiliations

 Corresponding author: Kanhao Xue, xkh@hust.edu.cn; Lei Ye, leiye@hust.edu.cn

DOI: 10.1088/1674-4926/44/4/042101

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Abstract: Inspired by the recently predicted 2D MX2Y6 (M = metal element; X = Si/Ge/Sn; Y = S/Se/Te), we explore the possible applications of alkaline earth metal (using magnesium as example) in this family based on the idea of element replacement and valence electron balance. Herein, we report a new family of 2D quaternary compounds, namely MgMX2Y6 (M = Ti/Zr/Hf; X = Si/Ge; Y = S/Se/Te) monolayers, with superior kinetic, thermodynamic and mechanical stability. In addition, our results indicate that MgMX2Y6 monolayers are all indirect band gap semiconductors with band gap values ranging from 0.870 to 2.500 eV. Moreover, the band edges and optical properties of 2D MgMX2Y6 are suitable for constructing multifunctional optoelectronic devices. Furthermore, for comparison, the mechanical, electronic and optical properties of In2X2Y6 monolayers have been discussed in detail. The success of introducing Mg into the 2D MX2Y6 family indicates that more potential materials, such as Ca- and Sr-based 2D MX2Y6 monolayers, may be discovered in the future. Therefore, this work not only broadens the existing family of 2D semiconductors, but it also provides beneficial results for the future.

Key words: two-dimensional materialsMgMX2Y6 monolayerIn2X2Y6 monolayersemiconductorfirst-principles calculations

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[37], 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[1316]. 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)[2224] or MIMIIGe2Y6 (MI and MII = 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 MIMIIX2Y6-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 MI2+ ion, a MII4+ 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 in Fig. 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 in Table 1. As shown in Table 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 in Table 1. The lattice constants, bond lengths and buckling heights of In2X2Y6 MLs are vary periodically and are close to that of MgHfX2Y6 MLs.

Fig. 1.  (Color online) (a) The top view and (b, c) side view of monolayer MgMX2Y6. (d) The corresponding first Brillouin zone of MgMX2Y6 monolayers.
Table 1.  Calculated lattice constant a/b (Å), bond length Mg (In)-Y/M-Y/X-X/X-Y (Å), buckling height d (Å), band gaps at HSE06+SOC (EHSE+SOCg, eV) levels, the valence band maximum (VBM, eV) and conduce band minimum (CBM, eV) at HSE06+SOC level.
Materiala/bMg/In-YM-YX-XX-YdVBMCBMEHSE+SOCg
MgTiSi2S66.1472.6542.4902.2312.1443.16−6.74−4.961.783
MgTiSi2Se66.4612.7982.6192.2542.2993.37−6.51−4.821.691
MgTiSi2Te67.0153.0262.8252.2952.5303.63−5.56−4.551.011
MgTiGe2S66.2692.6532.4872.3552.2573.15−6.61−4.961.654
MgTiGe2Se66.5722.7982.6202.3852.4013.37−6.36−4.841.517
MgTiGe2Te67.1073.0262.8252.4352.6173.64−5.47−4.600.870
MgZrSi2S66.2502.6812.6072.2332.1493.26−6.83−4.532.307
MgZrSi2Se66.5562.8202.7392.2612.3023.48−6.71−4.492.225
MgZrSi2Te67.1023.0452.9472.3032.5303.74−5.69−4.381.314
MgZrGe2S­66.3722.6782.6082.3542.2613.26−6.78−4.692.084
MgZrGe2Se66.6622.8182.7402.3882.4023.49−6.48−4.581.905
MgZrGe2Te67.1873.0422.9462.4402.6163.76−5.52−4.441.074
MgHfSi2S66.2182.6752.5802.2282.1513.25−6.92−4.422.500
MgHfSi2Se66.5302.8152.7112.2562.3033.46−6.69−4.302.398
MgHfSi2Te67.0843.0412.9192.3002.5323.72−5.64−4.181.456
MgHfGe2S66.3412.6742.5802.3482.2623.25−6.83−4.602.232
MgHfGe2Se66.6382.8152.7112.3832.4043.46−6.47−4.432.045
MgHfGe2Te67.1723.0412.9172.4362.6183.73−5.58−4.391.189
In2Si2S66.2782.7062.2702.1503.41−6.92−4.212.704
In2Si2Se66.6002.8352.2942.3033.59−6.44−4.252.185
In2Si2Te67.1533.0442.3292.5333.83−5.54−4.281.261
In2Ge2S66.3852.7082.3752.2593.43−6.93−4.472.459
In2Ge2Se66.6962.8362.4092.4013.61−6.44−4.491.952
In2Ge2Te67.2343.0432.4572.6163.85−5.48−4.261.220
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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 in Fig. 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 in Fig. 3. The corresponding crystal structures of MgMX2Y6 MLs after the simulation are also inserted in Fig. 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.

Fig. 2.  (Color online) Phonon dispersion of ML MgMX2Y6 and In2X2Y6.
Fig. 3.  (Color online) AIMD simulation results of MgMX2Y6 MLs at 300 K.

Furthermore, cohesive energy (Ecoh) is an important factor to judge the thermal stability. The Ecoh of MgMX2Y6 MLs is defined as Ecoh=(EMg+EM+2EX+6EYEMgMX2Y6)/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 Ecoh indicate higher thermal stability. The calculated cohesive energies of MgMX2Y6 MLs are shown in Fig. 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 (see Fig. 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 in Table 2. For a stable 2D material, it is required to meet the Born-Huang criterion[40], i.e., C11C22C212>0, C66 > 0. Since ML MgMX2Y6 and In2X2Y6 possess a hexagonal symmetry, there are C11 = C22. Therefore, for 2D MgMX2Y6 and In2X2Y6, it needs to be satisfied C211C212>0, C66 > 0. As shown in Table 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.

Table 2.  Calculated elastic constant C11/C22/C12/C66 (N/m), axial Young’s modulus Y11/Y22 (N/m), Poisson’s ratio v11/v22, ultimate strength σ* (N/m), ultimate strain ε*, corresponding to the ultimate strength for x, y, and biaxial tensions of MgMX2Y6 and In2X2Y6 MLs.
MaterialC11/C22C12C66C211C212Y11/Y22v11/v22xyBiaxial
σ*ε*σ*ε*σ*ε*
MgTiSi2S671.8820.3525.764752.3166.120.285.670.144.770.118.950.17
MgTiSi2Se659.6917.2721.213264.2554.690.294.930.163.700.106.230.12
MgTiSi2Te645.5613.5116.031893.4141.560.304.160.202.530.084.830.12
MgTiGe2S666.5519.8223.364036.0860.650.305.460.153.850.086.920.12
MgTiGe2Se655.2716.5419.372781.0450.320.304.630.183.320.106.070.14
MgTiGe2Te641.0911.1814.951563.4838.050.273.780.222.290.084.510.14
MgZrSi2S665.2818.1723.563932.0660.230.285.830.173.760.106.900.12
MgZrSi2Se655.8915.6320.132879.4351.520.285.080.193.130.095.700.11
MgZrSi2Te643.5812.3415.621746.9740.090.284.230.232.330.084.380.11
MgZrGe2S660.6617.3621.653378.0655.690.295.500.176.530.126.530.12
MgZrGe2Se651.8014.8318.492463.4047.560.294.740.202.900.095.380.12
MgZrGe2Te640.7611.5014.631529.2637.520.283.870.252.140.084.440.15
MgHfSi2S673.0420.6826.184907.9867.190.286.420.174.470.127.530.11
MgHfSi2Se660.6617.5521.553371.1755.580.295.980.113.630.105.540.19
MgHfSi2Te645.5613.5116.031893.4141.560.304.560.222.520.094.630.11
MgHfGe2S660.6617.3621.653378.0655.690.296.040.184.090.106.070.10
MgHfGe2Se651.8014.8318.492463.4047.560.295.140.203.270.095.660.12
MgHfGe2Te640.7611.5014.631529.2637.520.284.130.232.340.084.490.13
In2Si2S673.2722.1925.544876.2766.550.304.350.114.170.116.890.14
In2Si2Se662.4519.0321.713537.5056.650.303.780.113.280.115.810.14
In2Si2Te50.1614.9917.592291.5945.690.303.050.112.200.094.630.13
In2Ge2S668.5621.0123.774258.4662.110.314.220.104.020.126.970.15
In2Ge2Se658.5618.0920.233101.5052.960.313.700.113.170.116.040.15
In2Ge2Te647.1214.2816.422016.8842.800.303.030.122.100.094.610.14
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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 in Table 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.

Table 3.  Bader charge analysis of MgMX2Y6 and In2X2Y6 MLs.
MaterialMg/InTi/Zr/HfSi/GeS/Se/TeMaterialMg/InTi/Zr/HfSi/GeS/Se/Te
MgTiSi2S6+2+1.96+3.00−1.66MgHfSi2S6+2+3.98+2.98−1.99
MgTiSi2Se6+2+2.04+2.96−1.66MgHfSi2Se6+2+3.76+2.94−1.94
MgTiSi2Te6+2+1.98+0.65−0.88MgHfSi2Te6+2+2.88+0.74−1.06
MgTiGe2S6+2+1.98+3.02−1.67MgHfGe2S6+2+3.98+2.98−1.99
MgTiGe2Se6+2+1.72+2.97−1.61MgHfGe2Se6+2+3.96+2.96−1.98
MgTiGe2Te6+2+1.58+0.46−0.75MgHfGe2Te6+2+2.88+0.50−0.98
MgZrSi2S6+2+2.24+2.98−1.70In2Si2S6+3+3.00−2.00
MgZrSi2Se6+2+2.06+2.95−1.66In2Si2Se6+3+2.97−1.99
MgZrSi2Te6+2+1.78+0.75−0.88In2Si2Te6+3+0.54−1.18
MgZrGe2S6+2+2.24+2.98−1.70In2Ge2S6+3+2.97−1.99
MgZrGe2Se6+2+2.06+2.95−1.66In2Ge2Se6+3+2.97−1.99
MgZrGe2Te6+2+1.84+0.51−0.81In2Ge2Te6+3+0.48−1.16
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Fig. 4.  (Color online) Calculated cohesive energies of MgMX2Y6 and In2X2Y6 MLs.

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:

Y11=Y22=C211C212C11,
(1)
v11=v22=C12C11.
(2)

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 in Table 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 in Fig. 5 and the details are summarized in Table 2. As shown in Fig. 5 and Table 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.

Fig. 5.  (Color online) Tensile stress σ, as a function of uniaxial strain, ε, along the (a) x- and (b) y-directions and (c) of biaxial strain, respectively, for ML MgMX2Y6 and In2X2Y6.

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 (EHSE+SOCg) are summarized in Table 1, while the projected electronic band structures of MgMX2Y6 and In2X2Y6 MLs are shown in Fig. 6 and Fig. 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].

Fig. 6.  (Color online) Projected electronic band structures of MgMX2Y6 MLs based on HSE06+SOC calculation.
Fig. 7.  (Color online) Projected electronic band structures of In2X2Y6 MLs based on HSE06+SOC calculation.

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 in Fig. 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 in Fig. 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 (see Fig. 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 in Fig. 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 in Fig. 8, and the details are summarized in Table 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 in Fig. 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.

Fig. 8.  (Color online) The band edges of MgMX2Y6 and In2X2Y6 MLs. The vacuum level is set to zero. The work functions of Ag, Ti, Cu, Au and Pt have been marked in the figure.

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{ε21(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) and 9(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 (see Fig. 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 in Fig. 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.

Fig. 9.  (Color online) Optical absorption coefficient of MgMX2Y6 and In2X2Y6 MLs based on HSE06+SOC calculations.

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.

This work was supported by the National Natural Science Foundation of China (Grant No. 61974049, 62222404, 61974050). L. Y. acknowledges National Key Research and Development Plan of China (Grant No. 2021YFB3601200). J. H. Y. acknowledges the Postdoctoral Fund of Hubei Province.



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Fig. 1.  (Color online) (a) The top view and (b, c) side view of monolayer MgMX2Y6. (d) The corresponding first Brillouin zone of MgMX2Y6 monolayers.

Fig. 2.  (Color online) Phonon dispersion of ML MgMX2Y6 and In2X2Y6.

Fig. 3.  (Color online) AIMD simulation results of MgMX2Y6 MLs at 300 K.

Fig. 4.  (Color online) Calculated cohesive energies of MgMX2Y6 and In2X2Y6 MLs.

Fig. 5.  (Color online) Tensile stress σ, as a function of uniaxial strain, ε, along the (a) x- and (b) y-directions and (c) of biaxial strain, respectively, for ML MgMX2Y6 and In2X2Y6.

Fig. 6.  (Color online) Projected electronic band structures of MgMX2Y6 MLs based on HSE06+SOC calculation.

Fig. 7.  (Color online) Projected electronic band structures of In2X2Y6 MLs based on HSE06+SOC calculation.

Fig. 8.  (Color online) The band edges of MgMX2Y6 and In2X2Y6 MLs. The vacuum level is set to zero. The work functions of Ag, Ti, Cu, Au and Pt have been marked in the figure.

Fig. 9.  (Color online) Optical absorption coefficient of MgMX2Y6 and In2X2Y6 MLs based on HSE06+SOC calculations.

Table 1.   Calculated lattice constant a/b (Å), bond length Mg (In)-Y/M-Y/X-X/X-Y (Å), buckling height d (Å), band gaps at HSE06+SOC (EHSE+SOCg, eV) levels, the valence band maximum (VBM, eV) and conduce band minimum (CBM, eV) at HSE06+SOC level.

Materiala/bMg/In-YM-YX-XX-YdVBMCBMEHSE+SOCg
MgTiSi2S66.1472.6542.4902.2312.1443.16−6.74−4.961.783
MgTiSi2Se66.4612.7982.6192.2542.2993.37−6.51−4.821.691
MgTiSi2Te67.0153.0262.8252.2952.5303.63−5.56−4.551.011
MgTiGe2S66.2692.6532.4872.3552.2573.15−6.61−4.961.654
MgTiGe2Se66.5722.7982.6202.3852.4013.37−6.36−4.841.517
MgTiGe2Te67.1073.0262.8252.4352.6173.64−5.47−4.600.870
MgZrSi2S66.2502.6812.6072.2332.1493.26−6.83−4.532.307
MgZrSi2Se66.5562.8202.7392.2612.3023.48−6.71−4.492.225
MgZrSi2Te67.1023.0452.9472.3032.5303.74−5.69−4.381.314
MgZrGe2S­66.3722.6782.6082.3542.2613.26−6.78−4.692.084
MgZrGe2Se66.6622.8182.7402.3882.4023.49−6.48−4.581.905
MgZrGe2Te67.1873.0422.9462.4402.6163.76−5.52−4.441.074
MgHfSi2S66.2182.6752.5802.2282.1513.25−6.92−4.422.500
MgHfSi2Se66.5302.8152.7112.2562.3033.46−6.69−4.302.398
MgHfSi2Te67.0843.0412.9192.3002.5323.72−5.64−4.181.456
MgHfGe2S66.3412.6742.5802.3482.2623.25−6.83−4.602.232
MgHfGe2Se66.6382.8152.7112.3832.4043.46−6.47−4.432.045
MgHfGe2Te67.1723.0412.9172.4362.6183.73−5.58−4.391.189
In2Si2S66.2782.7062.2702.1503.41−6.92−4.212.704
In2Si2Se66.6002.8352.2942.3033.59−6.44−4.252.185
In2Si2Te67.1533.0442.3292.5333.83−5.54−4.281.261
In2Ge2S66.3852.7082.3752.2593.43−6.93−4.472.459
In2Ge2Se66.6962.8362.4092.4013.61−6.44−4.491.952
In2Ge2Te67.2343.0432.4572.6163.85−5.48−4.261.220
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Table 2.   Calculated elastic constant C11/C22/C12/C66 (N/m), axial Young’s modulus Y11/Y22 (N/m), Poisson’s ratio v11/v22, ultimate strength σ* (N/m), ultimate strain ε*, corresponding to the ultimate strength for x, y, and biaxial tensions of MgMX2Y6 and In2X2Y6 MLs.

MaterialC11/C22C12C66C211C212Y11/Y22v11/v22xyBiaxial
σ*ε*σ*ε*σ*ε*
MgTiSi2S671.8820.3525.764752.3166.120.285.670.144.770.118.950.17
MgTiSi2Se659.6917.2721.213264.2554.690.294.930.163.700.106.230.12
MgTiSi2Te645.5613.5116.031893.4141.560.304.160.202.530.084.830.12
MgTiGe2S666.5519.8223.364036.0860.650.305.460.153.850.086.920.12
MgTiGe2Se655.2716.5419.372781.0450.320.304.630.183.320.106.070.14
MgTiGe2Te641.0911.1814.951563.4838.050.273.780.222.290.084.510.14
MgZrSi2S665.2818.1723.563932.0660.230.285.830.173.760.106.900.12
MgZrSi2Se655.8915.6320.132879.4351.520.285.080.193.130.095.700.11
MgZrSi2Te643.5812.3415.621746.9740.090.284.230.232.330.084.380.11
MgZrGe2S660.6617.3621.653378.0655.690.295.500.176.530.126.530.12
MgZrGe2Se651.8014.8318.492463.4047.560.294.740.202.900.095.380.12
MgZrGe2Te640.7611.5014.631529.2637.520.283.870.252.140.084.440.15
MgHfSi2S673.0420.6826.184907.9867.190.286.420.174.470.127.530.11
MgHfSi2Se660.6617.5521.553371.1755.580.295.980.113.630.105.540.19
MgHfSi2Te645.5613.5116.031893.4141.560.304.560.222.520.094.630.11
MgHfGe2S660.6617.3621.653378.0655.690.296.040.184.090.106.070.10
MgHfGe2Se651.8014.8318.492463.4047.560.295.140.203.270.095.660.12
MgHfGe2Te640.7611.5014.631529.2637.520.284.130.232.340.084.490.13
In2Si2S673.2722.1925.544876.2766.550.304.350.114.170.116.890.14
In2Si2Se662.4519.0321.713537.5056.650.303.780.113.280.115.810.14
In2Si2Te50.1614.9917.592291.5945.690.303.050.112.200.094.630.13
In2Ge2S668.5621.0123.774258.4662.110.314.220.104.020.126.970.15
In2Ge2Se658.5618.0920.233101.5052.960.313.700.113.170.116.040.15
In2Ge2Te647.1214.2816.422016.8842.800.303.030.122.100.094.610.14
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Table 3.   Bader charge analysis of MgMX2Y6 and In2X2Y6 MLs.

MaterialMg/InTi/Zr/HfSi/GeS/Se/TeMaterialMg/InTi/Zr/HfSi/GeS/Se/Te
MgTiSi2S6+2+1.96+3.00−1.66MgHfSi2S6+2+3.98+2.98−1.99
MgTiSi2Se6+2+2.04+2.96−1.66MgHfSi2Se6+2+3.76+2.94−1.94
MgTiSi2Te6+2+1.98+0.65−0.88MgHfSi2Te6+2+2.88+0.74−1.06
MgTiGe2S6+2+1.98+3.02−1.67MgHfGe2S6+2+3.98+2.98−1.99
MgTiGe2Se6+2+1.72+2.97−1.61MgHfGe2Se6+2+3.96+2.96−1.98
MgTiGe2Te6+2+1.58+0.46−0.75MgHfGe2Te6+2+2.88+0.50−0.98
MgZrSi2S6+2+2.24+2.98−1.70In2Si2S6+3+3.00−2.00
MgZrSi2Se6+2+2.06+2.95−1.66In2Si2Se6+3+2.97−1.99
MgZrSi2Te6+2+1.78+0.75−0.88In2Si2Te6+3+0.54−1.18
MgZrGe2S6+2+2.24+2.98−1.70In2Ge2S6+3+2.97−1.99
MgZrGe2Se6+2+2.06+2.95−1.66In2Ge2Se6+3+2.97−1.99
MgZrGe2Te6+2+1.84+0.51−0.81In2Ge2Te6+3+0.48−1.16
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    Junhui Yuan, Kanhao Xue, Xiangshui Miao, Lei Ye. A family of flexible two-dimensional semiconductors: MgMX2Y6 (M = Ti/Zr/Hf; X = Si/Ge; Y = S/Se/Te)[J]. Journal of Semiconductors, 2023, 44(4): 042101. doi: 10.1088/1674-4926/44/4/042101
    J H Yuan, K H Xue, X S Miao, L Ye. A family of flexible two-dimensional semiconductors: MgMX2Y6 (M = Ti/Zr/Hf; X = Si/Ge; Y = S/Se/Te)[J]. J. Semicond, 2023, 44(4): 042101. doi: 10.1088/1674-4926/44/4/042101
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    Received: 01 November 2022 Revised: 06 December 2022 Online: Accepted Manuscript: 31 January 2023Uncorrected proof: 06 February 2023Published: 10 April 2023

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      Junhui Yuan, Kanhao Xue, Xiangshui Miao, Lei Ye. A family of flexible two-dimensional semiconductors: MgMX2Y6 (M = Ti/Zr/Hf; X = Si/Ge; Y = S/Se/Te)[J]. Journal of Semiconductors, 2023, 44(4): 042101. doi: 10.1088/1674-4926/44/4/042101 ****J H Yuan, K H Xue, X S Miao, L Ye. A family of flexible two-dimensional semiconductors: MgMX2Y6 (M = Ti/Zr/Hf; X = Si/Ge; Y = S/Se/Te)[J]. J. Semicond, 2023, 44(4): 042101. doi: 10.1088/1674-4926/44/4/042101
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      Junhui Yuan, Kanhao Xue, Xiangshui Miao, Lei Ye. A family of flexible two-dimensional semiconductors: MgMX2Y6 (M = Ti/Zr/Hf; X = Si/Ge; Y = S/Se/Te)[J]. Journal of Semiconductors, 2023, 44(4): 042101. doi: 10.1088/1674-4926/44/4/042101 ****
      J H Yuan, K H Xue, X S Miao, L Ye. A family of flexible two-dimensional semiconductors: MgMX2Y6 (M = Ti/Zr/Hf; X = Si/Ge; Y = S/Se/Te)[J]. J. Semicond, 2023, 44(4): 042101. doi: 10.1088/1674-4926/44/4/042101

      A family of flexible two-dimensional semiconductors: MgMX2Y6 (M = Ti/Zr/Hf; X = Si/Ge; Y = S/Se/Te)

      DOI: 10.1088/1674-4926/44/4/042101
      More Information
      • Junhui Yuan:received the B.S./M.S. degree from Wuhan University of Technology, P. R. China, in 2013/2016 and the Ph.D. degree in Microelectronics and Solid-State Electronics, Huazhong University of Science and Technology, P.R. China, in 2020. He is currently a postdoctoral fellow in the School of Integrated Circuits, Huazhong University of Science and Technology. He is working on the theory research of Hafnia-based ferroelectricity. His research interests include modeling of volatile and non-volatile resistive switching, novel shell DFT-1/2 computational method and prediction of novel functional materials
      • Kanhao Xue:received the B.S. and M.S. degrees in electronic engineering from Tsinghua University, Beijing, China, and the Ph.D. degree in electrical engineering from the University of Colorado at Colorado Springs, USA, in 2010. Hs is now a professor at School of Integrated Circuits, Huazhong University of Science and Technology. He is working on first-principles calculations and novel methods on electronic band structure calculation in semiconductors. He has authored and co-authored more than 120 articles in international refereed journals
      • Lei Ye:received the B.S. and M.S. degrees from Hunan University, and the Ph.D. degree in electrical engineering from the Chinese University of Hongkong. He is now a professor at School of Integrated Circuits, Huazhong University of Science and Technology. He is working on 2D materials and 2D materials-based devices. He has authored and co-authored more than 70 articles in international refereed journals, including: Science, Nature communications
      • Corresponding author: xkh@hust.edu.cnleiye@hust.edu.cn
      • Received Date: 2022-11-01
      • Revised Date: 2022-12-06
      • Available Online: 2023-01-31

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