Emerging versatile two-dimensional MoSi₂N₄ family

Since the extraordinary mechanical properties of graphene (103 GPa of the in-plane stiffness and nearly 1 TPa of the elastic modulus [50]), 2D materials with excellent mechanical properties have attracted great attentions. As for MoSi₂N₄, its experimentally measured tensile strength ($E$) and elastic modulus ($Y$) are 65.8$\pm$18.3 GPa and 491.4$\pm$139.1 GPa [28], respectively, which are nearly half of those in graphene and higher than those in most 2D TMDs (e.g., 22 GPa and 270$\pm$100 GPa of MoS₂) [3, 51, 52], MXene (e.g., 17 GPa and 333 GPa of Ti₃C₂T_x [53], 26 GPa and 386 GPa of Nb₄C₃T_x [54]), and black phosphorene [55] (18 GPa and 166 GPa). In detail, Ren et al. [28] measured the mechanical properties of monolayer MoSi₂N₄ via atomic force microscopy nanoindentation (Fig. 2(a)). With a diamond tip of 11.1 nm, the indentation of hole is about 23 nm. The elastic behavior of monolayer MoSi₂N₄ is demonstrated due to the well fitting force-displacement curves of loading and unloading states. Theoretical prediction of mechanical properties of 2D materials is mainly based on the linear elastic model. The theoretically calculated tensile strength and elastic modulus are 48.3–57.8 GPa and 479.1–487.0 GPa, respectively [47, 56, 57], in good agreement with experimental values.

Actually, the critical strain and ideal tensile stress of 2D materials are vital indicators for practical applications, which depend on the elastic limit and lattice vibration [58]. Li et al. [56] focused on the elastic limit and failure mechanism of monolayer MoSi₂N₄. Under biaxial and uniaxial (zigzag or armchair) strains, the ideal strengths of monolayer MoSi₂N₄ are similar, about 50 GPa, while the corresponding critical strain is 19.5$\%$ (biaxial strain), 26.5$\%$ (zigzag strain) and 17.5$\%$ (armchair strain). In Fig. 3, when the strains are below 20$\%$ ($\epsilon$ $<$ 20$\%$), the tensile stresses ($\sigma$) are in the order of $\sigma_{\text{bi}}$ $>$ $\sigma_{\text{arm}}$ $>$ $\sigma_{\text{zig}}$. There exists an obvious yield phenomenon under biaxial and armchair strains when $\epsilon$ $\geq$ 20$\%$, but the yield limit under a zigzag strain is about 25$\%$. By fitting the initial strain-stress curve based on the linear regression up to 1$\%$ strain (the inset of Fig. 3(a)), the elastic moduli are calculated as $E_{\text{zig}}$ = 448.3 $\pm$ 5.1 GPa and $E_{\text{arm}}$ = 457.8 $\pm$ 3.9 GPa, which are more than twice those of MoS₂ ($E_{\text{zig}}$ = 197.9 $\pm$ 4.3 GPa and $E_{\text{arm}}$ = 200 $\pm$ 3.7 GPa) [59]. The degeneracy of elastic moduli indicate a nearly elastic isotropy in monolayer MoSi₂N₄. On the other hand, the failure mechanism has been investigated on the aspect of lattice stability by phonon dispersion. When the tensile strength limit is reached under an armchair strain, the phonon dispersion has no imaginary frequency. This indicates lattice stability and further reveals that the failure phenomenon of monolayer MoSi₂N₄ is ascribed to the elastic failure of the SiN layer before reaching the critical strain. The obvious imaginary frequency of out-of-plane acoustic branch (ZA) before reaching tensile strength limit demonstrates that the failure mechanism of monolayer MoSi₂N₄ is attributed to phonon instability under the zigzag or biaxial strains.

The mechanical parameters of other members in MA₂Z₄ family are listed in Table 2. Mortazavi et al. [47] found in MA₂Z₄ family the mechanical properties are mainly affected by the terminating atom (Z-site) rather than the core atom (M-site). Three possible factors that influence the mechanical properties are proposed: Z-site atomic mass, structure and chemical bonds. Firstly, increasing Z-site atomic weight deteriorates the mechanical properties, but the elastic modulus has little change when M-site atomic mass is increased. For examples, MA₂N₄ has higher $Y$ and $E$ than MA₂P₄, while $Y$ and $E$ of MoSi₂N₄ and WSi₂N₄ are close. Secondly, since the M-A bonds are absolutely vertical, only M-Z and A-Z bonds participate in the deformation when applying an in-plane loading, indicating that the two bonds related to Z-site atoms determine the mechanical properties. Besides, chemical bonds formed with N atoms are always stronger than those with P/As atoms. Bonds formed with Si are sturdier than those with Ge, resulting in the highest elastic modulus of monolayer MSi₂N₄.

The mechanical properties of some other MA₂Z₄-derived materials have also been investigated, e.g., CrC₂N₄, SnSi₂N₄, SnGe₂N₄ and XMoSiN₂ (X = S/Se/Te) [60, 61, 62, 63]. The predicted elastic modulus and tensile strength of CrC₂N₄ are as high as 676 and 54.8 GPa, respectively, while those of SnSi₂N₄ (478 and 47 GPa) are close to those of MoSi₂N₄. This indicates that the MA₂Z₄ structure offers excellent mechanical features, but the intrinsic mechanisms related to the elements and structures require further studies.

Structure effect on piezoelectricity of MA₂Z₄ has been investigated [64]. The structures are divided into six configurations by different operations (translation, mirror and rotation) of A₂Z₂ layers, which are nominated as $\alpha_{i}$ ($i$ = 1–6), as shown in Fig. 4(a). It is observed that the values of $C_{11}-C_{12}$ of MSi₂N₄ (M = Mo, W) are close when $i$ = 1, 2, 4, 5, while those of $C_{11}-C_{12}$ are higher when $i$ = 3, 6, indicating $\alpha_{3}$ and $\alpha_{6}$ MSi₂N₄ (M = Mo/W) with high resistance to deformation. The structural sensibility of $d_{11}$ is homologous with that of $e_{11}$ from $\alpha_{1}$ to $\alpha_{6}$. $e_{11}$ of $\alpha_{5}$ exhibits the largest value among six structures, which is 13.95 $\times$ $10^{-10}$ C/m for MoSi₂N₄ and 12.17 $\times$ $10^{-10}$ C/m for WSi₂N₄. $d_{11}$ of $\alpha_{5}$-MoSi₂N₄ and $\alpha_{5}$-WSi₂N₄ is 3.53 and 2.91 pm/V, respectively. $d_{11}$ of $\alpha_{1}$-MoSi₂N₄ and $\alpha_{1}$-WSi₂N₄ (experimentally synthesized phases) is 1.15 and 0.78 pm/V, respectively.

Studies on the influence of different M/A/Z atoms reveal that $C_{11}-C_{12}$ of both $\alpha_{1}$ and $\alpha_{2}$ improves with the element periodicity increasing (Fig. 4(c)). For instance, $C_{11}-C_{12}$ of MSi₂N₄ is in the order of CrSi₂N₄¡MoSi₂N₄¡WSi₂N₄. The larger elastic constants of MA₂N₄, compared with those of MA₂P₄ and other 2D materials (e.g., TMDs, metal oxides, and III–V semiconductors [66, 67]), indicate that MA₂N₄ system is more rigid. The change trend of $e_{11}$ and $d_{11}$ follows the opposite regularity, compared with elastic coefficients. $d_{11}$ of MA₂P₄ with the same M and A atoms is larger than that of MA₂N₄ for both $\alpha_{1}$ and $\alpha_{2}$ phases. $d_{11}$ of $\alpha_{1}$- and $\alpha_{2}$-MA₂Z₄ is in the range of 0.78–6.12 pm/V and 0.25–5.06 pm/V, respectively. Monolayer MA₂P₄ nanosheets (e.g., $\alpha_{1}$-CrSi₂P₄, $\alpha_{1}$-MoSi₂P₄, $\alpha_{1}$-CrGe₂P₄, $\alpha_{1}$-MoGe₂P₄ and $\alpha_{2}$-CrGe₂P₄) show excellent piezoelectric response. In addition, the effect of A-site atoms on piezoelectric performance can be ignored so that $d_{11}$ of MoGe₂N₄ is close to that of MoSi₂N₄. $d_{11}$ of most monolayer MA₂P₄ nanosheets is even larger than that of 2D TMDs (e.g., $d_{11}$ = 3.65 pm/V of MoS₂, $d_{11}$ = 2.12 pm/V of WS₂, $d_{11}$ = 4.55 pm/V of MoS₂, and $d_{11}$ = 2.64 pm/V of WSe₂) [68] and $d_{33}$ (3.1 pm/V) of bulk piezoelectric wurtzite GaN [69].

The compressive and tensile biaxial strain are shown to obviously improve and deteriorate the piezoelectric performance of MoSi₂N₄, respectively (Fig. 4(d)) [65]. Piezoelectric stress and strain responses are improved with the in-plane strain from -4$\%$ to 4$\%$. $d_{11}$ can be enhanced by 107$\%$ if a rensile biaxial strain of 4$\%$ is applied. VSi₂P₄, a spin-gapless semiconductor (SGS), possesses a wide range of properties due to its strain sensitivity. With the increasing strain, it presents as ferromagnetic metal (FMM), SGS, ferromagnetic semiconductor (FMS), or ferromagnetic half-metal (FMHM) [70]. In the strain range of 1$\%$–4$\%$, the coexistence of ferromagnetism and piezoelectricity can be achieved in FMS VSi₂P₄. Its $d_{11}$ under 1$\%$, 2$\%$ and 3$\%$ strain is 4.61, 4.94 and 5.27 pm/V, respectively.

There exist both in-plane and out-of-plane piezoelectric polarizations in Janus MSiGeN₄ (M = Mo/W) owing to the broken reflection symmetry along the out-of-plane direction [71, 72]. The in-plane piezoelectric coefficients of Janus MSiGeN₄ ($d_{11}$ = 1.494 pm/V for MoSiGeN₄, $d_{11}$ = 1.050 pm/V for WSiGeN₄) are between those of MSi₂N₄ and MGe₂N₄, while the out-of-plane stress piezoelectric coefficients ($d_{31}$) are –0.014 and 0.011 pm/V for MoSiGeN₄ and WSiGeN₄, respectively. Under an in-plane biaxial strain, $d_{11}$ of MSiGeN₄ is improved with the increasing $e_{11}$, which is similar with MA₂N₄. A tensile strain of 10$\%$ can increase $d_{11}$ of MoSiGeN₄ and WSiGeN₄ by several times, with the values up to 8.081 and 7.282 pm/V, respectively. On the contrary, a compressive biaxial strain can effectively enhance the out-of-plane piezoelectric response. The MA₂Z₄-derived SrAlGaSe₄ has both in-plane ($d_{11}$ = –1.865 pm/V/) and out-of-plane ($d_{31}$ = –0.068 pm/V) piezoelectricity under uniaxial a tensile strain of 6$\%$ [73].

In addition to piezoelectricity, the intrinsic ferroelectricity and its electrical switching in MA₂Z₄ are still open issues. For example, the sliding ferroelectricity is found in vdW MoA₂N₄ bilayer and multilayer [74]. The interlayer inequivalence caused by the stacking order directly generates the out-of-plane polarization. Then, the induced vertical polarization can be switched via the interlayer sliding. The calculated vertical polarization is 3.36, 3.05, 2.49 and 3.44 pC/m for AB stacking bilayer MoSi₂N₄, MoGe₂N₄, CrSi₂N₄ and WSi₂N₄, respectively, which are higher than that of bilayer WTe₂ and BN [75, 76].

The flexoelectricity of monolayer MA₂Z₄ is also of interest, which is often induced by bending deformation of 2D materials with a strain gradient [47]. The out-of-plane bending flexoelectric coefficients of 12 kinds of MA₂Z₄ (M = Mo/Cr/W, A = Si/Ge, Z = N/P) are found in the range of 0.001–0.047 nC/m under a bending strain gradient of 0.3/Å. The highest flexoelectric coefficient of WGe₂N₄ is about 1.5 times higher than that of MoS₂ [77]. It is proposed that the enhancement of flexoelectricity is insufficient with the bending deformation, but is broadened by the construction of asymmetry structures (e.g., Janus counterparts) [78, 79].

Owing to the high tensile strength and thus strong bond interactions, the thermal conductivity of monolayer MA₂Z₄ family has attracted attentions. Based on thermal Boltzmann transport equations, the theoretical lattice thermal conductivity ($\kappa$) of MoSi₂N₄ is predicted as high as 400 Wm^-1K^-1 at room temperature [47, 49, 80, 81], which is larger than that of most other 2D materials, such as hydrogenated borophene (368 Wm^-1K^-1) [82], TMDs (23–142 Wm^-1K^-1) [83, 84, 85], group IVA and VIA compounds (0.26–9.8 Wm^-1K^-1) [86, 87]. But it is lower than that of graphene (3000–5000 Wm^-1K^-1) [88, 89], and h-BNs (600 Wm^-1K^-1) [90]. Such high thermal conductivity of monolayer MoSi₂N₄ is promising for heat conductors and thermal management in semiconductor devices.

In Slack’s classic rules [91, 92], crystals with high thermal conductivity follows four rules: simple crystal structure, light atomic masses, strong bonding and low anharmonicity. In the monolayer MA₂Z₄ family, by replacing M/A/Z atoms at different sites, the different contributions to $\kappa$ have been further examined [47, 49]. It is found that when Z- site atoms are replaced, $\kappa$ of monolayer MA₂Z₄ obeys the Slack’s rules (red lines in Fig. 5(a)). For instance, $\kappa$ of MoSi₂Z₄ decreases by one order of magnitude from Z = N to As, indicating that Z atom plays a critical role in controlling the thermal conductivity of MoSi₂Z₄. The similar variation phenomena are found in MoGe₂Z₄ and WGe₂Z₄ as well. Meanwhile, in A-site-replaced MA₂Z₄, $\kappa$ is decreased by about 40.3$\%$–50.4$\%$ from A = Si to Ge. $\kappa$ of MoGe₂N₄ (286 Wm^-1K^-1) is about 40.3$\%$ lower than that of MoSi₂N₄ (439 Wm^-1K^-1). $\kappa$ of WGe₂P₄ (64 Wm^-1K^-1) is only half that of WSi₂P₄ (129 Wm^-1K^-1). The decreasing phenomena are intrinsically attributed to the phonon properties. The Z- or A-site atoms of MA₂Z₄ determine the phonon frequency range. Monolayer MA₂N₄ shows wider frequency range than MA₂P₄ and MA₂As₄ (Fig. 5(b)). MSi₂Z₄ exhibits wider phonon dispersion than MGe₂Z₄. Thermal conductivity is related to the phonon group velocity that directly depends on phonon branches. Thus, wider phonon bands lead to higher phonon group velocity and result in higher thermal conduction.

When M atoms are from different group IVB or VIB atoms, the variation of $\kappa$ is abnormal and the conventional guideline for searching high $\kappa$ does not work. $\kappa$ exhibits an irregular oscillation with the increasing atomic mass and decreasing Debye temperature (green dash lines in Fig. 5(a)). Mortazavi et al. proposed that the weight of core atoms is the dominant factor on thermal conduction of MA₂Z₄ and $\kappa$ increases with the weight of core atoms. This violates the Slack’s rules but is consistent with the classical theory that stiffer systems favor a higher $\kappa$. However, in our recent work [49], we found that the influential factors on the variation of $\kappa$ is not limited to the atomic mass. For instance, the average mass of WSi₂N₄ is 1.5 times that of CrSi₂N₄, but $\kappa$ shows only 13$\%$ difference. We proposed that the abnormal thermal conductivity of M-site replaced MA₂Z₄ is relative to the group that M atoms belong to. $\kappa$ of MA₂Z₄ with group VIB M atom is about 3–4 times of that with group IVB M atom. These abnormal phenomena with respect to M atoms are attributed to the fundamental vibrational properties and phonon scattering behavior. The acoustic branches of MA₂Z₄ with group VIB M are more bunched and less flattened. Consequently, the phonon scattering rates (Fig. 5(c)) of group VIB M-site MA₂Z₄ are lower than that of group IVB M-site MA₂Z₄, and higher $\kappa$ presents in the former. As shown in Fig. 5(d), the thermal conductivity of monolayer MA₂Z₄ family is in a very wide range (10¹–10³ Wm^-1K^-1), locating between that of 2D TMDs and hBN.

In addition to the thermal conductivity, the thermoelectric performance of monolayer MA₂Z₄ has been also investigated. The dimensionless figure of merit ($ZT$) is always used to measure the efficiency of thermoelectric conversion of a thermoelectric material, which is expressed as:

where $S$, $1/\rho$, $T$, $\kappa_{e}$, and $\kappa_{l}$ are the Seebeck coefficient, electrical conductivity, working temperature, electronic thermal conductivity and lattice thermal conductivity, respectively. The higher $ZT$ indicates the better heat-to-electricity conversion efficiency. Thus, 2D materials for the thermoelectric field are prone to semiconductors with high Seebeck coefficient, electrical conductivity and low thermal conductivities.

The power factor ($PF$ = $\text{$S$}^{\text{$2$}}\text{$T$}/\rho$) of monolayer MoSi₂N₄ is studied to evaluate the capability of producing electricity [57]. The heat conversion into electricity is favored by the temperature since $PF$ increases considerably with temperature. The maximum $PF$ is located at a chemical potential range of 0.12–0.98 eV (upper in Fig. 6(a)). $PF$ in the negative chemical potential region is more sensitive to temperature than in the positive region. Meanwhile the maximum $ZT$ increases with temperature in the chemical potential range of -0.45–0.5 eV. The maximum $ZT$ of monolayer MoSi₂N₄ is up to 1.2 at 1200 K [57], while $ZT$ of MoGe₂N₄ is up to 1.0 at 900 K [95]. Guo et al. [93] studied the effect of strain on the thermoelectric performance of monolayer MoSi₂N₄, as shown in Fig. 6(b). They found that the compressive strain rather than the tensile strain has significant effect on $S$. Besides, $S$ with n-type doping has observable improvement under a compressive strain, which is ascribed to the strain-driven conduction band degeneracies.

As mentioned above, the high lattice thermal conductivity of monolayer MoSi₂N₄ brings negative effect on $ZT$ and further restrains the thermoelectric performance. It is imperative for seeking other MA₂Z₄ semiconductors with low thermal conductivity to satisfy the requirement of high $ZT$. Monolayer MoSi₂As₄ semiconductor has no doubt to be a good candidate due to its low lattice thermal conductivity. However, Huang et al. [94] found that $ZT$ of MoSi₂As₄ is 0.33 for n-type and 0.90 for p-type at 1200 K, which are even lower than that of MoSi₂N₄ (Fig. 6(c)). The underlying reason still remains unknown. Hence, whether there is any member with ultra-high thermoelectric properties needs to be further explored.

Till now, nearly hundred members in monolayer MA₂Z₄ family have been predicted. Wang et al. [29] systematically investigated 72 thermodynamically and dynamically stable MA₂Z₄ compounds with a septuple-atomic-layer which possess the same intercalated architecture as MoSi₂N₄. The electric properties can be classified according to the total numbers of valence electrons. Most of monolayer MA₂Z₄ nanosheets with 32 or 34 valence electrons are semiconductors, while those with 33 valence electrons are non-magnetic metals or ferromagnetic semiconductors. Ding et al. [98] identified 12 stable MSi₂N₄ with trigonal prismatic (H-phase) or octahedral (T-phase) structures, including six new members. The M-site atoms contain both the early transition metal element (group IIIB–VIB) and the late transition metal (group VIIIB, e.g., Pd and Pt). Group IIIB and IVB MSi₂N₄ with H- and T-phase geometries are all stable, while group VB and VIB ones are only stable with H-phase. Moreover, other MoSi₂N₄ derived structures have emerging in an overwhelming trend, such as, bilayer or multilayer MA₂Z₄, heterostructures, Janus MA₂Z₄, etc. These MA₂Z₄ nanosheets exhibit versatile electronic properties depending on the number of valence electrons and structural phases. In this subsection, we emphatically discuss the intrinsic electrical properties of these monolayers and their tunability.

H-phase monolayer MA₂Z₄ nanosheets, where M = Cr/Mo/W/Ti/Zr/Hf, A = Si/Ge, Z = N/P/As, are semiconductors with a bandgap around 0.04–1.79 eV (PBE) and 0.31–2.57 eV (HSE) calculated by DFT [47, 49]. Monolayer MA₂N₄ nanosheets are indirect bandgap materials ($\Gamma$-K), while monolayer MA₂P₄ nanosheets are direct gap semiconductors (K-K). The electronic contribution to CB and VB derived primarily from interior atomic orbitals leads to the robust band edge states [99]. From the projected band structure and charge density distribution (Fig. 7), valence band maximum (VBM) at $\Gamma$ point is mainly contributed by M-$d_{z^{2}}$ orbital and marginally contributed by Z-$p_{z}$ and A-$p_{z}$, while at K point it is contributed by M-($d_{x^{2}-y^{2}}$, $d_{xy}$) and Z-($p_{x}$, $p_{y}$, $p_{z}$) orbitals which locate at the centre area of the structure. Conduction band minimum (CBM) is solely occupied by M-($s,~{}d_{z^{2}}$) orbitals. The strong interaction of $d$ orbitals of M atom renders to the highly dispersion between VB and CB, indicating high charge carrier mobilities and small effective masses. The calculated electron and hole mobilities are 200 and 1100 cm²V^-1s^-1 for MoSi₂N₄, 490 and 2190 cm²V^-1s^-1 for MoGe₂N₄, 320 and 2026 cm²V^-1s^-1 for WSi₂N₄, 690 and 2490 cm²V^-1s^-1 for WGe₂N₄ [47]. The carrier mobility ($\mu$) of monolayer MoSi₂P₄ is comparable with that of MoSi₂N₄, where $\mu$ in zigzag (armchair) direction is 245.992 (257.985) cm²V^-1s^-1 for electrons and 1065.023 (1428.885) cm²V^-1s^-1 for holes [48], which are much higher than those of MoS₂ (72.16 and 200.52 cm²V^-1s^-1 [100]). Besides, SnGe₂N₄ [62], a MA₂Z₄-derived structure without transition metals, possesses a high electron mobility of 1061.66 cm²V^-1s^-1, but a low hole mobility of 28.35 cm²V^-1s^-1, owing to a strongly concave downward conduction band and a flat valence band. Recent DFT calculations also indicate 2D MoSi₂N₄ as an ideal platform for the exploration of exciton-involved physics [101].

The bandgap of bilayer MoSi₂Z₄ (Z = P/As) is similar with that of monolayers due to the weak vdW interaction between layers. In terms of carrier mobilities, there exist several interesting phenomena [102]. (1) In monolayer and bilayer MoSi₂Z₄, the hole carrier mobilities are about 3–4 times larger than those of electrons. The difference of carrier mobilities effectively facilitates the spatial separation of electrons and holes, restraining the recombination probability of photo-excited carriers. (2) Carrier mobility of bilayer MoSi₂Z₄, especially hole carrier mobility, exhibits anisotropic behaviors. (3) the carrier mobilities (electron and hole) of bilayer MoSi₂Z₄ are about 2 times that of monolayer MoSi₂Z₄. Monolayer and bilayer MoSi₂Z₄ with high carrier mobilities exhibit pronounced carrier polarization and are promising materials for high-performance nanoscale electronic and optoelectronic devices.

The electronic transition of monolayer MA₂Z₄ from semiconductor to metal can be triggered by applying in-plane strains [96, 48]. The bandgap of monolayer MoSi₂N₄ decreases with the increasing in-plane biaxial strain (Fig. 8(a)). The relation between bandgap ($E_{g}$) and strain ($\epsilon$) is fitted as [96] $E_{g}~{}=~{}0.004~{}\epsilon^{2}-0.17~{}\epsilon+1.87$. Under an in-plane biaxial strain of 4$\%$ (6$\%$), the effective mass of holes is –2.33 $m_{e}$ (–3.84 $m_{e}$) and the effective mass of electrons is 0.48 $m_{e}$ (0.43 $m_{e}$). This suggests that the in-plane biaxial strain simultaneously enhances the localization of holes and free electrons, which would achieve fascinating features like ferromagnetism and superconductivity. When the in-plane strain is over 10$\%$, there exists an inversion at the edge of VBM, known as a ‘Mexican hat’ in which two VBMs are induced and the transport properties would be improved. As the strain increases, the variation of ‘Mexican hat’ dispersion is more noticeable. When $\epsilon$ = 20$\%$, MoSi₂N₄ becomes semimetal. For MoSi₂P₄ under an in-plane armchair uniaxial strain, the bandgap transfers to be indirect at $\epsilon$ = 2$\%$, 3$\%$ and returns to be direct at $\epsilon$ = –10$\%$ [48]. Semiconductor-metal transition is predicted at $\epsilon_{\text{arm}}$ = –12$\%$ and $\epsilon_{\text{zig}}$ = –12$\%$ or 12$\%$. Under an in-plane compressive strain, the bandgap of bilayer MoSi₂N₄ and WSi₂N₄ is transferred from the indirect to direct state and the bandgap value changes slightly, while the bandgap rapidly decreases with the increasing tensile strain [103].

The vertical (out-of-plane) compressive strain is also demonstrated to effectively tune the electronic properties of bilayer vdW MA₂Z₄ [97]. With the increase of vertical compressive strain, the bandgap of bilayer MoSi₂N₄ monotonically decreases and reaches 0 eV at $\epsilon~{}=~{}22\%$ (Fig. 8(b)). This is attributed to the opposite energy shift of the states in different layers. This shift is driven by the asymmetric charge redistribution on the inner Z-Z sublayer at the interface. The similar transition is confirmed in other bilayer MA₂Z₄, and the pressure to realize such a transition ranges from 2.18 to 32.04 GPa.

The potential of 2D materials in industrial-grade low-dimensional nanodevices is further boosted by the enormous design flexibility offered by the vertical vdW heterostructures (vdWHs), in which physical properties can be customized by the vertically stacking of different 2D atomic layers. Interlayer coupling of semiconducting vdWHs form 3 types of band alignment: type I (straddling gap binds electrons/holes with the critical ratio of conduction band offset and valence band offset), type II (staggered gap promotes the separation of electrons and holes and suppresses the recombination of electrons and holes), and type III (broken gap). Several previous studies have revealed the electronic properties of vdWHs by stacking 2D semiconductors with MA₂Z₄.

TMDs/MA₂Z₄ vdWHs at the ground state belong to the type I alignment, due to the mismatch induced by strain and layer interaction between TMDs and MoSi₂N₄ (Fig. 9) [104, 106]. TMDs provide the main contribution to CBM and VBM. The carrier mobilities of MoSe₂/MoSi₂N₄ and WSe₂/MoSi₂N₄ are up to 10⁴ cm²V^-1s^-1, which are higher than those of monolayer MoSe₂ and WSe₂. Under an external electric field or strain, the band structures of TMDs/MA₂Z₄ vdWHs are transferred from type I to type II. Actually, some other vdWHs by stacking 2D materials (e.g., C₃N₄, ZnO, InSe and Cs₃Bi₂I₉) with MoSi₂N₄ exhibit semiconducting characteristic with type II band structure, indicating promising application in photocatalytic field [107, 108, 105, 109]. The carrier mobility of type II InSe/MoSi₂N₄ is up to 10⁴ cm²V^-1s^-1, indicating more suitable for photocatalytic nano devices when compared with type I TMDs/MA₂Z₄ vdWHs. In addition, BP/MoSi₂P₄ and BP/MoGe₂N₄ vdWHs possess direct bandgap and belong to type II alignment [110, 111]. Combined with high optical absorption properties, these MA₂Z₄-based vdWHs would be promising candidates for solar cell devices.

In metal and semiconductor contacts, the Schottky phenomenon with rectification effect in vdWHs occurs. Reducing the Schottky barrier height (SBH) or tuning the Schottky contacts to Ohmic contacts are key challenges for achieving energy efficient and high-performance power devices. MA₂Z₄-based vdWHs provide opportunities for reconfigurable and tunable nanoelectronic devices. In graphene/MA₂Z₄ vdWHs, there exists a tiny bandgap [112]. P-type Schottky contacts are formed at the interfaces of two heterojunctions, where the electrons are transferred from graphene to MoSi₂N₄ (WSi₂N₄). The n-type SBH ($\Phi_{\text{n}}$) and the p-type SBH ($\Phi_{\text{p}}$) are 0.922 eV and 0.797 eV, respectively, indicating the presence of a p-type Schottky contact [113]. In contrast, graphene/MoGe₂N₄ vdWH forms an n-type Schottky contact with a barrier of 0.63 eV [114]. Monolayer Janus MoSiGeN₄ maintains the semiconducting property with an indirect bandgap of 1.436 (2.124) eV obtained by PBE (HSE06) [115]. The vdWHs formed by graphene and Janus MoGeSiN₄ or MoSiGeN₄ have been investigated. The n-type SBH of graphene/MoGeSiN₄ is 0.63 eV, while the p-type SBH of graphene/MoSiGeN₄ is 0.74 eV. These SBHs are close to those of MA₂Z₄ (MoSi₂N₄ and MoGe₂N₄), but more adjustable, providing a useful guidance for the design of controllable Schottky nanodevices by using MA₂Z₄ family. Compared with graphene-based vdWHs, the vdWHs formed by MA₂Z₄ and other 2D materials possess better performance in Schottky contact [113, 116]. In NbS₂/MoSi₂N₄, $\Phi_{n}$ and $\Phi_{p}$ are 1.642 eV and 0.042 eV, respectively [113]. This ultralow p-type SBH of NbS₂/MoSi₂N₄ vdWH suggests the potential of NbS₂ as an efficient 2D electrical contact to MoSi₂N₄ with high charge injection efficiency, particularly at room-temperature.

In addition, the Schottky contact types of multilayer vdWHs vary with the order of stacking layers. For instance, the graphene/MoSi₂N₄/MoGe₂N₄ vdWH has an n-type Schottky contact with a SBH of 0.33 eV. While MoSi₂N₄/graphene/MoGe₂N₄ and MoSi₂N₄/MoGe₂N₄/graphene vdWHs are both p-type with a SBH of 0.41 and 0.46 eV, respectively [117]. The contact barriers in the multilayer vdWHs are smaller than those in the bilayer vdWHs, suggesting that the graphene/MoSi₂N₄/MoGe₂N₄ vdWHs provide an effective pathway to reduce the Schottky barrier. This is highly beneficial for improving the charge injection efficiency of contact heterostructures. The p-type Schottky contacts of MoSi₂N₄/graphene or WSi₂N₄/graphene at the interface can be transferred to n-type by a compressive strain. When a compressive strain of 10$\%$ is applied, the transition from Schottky to Ohmic contacts occurs in MoSi₂N₄/graphene and WSi₂N₄/graphene. However, no transition is induced by a tensile strain [112, 114].

The optical absorption of monolayer MA₂Z₄ system is up to 10⁵ cm^-1 in the visible range [28, 57, 121, 122], which is comparable with that of graphene, phosphorene and MoS₂. In monolayer MoSi₂N₄ (Fig. 10(a)), the experimental results show that a strong peak of optical absorption appears at about 320 nm and a broad peak in the range of 500–600 nm [28]. The calculated results are in good agreement with the experimental data.

Different atomic compositions bring the tunable optical properties to this system [47, 102, 72, 123]. The obvious redshift phenomena of optical absorption spectra are triggered by heavy atoms in M-site MA₂N₄ (Fig. 10(b)), while the absorption coefficient is insensitive to the M-site counterparts. The absorption of MoSi₂N₄ and WSi₂N₄ is both 10⁵ cm^-1 in the visible region. The similar red-shift of absorption spectra exhibits in A- or Z-site MoA₂Z₄ as well (Fig. 10(c)). The absorption coefficient, however, is improved with the increasing mass of A or Z atoms. For instance, the absorptions of MoSi₂Z₄ (Z = N/P/As) are in the order of MoSi₂N₄¡MoSi₂P₄¡MoSi₂As₄. In Janus MA₂Z₄ (Fig. 10(d)), the redshift of absorption spectra and the absorption enhancement exist in the lighter MoSiGeN₄, compared with WSiGeN₄. As for MA₂Z₄-derived materials (e.g., CrC₂N₄ [60], SnGe₂N₄ [62] and XMoSiN₂ [63]), the absorption of visible light can be up to 10⁵ cm^-1 as well. When Cr is replaced by Mo or W in CrC₂N₄, the first absorption peak appears in ultraviolet spectra and high frequencies.

Furthermore, optical properties of MA₂Z₄ are shown to be manipulated by strain, surface functionalization, and heterostructure. Firstly, the optical absorption spectrum of MoSi₂N₄ shows redshift (blueshift) under a tensile (compressive) strain [96, 124, 125]. The absorption edges decrease with the increasing tensile strain. A 4–10$\%$ tensile strain enhances the optical absorption capacity in the visible region up to 43–70$\%$. The reflectance ability (average reflectance rate), meanwhile, is improved from 16$\%$ to 23$\%$ with the tensile strain from 0$\%$ to 10$\%$. Yang et al. [125] found monolayer MoSi₂N₄ exhibits more outstanding optical absorption capacity in the ultraviolet range than that in the visible range, especially under biaxial compressive strain. The optical bandgap is evaluated by the slope of absorption peak, in which the corresponding value is 2.47 eV without strain, 2.9 eV with –3$\%$ strain and 3.05 eV with –4$\%$ strain. Furthermore, compared with monolayer MA₂Z₄, the bilayer or multilayer MA₂Z₄ shows strong optical absorbance and broad absorption areas as well [102, 126, 127]. When the vertical (out-of-plane) compressive strain increases from 0 to 12$\%$, there exists strong blueshift but only slight decrease of absorption and reflectivity. This manifests that MA₂Z₄ possesses stable optical absorption capacity independent of vertical strain and the number of layers, which will be more convenient for experimental fabrication of 2D optoelectronic devices [127].

Secondly, it has been reported that surface functionalization plays an active role in regulating the optical properties of MoSi₂N₄, where the adatoms can be Au, F, and Alkali elements (Li, Na, K) [128, 129, 130]. In Au-MoSi₂N₄, the optical absorption capacity is significantly improved with the increment of Au concentration (6.25–56.3$\%$/unit cell). It can be increased by 1–2 times in the visible light region and by 52$\%$ in the ultraviolet region. The results show that Au absorption is beneficial for the photocatalytic activity, making Au-MoSi₂N₄ a potential candidate for photoelectrochemical applications and short-wavelength optoelectronic devices. Alkali-metal adsorption not only enhances the optical absorption coefficient but enlarges the absorption area [129]. The absorption coefficient at a wavelength of 380 nm is 0.34$\times$10⁵ cm^-1 for pristine MoSi₂N₄, 0.57$\times$10⁵ cm^-1 for Li-MoSi₂N₄, 1.09$\times$10⁵ cm^-1 for Na-MoSi₂N₄, and 0.89$\times$10⁵ cm^-1 for K-MoSi₂N₄. The light at wavelength of 780 nm is almost unabsorbable by pristine MoSi₂N₄. However, the absorption at this wavelength is 0.92$\times$10⁵ cm^-1 for Li-MoSi₂N₄, 1.34$\times$10⁵ cm^-1 for Na-MoSi₂N₄, and 1.14$\times$10⁵ cm^-1 for K-MoSi₂N₄. Na-MoSi₂N₄ exhibits the strongest optical response among these three alkali-metal-decorated MoSi₂N₄.

Thirdly, in Fig 11, the optical absorption capacity of monolayer MoSi₂N₄ is comparable with that of monolayer MoS₂ and MoSe₂, and better than that of other monolayer 2D materials, such as BlueP, InSe and Cs₃Bi₂I₉. It is interesting that MA₂Z₄-based vdWHs have more excellent optical response than the monolayer MA₂Z₄ and other monolayer 2D materials [118, 104, 109, 105, 119, 120, 111, 131]. These enhancements mainly appear in the visible and ultraviolet light range. The high performance of BlueP/MoSi₂N₄ vdWH is in the visible light range of 460–780 nm (the insert of Fig. 11(a)), which is attributed to the overlap of electronic states between valence bands caused by the interlayer coupling and charge transfer [118]. InSe/MoSi₂N₄ vdWH shows great improvement of absorption in the ultraviolet range [105]. The absorption of monolayer Cs₃Bi₂I₉ is superior than that of monolayer MoSi₂N₄ in the photon energy range of 1–2.6 eV, while the relationship reverses in the ultraviolet range (high photon energy), as shown in Fig.11(c). Surprisingly, the absorption is enhanced in the whole range by constructing the Cs₃Bi₂I₉/MoSi₂N₄ vdWH [109]. Similarily, this increasing phenomena appear in Janus MoSTe/MoGe₂N₄, MoS₂/MoSi₂N₄ and MoSe₂/MoSi₂N₄ vdWHs (Fig.11(d–f)) [104, 119, 120], indicating the tunability of optical absorption by forming vdWHs. As for strain effect on the absorption of MA₂Z₄-based heterostructures, the obvious redshift (blueshift) of optical spectrum appears under a tensile (compressive) strain. In BlueP/MoSi₂N₄ vdWH, it is found that a compressive strain restrains the absorption on coefficient in the visible region, while a tensile strain (0–8$\%$) promotes the optical absorption ability [118]. In Janus MoSTe/MoGe₂N₄ vdWH, a compressive strain increases the absorption coefficient in the visible and ultraviolet regions, while a tensile strain enhances it in the infrared region, regardless of which side (S or Te atoms) approaching to MoSi₂N₄ [120].

It has been reported that group-VB (V/Nb/Ta) MA₂Z₄ nanosheets exhibit robust intrinsic magnetism, and group-VIB (Cr/Mo/W) MA₂Z₄ ones show antiferromagnetic ground states [29, 132, 133, 134]. In Table 2, NbSi₂N₄, NbSi₂As₄, NbGe₂Z₄, NbGe₂P₄, TaSi₂N₄ and TaGe₂P₄ are ferromagnetic metals with the magnetic moment from 0.37 to 0.78 $\mu_{\text{B}}$ per metal atom, while VA₂Z₄ (except VGe₂N₄) are ferromagnetic semiconductors with a magnetic moment of 1 $\mu_{\text{B}}$ per V atom. On the other hand, NbSi₂P₄, TaSi₂P₄, TaSi₂As₄, TaGe₂N₄ and group-VIB MA₂Z₄ (e.g., CrSi₂N₄, CrSi₂P₄, CrGe₂N₄, CrGe₂As₄, MoGe₂As₄, WGe2As₄ and WGe₂P₄) are antiferromagnetic.

For ferromagnetic semiconductors, there exist three types: type-I spin gapless semiconductors (SGSs) (a zero bandgap in one spin channel but a bandgap in another spin channel), type-II SGSs (bandgaps of spin-up or spin-down channel but zero gap for the valence band and conduction band in opposite spin channels), and bipolar magnetic semiconductors (BMSs) (the valence band and conduction band in opposite spin channels approaching to Fermi level). 2D SGSs and BMSs with high Curie temperatures are highly desirable for advanced spintronic applications due to their unique electronic structure and high spin polarization [137, 138]. The intrinsic magnetic mechanisms of monolayer MA₂Z₄ nanosheets (e.g., VSi₂N₄, VSi₂P₄ and NbSi₂N₄) have been investigated by dissecting the spin-up and spin-down band structures, magnetic anisotropic energy (MAE) and Curie temperature ($T_{\text{C}}$).

The magnetic ground states of monolayer VSi₂Z₄ are ferromagnetic (FM) by considering the spin polarization, as shown in Fig. 12(a) [135]. Based on GGA-PBE functional calculations, for the spin-up channel of monolayer VSi₂P₄ both VBM and CBM are located at K point and a direct bandgap of 0.36 eV presents, while VBM at $\Gamma$ point and CBM at M point indicate an indirect bandgap of 0.42 eV for the spin-down channel (Fig 12(c)). The zero indirect bandgap between spin-up VBM and spin-down CBM implies that monolayer VSi₂P₄ and VSi₂N₄ are type-II SGSs via PBE calculations. In VSi₂As₄, the VBM and CBM for spin channels nearby the Fermi level are in the opposite direction, indicating that the spin-polarized currents would be easily generated with tunable spin polarization by applying a small gate voltage. The results show that the intrinsic ferromagnetism is attributed to the unpaired electron from the metal atom. It can be noticed that in VSi₂Z₄ (Fig 12(b)), the $d$ orbital electrons from transition metal occupy the bands approaching to the Fermi level, accompanying with the mixed orbital (e.g., $d_{z^{2}}$+ $d_{x^{2}+y^{2}}$) or transition of orbital composition (e.g., the transition between $d_{z^{2}}$ and $d_{x^{2}+y^{2}}$) [136, 121, 139].

The increment of bandgap is 0.3 eV under the consideration of Coulomb interaction ($U$) effect [29, 140]. The bandgap of VSi₂Z₄ by HSE functional is larger than that by PBE and PBE+$U$. For instances, by HSE functional, VSi₂N₄ possesses a direct bandgap of 0.78 eV at K point, VSi₂P₄ has an indirect bandgap of 0.84 eV with spin-up CBM at $\Gamma$ point and spin-down VBM at M point, and NbSi₂P₄ exhibits an indirect bandgap of 0.54 eV with spin-up VBM at $\Gamma$ point and spin-down CBM at M point. The comparison with the three calculation method shows that the bandgap increases from PBE, PBE+$U$ to HSE. However, the spin polarizations under these methods are still high since there exists only one spin channels around the Fermi level.

Magnetic anisotropic energy (MAE) of NbSi₂N₄, VSi₂N₄ and VSi₂P₄ are summarized in Table 2. The positive values indicate the in-plane alignment of magnetic moments. Therefore, these ferromagnetic monolayer MA₂Z₄ nanosheets have easy magnetization plane, i.e., none energy consumes when magnetization rotates in the plane. In the finite energy resolution (1 eV), it is found that the total energy has no dependence on the angle of the magnetic moment within the plane, implying the weak in-plane anisotropy of MAE [140], similar to the recently reported 2D MnPS₃ [141]. MAE of VSi₂Z₄ (Z = N/P) in the range of 0.11 to 0.25 meV per magnetic atom is lower than other 2D materials (e.g., 0.8 meV of CrI₃ [142], 0.72 meV of Fe₃P [143] and 1 meV of Fe₃GeTe₂ [144]), and close to CrCl₃ (0.02 meV), CrBr₃ (0.16 meV), NiI₂ [145] (0.11 meV) and FeCl₂ [146] (0.07 meV).

Curie temperature ($T_{\text{C}}$) of VSi₂N₄, VSi₂P₄ and VSi₂As₄ is 230, 235, and 250 K, respectively. [135], which is much higher than 45 K for monolayer CrI₃  [147] and 130 K for monolayer Fe₃GeTe₂ [148]. In Akanda’s paper [140], the normalized magnetization of VSi₂N₄ and VSi₂P₄ based on Monte Carlo calculations follows to the analytic expression: $m(T)~{}=~{}(1-T/T_{\text{C}})^{\beta}$, as shown in Fig. 14. The results from PBE+$U$ show that $T_{\text{C}}$ of VSi₂N₄ and VSi₂P₄ are 350 and 452 K, respectively, which are above room temperature. The difference of predicted $T_{\text{C}}$ is ascribed to the unequal calculation method using different DFT functionals. $T_{\text{C}}$ of VSi₂N₄ by HSE functional is up to 506–687 K. $T_{\text{C}}$ of Janus VSiGeN₄ and VSiSnN₄ is 507 K and 347 K, respectively [136]. With the same functional (e.g., PBE+$U$), the transition temperature of monolayer MA₂Z₄ is higher than that of other 2D magnets (e.g., TcGeSe₃, TcGeTe₃, ScCl, YCl, LaCl, LaBr₂, CrSBr, etc.) with high $T_{\text{C}}$ over room temperature [149, 150].

The magnetic properties of MA₂Z₄ family can be tuned by strain, adatoms and defects [140, 136, 139, 151, 152, 153]. Similar to the case of magnetic thin films [154, 155, 156], strain or stress also has a significant effect on magnetic behaviors of monolayer MA₂Z₄. The strain coefficient ($\alpha_{\epsilon}$) is defined as the sensitivity of MAE to strain, which is expressed as $\alpha_{\epsilon}$ = $dE_{\text{MAE}}/d\epsilon$. In VSi₂N₄, $\alpha_{\epsilon}$ is –10 $\mu$eV/$\%$ under a compressive uniaxial strain and –6 $\mu$eV/$\%$ under a biaxial strain, while in VSi₂P₄, $\alpha_{\epsilon}$ = –6 $\mu$eV/$\%$ under a tensile uniaxial strain, –10 $\mu$eV/$\%$ under a tensile biaxial strain, and 17 $\mu$eV/$\%$ under a compressive biaxial strain (Fig. 13). The strain sensitivity of both two materials is lower than that of CrSb (32 $\mu$eV/$\%$) [157]. When the biaxial strain is up to 3–4$\%$, the spin direction of VSi₂N₄ rotates from in-plane to out-of-plane state [140, 139].

Additionally, substitutional doping and atomic vacancies induce the transition of nonmagnetic state in MoSi₂N₄ to magnetic state. Transition metal substitutional doping (e.g., Ag, Au, Bi, Fe, Mn, Pb and V) induces the asymmetric spin channels in MoSi₂N₄ with generating a magnetic moment of 1.00–5.87 $\mu_{\text{B}}$ [153, 159]. Schwingenschl$\ddot{\text{o}}$gl et al. [152] reported that N-site vacancy in MoSi₂N₄ leads to the spin-majority bands crossing the Fermi level. N- and Si-site vacancies generate a magnetic moment of 1.0 and 2.0 $\mu_{\text{B}}$, respectively. The nonmagnetic state in MA₂Z₄-derived MoN₂X₂Y₂ (XY = AlO, GaO, InO) nanosheets is transferred to ferromagnetic state by a hole doping [134]. Li et al. [160] found the element substitution of nitrogen by carbon causes different magnetic moments in monolayer MSi₂C_xN_4-x (M = Cr/Mo/W, $x$ = 1 or 2). The position of carbon atoms determines the ground state of magnetic moments. Monolayer CrSi₂CN₃ (C bridged Si and Mo) is predicted as ferromagnetic half-metal, where the magnetic moments is mainly originated from Cr atoms (0.72 $\mu_{\text{B}}$ per atom). There exist three AFM structures, i.e., CrSi₂N₂C₂ (two C atoms located at two outermost sites), MoSi₂N₂C₂ and CrSi₂N₃C (one C atoms located at outermost site). The magnetic moments are ascribed to the C and metal atoms. The FM states in half-metallic 2D systems are attributed to the hole-mediated double exchange, while the AFM states are originated from the super-exchange.

MA₂Z₄ family also has spin-valley properties. As an emerging degree of freedom, valley refers to the presence of multiple energy extremal points in the Brillouin zone (BZ) for low-energy carriers in a semiconductor. Analogous to charge and spin, the valley degree of freedom can be exploited for information encoding and processing, leading to the concept of valleytronics [161]. The spin-valley feature of semiconductors in MA₂Z₄ family has been systematically studied. Taking into account the spin-orbit coupling (SOC), valley spin splittings of MA₂Z₄ appear near both the CBM and VBM. The remarkable splitting near VBM is 139–500 meV, but the spin splitting near CBM is tiny [162, 158, 163, 164].

Six kinds of monolayer MA₂Z₄ (M = Mo/W, A = C/Si/Ge, Z = N/P/Ge) nanosheets, i.e., MoSi₂P₄, MoSi₂As₄, WSi₂P₄, WSi₂As₄, WGe₂P₄ and WGe₂As₄, possess strong spin-valley coupling [158, 165]. They are all direct bandgap semiconductors with band extrema located at the inequivalent K and K’ point (Fig. 15(a)). A remarkable spin splitting near the VBM is shown in all the six monolayers with SOC, and the splitting in W-based compounds is larger than that in Mo-based ones. The spin valleys are attributed to the MZ₂ layer by analysing DOS and charge distribution (Fig. 15(b)). Furthermore, the comparison of spin band structures between monolayer MoP₂ (Fig. 15(c)) and MoSi₂P₄ (Fig. 15(e)) shows that there exist other interfering states in the energy range of the valleys of MoP₂. This manifests that the AZ layers play the role of structural stabilizer and protect the valley states from the interference of the $p$ orbitals of Z atoms and $d$ orbitals of M atom. Time-reversal symmetry always constrains the valley-polarized current. Nonzero Berry curvature of MA₂Z₄ in Fig. 15(f) indicates the abnormal transverse velocity proportional to the Berry curvature would be generated, leading to a spatial separation of the carriers coupled to two different valleys. The carriers from different valleys are accumulated at opposite sides, resulting in an anomalous valley Hall effect. [158, 166]

Monolayer MoSi₂As₄ is found to exhibit ‘perfect valleys’ with no interference from other part of Brillouin zone, as well as multiple-folded valleys (Fig. 16) [166]. The another valley is originated from the second unoccupied conduction band, which enlarges the degree of freedom in energy. The calculation results show that the electron can be selectively pumped from VBM to CBM with a low photon energy of 0.41 eV. With 1.00 eV energy input, the electron can be excited to the VBM and second unoccupied conduction band, indicating the potential application of MoSi₂As₄ for multiple-information operator and storage in valleytronic devices.

The experimentally synthesized MoSi₂N₄ and WSi₂N₄ are indirect bandgap semiconductors in which CBM is located at K (or K’) and VBM at $\Gamma$ point, limiting their potential utilization in valleytronic devices. Nevertheless, the difference between the uppermost valence band at K and $\Gamma$ point is small (e.g., 144.4 meV in MoSi₂N₄). Strain engineering could eliminate this small difference. The indirect bandgap of MoSi₂N₄ is transferred to the direct bandgap under a compressive strain of 2$\%$ and the direct-gap state maintains until a compressive strain up to 4$\%$ [162, 161]. Monolayer MoSi₂N₄ with –4$\%$ strain possesses a large spin splitting and strong spin valley coupling (valleys K and K’) and thus would be an ideal valleytronic materials.

For other magnetic members in MA₂Z₄ family, the coupling between magnetism and valley provokes the quantum anomalous Hall (QAH) effect. The spin splitting of VSi₂N₄ is 102.3 meV (27.3 meV) at VBM (CBM) [37]. In contrast, the spin splitting of VSi₂P₄ is 49.4 meV at CBM but can be neglected at VBM [167]. Ma et al. [167] separately considered the effect of SOC or magnetic exchange interaction on spin splitting of VSi₂P₄, and proposed that the combined effect causes its spontaneous valley polarization (Fig. 17). The spin splitting energy at K and K’ points is expressed as $\Delta^{\text{K/K'}}$ = $E^{\text{K/K'}}_{\uparrow}$-$E^{\text{K/K'}}_{\downarrow}$, $\uparrow$ and $\downarrow$ mean spin-up and spin-down channels. With the magnetic exchange interaction, the broken time-reversal symmetry leads to $\Delta^{\text{K}}_{\text{mag}}$ = $\Delta^{\text{K'}}_{\text{mag}}$; with SOC, the time-reversal symmetry results in $\Delta^{\text{K}}_{\text{SOC}}$ = $-\Delta^{\text{K'}}_{\text{SOC}}$; with the synergistic effect, the net spin splitting should be the $\Delta^{\text{K}}_{\text{mag}}$+$\Delta^{\text{K}}_{\text{SOC}}$ at K point and $\Delta^{\text{K'}}_{\text{mag}}$+$\Delta^{\text{K'}}_{\text{SOC}}$ at K’ point. Li et al. [168] found Hubbard-$U$ can effectively tune the phase diagram of MA₂Z₄ family, and result in intriguing magnetic, valley and topological features. When $U$ = 2.25 and 2.36 eV, the QAH phase and valley structure coexist, as shown in Fig. 17(e) and (f). Besides, strain effect promotes the trivial topology in VSi₂P₄ and VA₂Z₄-derived materials (VN₂X₂Y₂) [167, 168, 169]. There exists an edge state connecting the valence and conduction bands under a compressive strain of 2.1 $\%$ [167] and a biaxial tensile strain [168]. Recent first-principles calculations reveal 2D MSi₂Z₄ with 1T’ structure as larg-bandgap and tunable quantum spin Hall insulator, which possesses a protected spin-polarized edge state and a large spin-Hall conductivity [170].

Semiconductors in 2D MA₂Z₄ family have atomic-scale thickness, ambient stability, suitable bandgap and excellent electronic mobilities, indicating that this family could be promising candidates for the new-generation miniaturized field-effect transistors (FETs) [171, 172, 173, 174, 175]. Double-gate metal-oxide-semiconductor FET (DG MOSFET) based on monolayer MoSi₂N₄ has been designed and investigated [171, 172]. This kind of FET has advantages of high operating frequency, good gain controllability and low feedback capacitance. There are three factors to evaluate the performance of DG MOSFET, i.e., on-current ($I_{\text{on}}$), gate control, and intrinsic delay time and power consumption.

$I_{\text{on}}$ represents the device operating speed, while off-current ($I_{\text{off}}$) means the static power dissipation. According to the International Technology Roadmap for Semiconductors (ITRS) published in 2013, the standard of $I_{\text{on}}$ ($I_{\text{on}}/I_{\text{off}}$) is 900 $\mu$A$\mu$m^-1 (9.0$\times$10³). The highest $I_{\text{on}}$ of n-type (p-type) DG monolayer MoSi₂N₄ MOSFET is up to 1813 $\mu$A$\mu$m^-1 (1690 $\mu$A$\mu$m^-1), as shown in Fig. 18(b) and (c) [171], which is 70$\%$ (80$\%$) higher than n-type (p-type) DG monolayer MoS₂ MOSFET [176]. Subthreshold swing ($SS$) can reflect the switching rate of MOSFET between the on and off states, which always describes the gate-control ability in the subthreshold region. $SS$ of DG monolayer MA₂Z₄ MOSFET is predicted as 69 mV dec^-1 for the n-type and 52 mV dec^-1 for the p-type when the optimum $I_{\text{on}}$ is reached. As for sub-5 nm scale MOSFETs, the value of $SS$ should be below the Boltzmann limit at room temperature (60 mV dec^-1) since the short channel effect would cause the coexist of tunneling current and thermoionic current, and further influence the overall performance of devices. When gate length ($L_{\text{g}}$) is below 5 nm, $SS$ of p-type and n-type MA₂Z₄ MOSFETs is 46–52 mV dec^-1 (Fig. 18(d)) [171, 172]. Switching speed in MOSFETs is described by the intrinsic delay time and power consumption. The ITRS standard of delay time for the high performance and low power is 0.423 and 1.493 ps, respectively. Besides, ITRS requirements of power consumption for the high performance and low power are 0.24 and 0.28 fJ$\mu$m^-1, respectively. DG monolayer MA₂Z₄ MOSFET with an optimum assembly can meet the standards. In brief, the design of device determines whether DG monolayer MA₂Z₄ MOSFET satisfies the ITRS standard in terms of the structural parameters, like gate length, doping concentration to the source and drain, and underlap length between gate and electrodes.

Photocatalytic process can be described by four important steps: (1) light absorption to generate electron-hole pairs, (2) separation of excited charges to suppress the electron-hole recombination, (3) transfer of electrons and holes to the surface of photocatalysts, and (4) utilization of charges on the surface for redox reactions [177, 178]. The typical reactions are water splitting and CO₂ reduction, which are important for the environment and dual-carbon confinement strategy. 2D MA₂Z₄ systems with high optical absorption could be prospective candidates for photocatalytic applications.

The band edges of photocatalyst candidate in 2D MA₂Z₄ systems must straddle the standard redox potentials. The conduction band edge (CBE) of monolayer MoSi₂N₄ is about 0.71 eV, higher than the H⁺/H₂ reduction level. The valence band edge (VBE) is about 0.19 eV, lower than the O₂/H₂O oxidation level. Band edges of monolayer MoSi₂N₄ straddle the water redox potentials in both highly acidic and neutral conditions (Fig. 19(a)). In contrast, the band edges of monolayer WSi₂N₄ (WGe₂N₄) only satisfy the standard in the highly acidic (neutral) condition [47, 124]. Group-IVB MSi₂N₄ nanosheets with appropriate band edges are considered as the most suitable materials among 2D MA₂Z₄ family for both water splitting and CO₂ reduction application [121]. It is revealed that adatoms reduce the bandgap of MoSi₂N₄ and further promote the possibility of photocatalytic process [128, 129, 179]. The CBE of Au-MoSi₂N₄ is closer to the level for CO₂/HCOOH reduction and thus can easily transfer charges to CO₂ and produce natural gas (HCOOH, HCHO, CO, CH₄, and CH₃OH). The VBE of Au-MoSi₂N₄ approaching to the O₂/H₂O redox potential is beneficial for the interaction between holes and H₂O, and restrains the electron-hole recombination (Fig. 19(b)) [128]. Meanwhile, Li and Na absorption can improve the capacity of water splitting by using MoSi₂N₄ [129]. The adjustable CBE and VBE are observed in MA₂Z₄-based heterostructures (e.g., BlueP/, InSe/ and MoSiGeN₄/MoSi₂N₄) [118, 105, 126, 180, 181, 182], providing more possibility of this family for photocatalytic devices.

The other important factor of photocatalysis is the absorption of reactive atoms or molecules, which determines the electron transition to the surface and the charge utilization for redox reactions. The optimal site for hydrogen or CO₂ absorption in monolayer MA₂Z₄ is reported as Z site due to the lowest spontaneous binding energy compared with other sites [183, 121]. The Gibbs free energy for the hydrogen adsorption ($\Delta\!G_{\text{H}^{\ast}}$) on monolayer MoSi₂N₄ and WSi₂N₄ is calculated to be 2.51 and 2.79 eV, respectively, which is much larger than the ideal value ($\Delta\!G_{\text{H}^{\ast}}$ = 0 eV). This indicates the weak binding of hydrogen in pristine monolayer MSi₂N₄ (M = Mo/W) and inertial hydrogen evolution reaction (HER) activity [183]. The HER performance of MSi₂N₄, as well as N₂ reduction reaction (NRR), can be triggered by introducing N-site vacancy [183, 184, 185, 186]. As shown in Fig. 20, it is obvious that the vacancy significantly influence $\Delta\!G_{\text{H}^{\ast}}$. $\Delta\!G_{\text{H}^{\ast}}$ is decreased about 2–3 times by M-site vacancy and approaches to zero by N-site vacancy ($\Delta\!G_{\text{H}^{\ast}}$ = –0.14 eV in MoSi₂N₄ and –0.02 eV in WSi₂N₄). This manifests that the HER performance of MSi₂N₄ with the outermost N-site vacancy is comparable with and even better than that of Pt. On the other hand, transition metallic atomic doping and strain engineering are verified as the effective strategies to tune $\Delta\!G_{\text{H}^{\ast}}$ and trigger HER, oxygen evolution reaction (OER) or oxygen reduction reaction (ORR) activity [185, 187, 188]. Particularly, $|\Delta\!G_{\text{H}^{\ast}}|$ is calculated as only 0.05 eV in WSi₂N₄ with Fe-doping at Si-site. A 3$\%$ tensile strain results in $\Delta\!G_{\text{H}^{\ast}}$ = 0.015 eV in NbGe₂N₄.

In the MA₂Z₄ family, there exist other good photocatalysts with appropriate $\Delta\!G$ [132, 189, 190, 123], for examples TiSi₂N₄, HfA₂Z₄, ZrA₂Z₄, Janus MSiGeN₄, etc. Recently, a theoretical method in multilevel is proposed to screen appropriate candidates for hydrogen evolution in MA₂Z₄ family [189]. There exist four screening criteria: (1) small structural deformation after hydrogen absorption to maintain the potential stability during the HER processes, (2) low absolute value of Gibbs free energy ($|\Delta G|$ $\rightarrow$ 0) with hydrogen adsorption, (3) suitable bandgaps, (4) high environmental stability. Taking these factors in mind, the multilevel screening workflow discovers seven MA₂Z₄ with great stability and highly active HER among 144 MA₂Z₄ structures, i.e., $\alpha_{1}$-VGe₂N₄, $\alpha_{1}$-NbGe₂N₄, $\alpha_{1}$-TaGe₂N₄, $\alpha_{1}$-NbSi₂N₄, $\alpha_{2}$-VGe₂N₄, $\alpha_{2}$-NbGe₂N₄, and $\alpha_{2}$-TiGe₂P₄. Monolayer $\alpha_{1}$-NbSi₂N₄ with the lowest formation energy is considered as the most promising 2D MA₂Z₄ material for the HER application along the synthesis routine. At low H coverage ($\theta<25\%$), the optimal $|\Delta G|$ of these seven MA₂Z₄ is lower than 0.1 eV, which is comparable with or even superior to that of Pt (–0.09 eV at $\theta$ = 25$\%$). A descriptor ($E_{\text{LUS}}$) is proposed as energy level of the lowest unoccupied state to evaluate the capacity of H adsorption, since hydrogen adsorption is often accompanied with the electron transition to the CBM (for semiconductors) or the Fermi level (for metals). A higher $E_{\text{LUS}}$ is prone to restraining electron filling and causing weaker H adsorption. On the contrary, a lower $E_{\text{LUS}}$ represents the higher ability of H adsorption. Compared with $\alpha_{1}$-MoSi₂N₄, $\alpha_{1}$-NbSi₂N₄ with lower $E_{\text{LUS}}$ shows higher activity toward HER.

HER performance of monolayer MA₂Z₄ family is also examined by the combination of DFT calculations and machine learning algorithms including support vector regression (SVR), kernel ridge regression (KRR), random forest regression (RFR), extreme gradient boosting regression (XGBR), least absolute shrinkage, and selection operator (LASSO) [190]. $\Delta G_{\text{H}^{\ast}}$ and Gibbs free energy of deuterium ($\Delta G_{\text{D}^{\ast}}$) can be accurately and rapidly predicted via XGBR by using only simple genetic programming processed elemental features, with a low predictive root-mean-square error of 0.14 eV. $\Delta G_{\text{H}^{\ast}}$ of group-VB MA₂Z₄ is closer to zero, indicating the excellent HER capacity. For example, TaSn₂P₄ (0.07 eV) has a similar absolute value of $\Delta G_{\text{H}^{\ast}}$ as NbSn₂P₄ (–0.05 eV), while $\Delta G_{\text{H}^{\ast}}$ of CrSn₂P₄ (0.23 eV) is nearly 3.5 times higher than that of TaSn₂P₄ (0.07 eV). It can be concluded that M element is the crucial factor for the HER performance of MA₂Z₄ materials. In addition, NbSi₂N₄ with $\Delta G_{\text{H}^{\ast}}$ = –0.041 eV and $\Delta G_{\text{D}^{\ast}}$ = –0.102 eV, as well as VSi₂N₄ with $\Delta G_{\text{H}^{\ast}}$ = 0.024 eV and $\Delta G_{\text{D}^{\ast}}$ = –0.033 eV is screened as the best HER and deuterium evolution reaction (DER) catalysts among the MA₂Z₄ family.

As for ORR, the four-electron (4e^-) mechanism favors the production of H₂O. The order of ORR activity is MGe₂As₄ $>$ MSi₂As₄ $>$ MSi₂N4 $>$ MSi₂2P₄ $>$ MGe₂P₄ $\approx$ MGe₂N₄ [191]. Among them, VGe₂As₄, CrGe₂As₄, VSi₂As₄ and NbSi₂As₄ are screened out to be highly promising electrocatalysts with a small overpotential around 0.5–0.6 V. The topmost surface As acts as the active site, and the p-band center of the As atom shows correlation with the adsorption strength of the critical intermediate. Zhang et al. [192] efficiently screened photocatalytic OER catalysts in MA₂Z₄ family via an automated high-throughput workflow. They found the adsorption ability of O atoms determines the catalytic effect. $\beta_{2}$-ZrSi₂N₄ and $\beta_{2}$-HfSi₂N₄ are considered as the efficient photocatalytic OER catalysts. In particular, CrGe₂As₄ exhibits outstandingly high ORR activity with ultralow overpotential (0.49 V), which is comparable with the Pt-based catalysts. The metallic conductivity, as well as the moderate adsorption and orbital hybridization between As and O* intermediate, is responsible for the exceptional activity [191].

Metal-air batteries have great advantages of high-energy-density metal anodes, active air cathodes, light weight, and simple structure, which are convenient to utilize in portable equipments. Bilayer or multilayer vdW MoSi₂N₄ shows great potential application as electrodes (both anode and cathode) of Zn-air batteries [194]. On the anode side, the maximum theoretical capacity of Zn in MoSi₂N₄ is up to 257 mAh/g. While on the cathode side, O₂ reduction reaction on the MoSi₂N₄ surface is more efficient than the general sluggish four-electron aqueous O₂ redox reactions. Furthermore, VSi₂N₄ provides two critical specifications (high specific capacity and full battery open-circuit voltage) for the high-performance secondary Li-ion or Na-ion batteries (LIBs/NIBs) (Fig. 21(a)) [193]. The ionic capacity is up to 1312 mAh/g for Li and 492 mAh/g and Na, while the average open-circuit voltages are 0.02–0.06 V for LIBs, and 0.02–0.17 V for NIBs.

Environmental dependence of physical properties of 2D materials promotes the development of potential applications, e.g., sensors [195, 196, 197, 198, 199]. Gas sensor is one of the significant device for detecting gas molecules, especially contaminated and poisonous ones, which can be utilized in the field of industrial harm evaluation, cultivation of agricultural products, assessment of medical drugs, etc. The physisorption behaviors of gas molecules of monolayer MoSi₂N₄ by spin-polarized DFT calculations have been investigated [45, 200]. Due to the weak interaction and small charge transfer, the gas molecules are physically adsorbed on the MoSi₂N₄ surface (Fig. 21(b)). The results show that H₂, N₂, CO, CO₂, NO, NO₂, H₂O, H₂S, NH₃ and CH₄ molecules reduce the bandgap of MoSi₂N₄ (from 1.73 eV to 1.50 eV) [200]. While the absorption of O₂, NO₂ and SO₂ molecules obviously influences the electronic properties of MoSi₂N₄ and even induces the spin polarization with magnetic moments (1–2 $\mu_{\text{B}}$). The magnitude of magnetic moments is sensitive to the concentration of gas molecules, which increases with the increment of concentration of NO₂ but decreases with the increment of concentration of SO₂. This indicates that MoSi₂N₄-based gas sensor has a high application potential for O₂, NO, NO₂ and SO₂ detection. Furthermore, the introduction of N vacancy into MoSi₂N₄ improves the absorption performance [45], resulting in MoSi₂N₄ with promising prospects in highly sensitive and reusable gas sensors of H₂O and H₂S molecules.