Janus 2D monolayers have currently attracted increasing attention, which possess unique physical and chemical properties caused by their special crystal structure, such as strong Rashba spin splitting, the second harmonic generation response, and out-of-plane piezoelectric polarizations^[1]. Experimentally, Janus monolayer MoSSe, which is constructed from MoS₂/MoSe₂, has been achieved by different experimental methods with an out-of-plane structural asymmetry^{[2, 3]}. In theory, many Janus 2D materials have been predicted by the first-principle calculations^[4–12], such as Janus transition metal chalcogenides (TMDs), Janus transition-metal oxides, PtSSe, Janus MA₂Z₄, TiXY (X/Y = S, Se and Te), VSSe, SnSSe and Janus group-III monochalcogenide M₂XY (M = Ga, In; X/Y = S, Se and Te). Some unique properties have also been predicted, such as an out-of-plane piezoelectric coefficient^{[13, 14]} and a significant Rashba spin splitting induced by intrinsic out-of-plane built-in electric field^[10]. The Janus monolayer BiN and AsP field-effect transistors (FETs) possess sufficient competitiveness against other 2D FETs, which is confirmed by the benchmarking of the energy-delay product^{[15, 16]}.

Recently, the high-quality 2D MoSi₂N₄ and WSi₂N₄ are successfully synthesized with excellent ambient stability by chemical vapor deposition (CVD)^[17], which starts a new 2D material family. In the wake of MoSi₂N₄ and WSi₂N₄, twelve kinds of 2D family WSi₂N₄ are proposed with and ( = 1 to 6) phases by intercalating MoS₂-type MZ₂ monolayer into InSe-type A₂Z₂ monolayer^[18]. The MA₂Z₄ with 32 or 34 valence electrons (VEC) are mostly semiconductors. Moreover, -WSi₂P₄ for the spin-valley polarization, -TaSi₂N₄ for Ising superconductor and -SrGa₂Se₄ for topological insulator are predicted^[18]. And then, spin-valley polarization are also investigated in MoSi₂N₄, WSi₂N₄ and MoSi₂As₄^{[19, 20]}. Intrinsic piezoelectricity of the MA₂Z₄ family along structure effect has been predicted by the first-principle calculations^{[21, 22]}. Coexistence of intrinsic piezoelectricity and ferromagnetism induced by small biaxial strain in the septuple-atomic-layer VSi₂P₄ and VSi₂N₄ has been achieved, which offers the opportunities for achieving multifunctional electronic devices^[23]. 2D van der Waals electrical contact to monolayer MoSi₂N₄ has been reported^[24]. For example, the MoSi₂N₄/NbS₂ contact exhibits an ultralow Schottky barrier height^[24]. High intrinsic lattice thermal conductivity in monolayer MoSi₂N₄ has also been predicted by solving the phonon Boltzmann transport equation^[25].

It is very worthy of intensive study to construct a Janus monolayer from the MA₂Z₄ family. In our previous work, Janus MSiGeN₄ (M = Mo and W) monolayers have been predicted from -MA₂Z₄^[12]. The MSi₂N₄ and MGe₂N₄ (M = Zr and Hf) monolayers with a phase are semiconductors^[18]. And then, in this work, Janus monolayer MSiGeN₄ (M = Zr and Hf) are built by replacing the Si/Ge atoms of the top SiN/GeN bilayer in a MSi₂N₄/MGe₂N₄ monolayer with Ge/Si atoms. Their crystal stability, electronic structures, piezoelectric and optical properties are investigated by the first-principle calculations. Calculated results show that MSiGeN₄ (M = Mo and W) monolayers are indirect gap semiconductors with dynamic, mechanical and thermal stabilities, and that tensile strain can induce weak Rashba-type spin splitting near CBM, and that MSiGeN₄ (M = Mo and W) monolayers have observable piezoelectric response and high absorption coefficients in the visible light region.

Within the density functional theory (DFT)^[26], the projected augmented wave (PAW) method and GGA with the Perdew–Burke–Ernzerh (GGA-PBE) as the exchange-correlation functional are used, as implemented in the VASP code^[27–30]. It has been proved that GGA is more suitable for a gap of experimentally synthesized MoSi₂N₄ monolayer than HSE06^[17]. So, we use GGA to study the electronic and optical properties of MSiGeN₄ (M = Zr and Hf) monolayers. The SOC is included to investigate electronic structures of MSiGeN₄ (M = Zr and Hf) monolayers. The plane-wave cutoff energy of 500 eV is used with a 16 × 16 × 1 k-mesh. We relax both the lattice constants and internal atomic positions until the maximal residual Hellmann–Feynman forces are less than 0.0001 eV/Å. We use the total energy as convergence criterion, which is set to eV. A vacuum space of larger than 30 Å is used to avoid interactions with other neighboring layers. After optimizing the crystal structure, 0.0001 eV/Å. We use the total energy as convergence criterion, which is set to with a supercell of 5 × 5 × 1, and the phonon dispersions of MSiGeN₄ (M = Zr and Hf) monolayers are calculated by using the PHONOPY code^[31]. The ab-initio molecular dynamics (AIMD) simulations are performed with a supercell of size 4 × 4 × 1 for more than 3000 fs with a time step of 1 fs. The in-plane spin-textures are computed by using PYPROCAR code^[32].

The elastic stiffness tensor are calculated by using strain-stress relationship (SSR), and the piezoelectric stress coefficients are attained by density functional perturbation theory (DFPT) method^[33]. A Monkhorst–Pack mesh of 16 × 16 × 1 is used to calculate , and 9 × 16 × 1 for . Due to 3D periodic boundary conditions, the 2D elastic coefficients and piezoelectric stress coefficients have been renormalized: = and = , where is the length of the unit cell along the z direction. It is noted that the calculated optical properties also depend on ^[34], which should be normalized by multiplying , where for the thickness of 2D material. However, the is not well defined like graphene. In this work, the = 35.8 Å is used as .

The crystal structure of the Janus monolayer MSiGeN₄ (M = Zr and Hf) is shown in Fig. 1 with the rhombus primitive cell and the rectangle supercell marked by black and red frames. With MN₂ (M = Zr and Hf) triple layers sandwiched between the SiN and GeN bilayers, the symmetry of Janus monolayer MSiGeN₄ (M = Zr and Hf) is lower than that of the MSi₂N₄ (M = Zr and Hf) or the MGe₂N₄ (M = Zr and Hf) monolayer due to the lack of the vertical reflection symmetry. In other words, Janus monolayer MSiGeN₄ (M = Zr and Hf) can be built by replacing the Si/Ge atoms of the top SiN/GeN bilayer in the MSi₂N₄/MGe₂N₄ monolayer with Ge/Si atoms. We use GGA to optimize lattice constants of monolayer MSiGeN₄ (M = Zr and Hf), and they are 3.110 and 3.097 Å, which are between the ones of MSi₂N₄ (3.05 and 3.19 Å) and MGe₂N₄ (3.04 and 3.18 Å)^[18].

The dynamical stability of monolayer MSiGeN₄ (M = Zr and Hf) is studied by calculating their phonon band dispersions, which are shown in Fig. 2. It is clearly seen that the phonon branches are free from any imaginary frequencies, indicating the dynamical stability of these monolayers. They exhibit three acoustic and eighteen optical phonon branches due to seven atoms in the unit cell. In addition, the thermal stability is examined by AIMD simulations. During the simulations, the temperature is kept at 600 K, and the temperature and total energy fluctuations of only ZrSiGeN₄ monolayer are plotted in Fig. 3 due to similar results between ZrSiGeN₄ and HfSiGeN₄. The crystal structures of ZrSiGeN₄ at 600 K after the simulation for 3 ps are also plotted in Fig. 3. It is found that monolayer ZrSiGeN₄ undergoes no structural reconstruction around 600 K, which indicates the thermal stability of these monolayers with the temperature and total energy fluctuating in the acceptable range.

To verify the mechanical stability of MSiGeN₄ (M = Zr and Hf), the investigation of the mechanical properties are carried out. Due to hexagonal symmetry, using Voigt notation, the elastic tensor can be expressed as:

$C = \left( {\begin{array}{*{20}{c}} {{C_{11}}} & {{C_{12}}} & 0\\ {{C_{12}}} & {{C_{11}}} & 0\\ 0 & 0 & {({C_{11}} - {C_{12}})/2} \end{array}} \right). $ (1)

The two independent elastic constants of ZrSiGeN₄/HfSiGeN₄ are = 403.92/465.93 N/m and = 110.58/135.19 N/m, and the = ( )/2 = 146.67 /165.37 N/m as shear modulus , which satisfy the Born criteria of mechanical stability^[35]:

$ C_{11}>0,\; \; C_{66}>0. $ (2)

The 2D Young's moduli is given^[35]:

$ C_{\rm{2D}} = \frac{C_{11}^2-C_{12}^2}{C_{11}}. $ (3)

The calculated is 373.65/426.71 N/m for monolayer ZrSiGeN₄/HfSiGeN₄, which are larger than ones of many 2D materials^{[13, 36–38]}, indicating that these monolayers are rigid. However, they are smaller than ones of our predicted MSiGeN₄ (M = Mo and W) monolayers^[12]. The Poisson's ratio is also calculated by / , and they are 0.274/0.290 for monolayer ZrSiGeN₄/HfSiGeN₄.

The MGe₂N₄ (M = Zr and Hf) has a lower formation energy than MSi₂N₄^[18]. So, we calculate the cohesive energy ( ) of MSiGeN₄ with respect to MGe₂N₄, which is defined as = – + > – , where and are the energies of the MSiGeN₄ and MGe₂N₄ monolayers, and and are the energies of the isolated atoms of Si and Ge. The calculated is −1.85 eV/−1.90 eV for ZrSiGeN ₄/HfSiGeN₄ monolayer, which implies the feasibility of the experimental synthesis.

Phonon band dispersion calculations, elastic constants and AIMD simulations reveal the dynamical, mechanical and thermal stabilities of predicted monolayers. Monolayer MSi₂N₄ (M = Mo and W) have been synthesized by introducing Si during CVD growth of M₂N (M = Mo and W)^[17]. We have proposed that the MSiGeN₄ (M = Mo and W) monolayers can be achieved by introducing Si and Ge during CVD growth of M₂N (M = Mo and W)^[12]. The same way may be used to achieve the MSiGeN₄ (M = Zr and Hf) monolayers.

For Janus MSiGeN₄ (M = Mo and W) monolayers, the SOC can produce important effects on their electronic structures, like Rashba spin splitting and valley polarization^[12]. Thus, the SOC is considered to investigate electronic structures of MSiGeN₄ (M = Zr and Hf) monolayers, and the corresponding energy bands without/with SOC are shown in Fig. 4. The results without/with SOC show that MSiGeN₄ (M = Zr and Hf) monolayers are indirect gap semiconductors with VBM at the Γ point and CBM at the M point. The GGA gap of ZrSiGeN₄ is 1.134, and 1.336 eV for HfSiGeN₄. When including SOC, the gap of ZrSiGeN₄/HfSiGeN₄ reduces to 1.115 eV/1.282 eV. The SOC can also remove band degeneracy, and we take spin-orbit splitting (Δ) at VBM to illustrate the strength of SOC. The Δ is 35 meV/81 meV for ZrSiGeN₄/HfSiGeN₄, which means that the SOC has more strong effects on HfSiGeN₄.

Strain can be used to tune the physical and chemical properties of 2D materials^[39–41]. Here, we examine the biaxial strain effects on the electronic properties of MSiGeN₄ (M = Zr and Hf) monolayers. The is used to simulate compressive/tensile strain with / 1$?>, where and are the strained and unstrained lattice constants, respectively. Due to similar dependence on strain for electronic structures of ZrSiGeN₄ and HfSiGeN₄, we only show energy bands of HfSiGeN₄ as a function of in Fig. 5, and the the energy band gaps of MSiGeN₄ (M = Zr and Hf) monolayers with from 0.90 to 1.10 are plotted in Fig. 6. With strain from 0.90 to 1.10, the gap of monolayer MSiGeN₄ (M = Zr and Hf) firstly increases, and then decreases, which can be found in many 2D materials^{[12, 41]}. In the considered strain range, it is found that both compressive and tensile strains can induce semiconductor to metal transition. The critical point is about 0.92 for compressive strain, and about 1.10 for tensile one. For MSiGeN₄ (M = Mo and W) monolayers, no semiconductor to metal transition is observed in the same strain range^[12]. Another difference is that the maximum of gap can be induced by compressive strain for MSiGeN₄ (M = Mo and W)^[12], and tensile strain for MSiGeN₄ (M = Zr and Hf).

It is found that tensile strain can tune the positions of VBM and CBM of MSiGeN₄ (M = Zr and Hf). Tensile strain can make CBM change from M point to Γ point, and VBM move from Γ point to one point along the M–K direction. These lead to that MSiGeN₄ (M = Zr and Hf) change from an indirect gap semiconductor to a direct one to an indirect one. For example, a direct gap semiconductor can be observed at 1.04 strain. The Rashba-type spin splitting near the CBM can be induced by tensile strain. At 1.04 strain, the enlarged views of the conduction bands near the Fermi level for MSiGeN₄ (M = Zr and Hf) monolayers using GGA+SOC are plotted in Fig. 7. It is clearly seen that a very weak Rashba-type spin splitting is produced. The constant energy 2D contour plots of spin texture of HfSiGeN₄ as a representative centered at the Γ point are plotted in Fig. 8 at an energy surface of 1.5 eV above the Fermi level. The spin-up (red) and spin-down (blue) electronic bands can be distinctly observed, which shows the Rashba-type spin splitting. Due to the pure 2D Rashba spin splitting, the concentric spin-texture circles are observed. In the Rashba spin split bands, the only in-plane and spin components are observed with the lack of out-of-plane component. The tensile strain can tune the relative positions of valence band extrema (VBE), and can induce valence bands convergence, for example at = 1.06 point, which can produce an important influence on their electronic transport properties. However, the compressive strain produces small changes on electronic structures of MSiGeN₄ (M = Zr and Hf) monolayers except for the energy band gap.

The -MSi₂N₄ or MGe₂N₄ (M = Zr and Hf) monolayers (No. 164) are centrosymmetric, which show no piezoelectricity. The MSiGeN₄ (M = Zr and Hf) monolayers (No. 156) lack inversion symmetry, and a reflection symmetry with respect to the central M atomic layer is broken. This leads to both / and / being nonzero, and the piezoelectric stress and strain tensors become:

$ e = \left( {\begin{array}{*{20}{c}} {{e_{11}}} & { - {e_{11}}} & 0\\ 0 & 0 & { - {e_{11}}}\\ {{e_{31}}} & {{e_{31}}} & 0 \end{array}} \right), $ (4)

$ d = \left( {\begin{array}{*{20}{c}} {{d_{11}}} & { - {d_{11}}} & 0\\ 0 & 0 & { - 2{d_{11}}}\\ {{d_{31}}} & {{d_{31}}} & 0 \end{array}} \right). $ (5)

For MSiGeN₄ (M = Mo and W) monolayers, the same reduced piezoelectric stress and strain tensors can be attained^[12], but the original -MSi₂N₄ or MGe₂N₄ (M = Mo and W) monolayers (No.187) lack inversion symmetry, and have vertical reflection symmetry, which gives rise to only in-plane piezoelectricity / . The two independent and are derived by :

$ d_{11} = \frac{e_{11}}{C_{11}-C_{12}} ,\quad d_{31} = \frac{e_{31}}{C_{11}+C_{12}}. $ (6)

We use nonprimitive orthorhombic supercells as the computational unit cell (in Fig. 1) to calculate the of MSiGeN₄ (M = Zr and Hf) monolayers. Based on Eq. (6), their can be attained from previous calculated and . The calculated and of MSiGeN₄ (M = Zr and Hf) monolayers are listed in Table 2, along with ones of MSiGeN₄ (M = Mo and W) monolayers^[12]. Calculated results show that the / of -MSiGeN₄ (M = Zr and Hf) are lower than ones of -MSiGeN₄ (M = Mo and W). Similar results can be found in Janus monolayer SnSSe and MoSSe^[5]. The monolayer SnSSe is built from centrosymmetric monolayer or , while MoSSe can be attained from noncentrosymmetric or . And, the calculated / of SnSSe is lower than ones of MoSSe^[5]. However, for both MSiGeN₄ (M = Zr and Hf) and MSiGeN₄ (M = Mo and W), their are very small.

By calculating the complex dielectric function of MSiGeN₄ (M = Mo and W) monolayers, their optical properties can be attained. The imaginary part of is determined by a summation over empty band states as follows^[42]:

$ \varepsilon_2(\omega) = \frac{2\pi e^2}{\Omega \epsilon_0}\sum\limits_{k,v,c}\delta(E^c_k-E^v_k-\hbar\omega)|<\psi^c_k|u.r|\psi^v_k>|^2, $ (7)

where the vacuum dielectric constant, the volume and the energy of the incident phonon are marked by , and ; The and represent the momentum operator and the wave function at the k point; The v, and mean the valence bands, conduction bands and the polarization vector in the incident electric field. By the Kramers–Kronig relation, the real part can be calculated. According to the calculated and , the absorption coefficient can be expressed as^[43]:

$ \alpha(\omega) = \frac{\sqrt{2}\omega}{c}\left\{\left[\varepsilon_1^2(\omega)+\varepsilon_2^2(\omega)\right]^{1/2}-\varepsilon_1(\omega)\right\}^{\frac{1}{2}}. $ (8)

Here, the rhombus primitive cell is used to calculate the optical properties of MSiGeN₄ (M = Mo and W) monolayers. It is well known that the unit-cell volume of a 2D material is not well-defined, which leads to some calculated physical properties of a 2D material depending on the the length of the unit cell along the z direction, for example optical properties^[34]. The and of MSiGeN₄ (M = Mo and W) monolayers are calculated along xx/yy and zz directions, and are shown in Figs. 9 and 10. Because of hexagonal symmetry, the optical spectra between x and y directions are isotropic for light polarization along the in-plane directions. Due to similar energy band structures, it is clearly seen that the ZrSiGeN₄ and HfSiGeN₄ show very similar optical spectra. Due to distinct optical selection rules, there is a strong anisotropy in the optical spectra along xx/yy and zz directions. The static dielectric constants for ZrSiGeN₄/HfSiGeN₄ are 2.94/2.74 along xx/yy direction and 2.74/2.56 along zzdirection, which can be attained from the real part of the dielectric constant at zero energy. It is found that a sharp increase for at the first onset of the optical transitions between xx/yyand zzdirections is almost the same, and is at about 1.31 eV/1.71 eV for ZrSiGeN₄/HfSiGeN₄. From the ultraviolet to the visible light region, the calculated results imply that both xx/yyand zzdirections show strong absorption intensity with the absorption coefficient values being more than 10⁴ cm⁻¹ in the visible region, which means high efficiency in the utilization of solar energy. It is found that the absorption intensity along the xx/yydirection has more smaller changes than one along zzdirection in the visible light region, especially for ZrSiGeN₄.

The Janus -MSiGeN₄ (M = Mo and W) and -MSiGeN₄ (M = Zr and Hf) monolayers have been predicted by the reliable first-principle calculations. In fact, our works provide an idea to achieve Janus structures from the MA₂Z₄ family, and many Janus MA₂Z₄ can be further investigated. For example, -MSiGeP₄ (M = Cr, Mo and W) can be built, based on semiconducting MSi₂P₄ and MGe₂P₄ (M = Cr, Mo and W)^[18]. In these Janus MA₂Z₄, both in-plane and out-of-plane piezoelectricity, Rashba-type spin splitting, valley polarization and second harmonic generation responses can be explored.

In summary, the electronic structures, piezoelectric properties and optical properties of Janus MSiGeN₄ (M = Zr and Hf) monolayers are systematically investigated by DFT. They exhibit mechanical, dynamic and thermal stabilities, and show indirect gap properties. It is found that strain can effectively tune their electronic structures, and can induce semiconductors to metal transition in the considered strain range. A very weak Rashba-type spin splitting near CBM can be induced by tensile strain. Both in-plane and out-of-plane piezoelectricity are predicted in MSiGeN₄ (M = Zr and Hf) monolayers. The high absorption coefficients in the visible light region can be found along both xx/yy and zz directions. Our works can stimulate further experimental works to synthesize Janus MA₂Z₄ monolayers, and will motivate farther theoretical studies of other Janus MA₂Z₄, like -MSiGeP₄ (M = Cr, Mo and W) monolayers.

This work is supported by Natural Science Basis Research Plan in Shaanxi Province of China (2021JM-456). We are grateful to the Advanced Analysis and Computation Center of China University of Mining and Technology (CUMT) for the award of CPU hours and WIEN2k/VASP software to accomplish this work.