Tuning transport coefficients of monolayer $\mathrm{MoSi_{2}N_{4}}$ with biaxial strain

The successful exfoliation of grapheneq6 induces increasing attention on two-dimensional (2D) materials. Many of them have semiconducting behaviour, which has various potential application in electronics, optoelectronics and piezoelectronicsm4-1 ; m5-1 ; xzq ; m2-1 . Their electronic structures, heat transport and piezoelectric properties have been widely investigatedm1 ; m2 ; m3 ; m4 ; m5 ; m6 ; m7 ; m8 ; m9 ; m10 ; m11 . It has been proved that the strain can effectively tune electronic structures, transport and piezoelectric properties of 2D materialsm12 ; m10 ; m11 ; m13 ; m14 ; m15 ; m16 ; m17 ; m18 , which shows great potential for better use in the nanoelectronic, thermoelectric and piezoelectric applications. For example, both compressive and tensile strain can induce the semiconductor to metal transition in monolayer $\mathrm{MoS_{2}}$m12 . In many transition metal dichalchogenides (TMD) monolayers, the power factor can be enhanced by strain due to bands convergem10 ; m11 ; m13 . With increased tensile strain, the lattice thermal conductivity shows monotonous decrease, up-and-down and jump behavior with similar penta-structuresm14 . Strain can also improve the piezoelectric strain coefficient by tuning the elastic and piezoelectric stress coefficientsm15 ; m16 ; m17 ; m18 .

Recently, the layered 2D $\mathrm{MoSi_{2}N_{4}}$ and $\mathrm{WSi_{2}N_{4}}$ have been experimentally achieved by chemical vapor deposition (CVD)m19 . The septuple-atomic-layer $\mathrm{MA_{2}Z_{4}}$ monolayers with twelve different structures are constructed by intercalating $\mathrm{MoS_{2}}$-type $\mathrm{MZ_{2}}$ monolayer into InSe-type $\mathrm{A_{2}Z_{2}}$ monolayerm20 . The 66 thermodynamically and dynamically stable $\mathrm{MA_{2}Z_{4}}$ are predicted by the first principle calculations. They can be common semiconductor, half-metal ferromagnetism or spin-gapless semiconductor (SGS), Ising superconductor and topological insulator, which depends on the number of valence electronsm20 . We predict intrinsic piezoelectricity in monolayer $\mathrm{MA_{2}Z_{4}}$m21 , which means that $\mathrm{MA_{2}Z_{4}}$ family may have potential application in piezoelectric field. Structure effect on intrinsic piezoelectricity in monolayer $\mathrm{MSi_{2}N_{4}}$ (M=Mo and W) has also been reported by the first principle calculationsm21-1 . By applied strain, the $\mathrm{VSi_{2}P_{4}}$ monolayer undergoes ferromagnetic metal (FMM) to SGS to ferromagnetic semiconductor (FMS) to SGS to ferromagnetic half-metal (FMHM) with increasing strainm22 . Some materials of $\mathrm{MA_{2}Z_{4}}$ lack inversion symmetry with a strong SOC effect, which are expected to exhibit rich spin-valley physicsm20 . The valley-dependent properties of monolayer $\mathrm{MoSi_{2}N_{4}}$, $\mathrm{WSi_{2}N_{4}}$ and $\mathrm{MoSi_{2}As_{4}}$ have been predicted by the first-principles calculationsm20 ; m23 ; m24 . Recently, Janus 2D monolayer in the new septuple-atomic-layer 2D $\mathrm{MA_{2}Z_{4}}$ family has been achievedm25 , which shows Rashba spin splitting and out-of-plane piezoelectric polarizations.

In nanoscale devices, the residual strain usually exists in real applicationsl111 . In our previous work, the small strain effects (0.96 to 1.04) on piezoelectric coefficients of monolayer $\mathrm{MoSi_{2}N_{4}}$ have been investigatedm21 . In this work, the large (0.90 to 1.10) biaxial strain-tuned electronic structures and transport coefficients of monolayer $\mathrm{MoSi_{2}N_{4}}$ are studied by the first principle calculations. With $a/a_{0}$ from 0.90 to 1.10, the energy band gap of monolayer $\mathrm{MoSi_{2}N_{4}}$ firstly increases, and then decreases. In n-type doping, the Seebeck coefficient S can be effectively enhanced by applying compressive strain, and then the $ZT_{e}$ can be improved. The tensile strain can induce flat valence bands around the $\Gamma$ point near the Fermi level, producing large p-type S. Therefore, our works give an experimental proposal to improve transport coefficients of monolayer $\mathrm{MoSi_{2}N_{4}}$.

The rest of the paper is organized as follows. In the next section, we shall give our computational details and methods about transport coefficients. In the third and fourth sections, we will present main results of monolayer $\mathrm{MoSi_{2}N_{4}}$ about strain-tuned electronic structures and transport coefficients. Finally, we shall give our conclusions in the sixth section.

To avoid interactions between two neighboring images, a vacuum spacing of more than 32 $\mathrm{{\AA}}$ along the z direction is added to construct monolayer $\mathrm{MoSi_{2}N_{4}}$. The elastic stiffness tensor $C_{ij}$ are calculated by using strain-stress relationship (SSR), which are performed by using the VASP codepv1 ; pv2 ; pv3 within the framework of DFT1 . A kinetic cutoff energy of 500 eV is adopted, and we use the popular generalized gradient approximation of Perdew, Burke and Ernzerh of (GGA-PBE)pbe as the exchange-correlation potential to calculate elastic and electronic properties. The total energy convergence criterion is set to $10^{-8}$ eV, and the Hellmann-Feynman forces on each atom are less than 0.0001 $\mathrm{eV.{\AA}^{-1}}$. The Brillouin zone (BZ) sampling is done using a Monkhorst-Pack mesh of 15$\times$15$\times$1 for elastic constants $C_{ij}$. The 2D elastic coefficients $C^{2D}_{ij}$ have been renormalized by the the length of unit cell along z direction ($Lz$): $C^{2D}_{ij}$=$Lz$$C^{3D}_{ij}$ .

The electronic transport coefficients of $\mathrm{MoSi_{2}N_{4}}$ monolayer are calculated through solving Boltzmann transport equations within the constant scattering time approximation (CSTA), which is performed by BoltzTrapb code. To include the SOC, a full-potential linearized augmented-plane-waves method is used to calculate the energy bands of $\mathrm{MoSi_{2}N_{4}}$ monolayer, as implemented in the WIEN2k package2 . To attain accurate transport coefficients, a 35 $\times$35$\times$ 1 k-point meshes is used in the first BZ for the energy band calculation, make harmonic expansion up to $\mathrm{l_{max}=10}$ in each of the atomic spheres, and set $\mathrm{R_{mt}*k_{max}=8}$.

The $\mathrm{MoSi_{2}N_{4}}$ monolayer can be considered as the insertion of the 2H $\mathrm{MoS_{2}}$-type $\mathrm{MoN_{2}}$ monolayer into the $\alpha$-InSe-type $\mathrm{Si_{2}N_{2}}$, and the side and top views of the structure of the $\mathrm{MoSi_{2}N_{4}}$ monolayer are plotted in Figure 1. The structure breaks the inversion symmetry, but preserves a horizontal mirror corresponding to the plane of the Mo layer. This leads to that $\mathrm{MoSi_{2}N_{4}}$ monolayer only has in-plane piezoelectric response, and has not out-of-plane piezoelectric polarizations. Using optimized lattice constantsm21 , the energy bands of $\mathrm{MoSi_{2}N_{4}}$ monolayer using GGA and GGA+SOC are shown in Figure 2, and exhibit both the indirect band gaps with valence band maximum (VBM) at $\Gamma$ point and CBM at K point. Due to lacking inversion symmetry and containing the heavy element Mo, there exists a SOC induced spin splitting of about 0.13 eV near the Fermi level in the valence bands at K point. This may provide a platform for spin-valley physicsm20 ; m23 ; m24 , but the VBM is not at K point, which can be tuned by strain. According to orbital projected band structure, it is found that the states near the Fermi level are dominated by the Mo-$d$ orbitals. More specifically, the states around both CBM and VBM are dominated by the Mo $d_{z^{2}}$ orbital.

It is proved that the electronic structures, topological properties, transport and piezoelectric properties of 2D materials can be effectively tuned by strainm12 ; m10 ; m11 ; m13 ; m14 ; m15 ; m16 ; m17 ; m18 ; t9 . The biaxial strain can be simulated by $a/a_{0}$ or $(a-a_{0})/a_{0}$, where $a$ and $a_{0}$ are the strained and unstrained lattice constant, respectively. The $a/a_{0}$$<$1 or $(a-a_{0})/a_{0}$$<$0 means compressive strain, while $a/a_{0}$$>$1 or $(a-a_{0})/a_{0}$$>$0 implies tensile strain. With $a/a_{0}$ from 0.90 to 1.10, the energy band structures are plotted in Figure 2, and the energy band gap and spin-orbit splitting value $\Delta$ at K point are shown in Figure 3.

It is found that the energy band gap firstly increases (0.90 to 0.96), and then decreases (0.96 to 1.10), which is due to transformation of CBM. Similar phenomenon can be observed in many TMD and Janus TMD monolayersm11 ; m11-1 . With strain from compressive one to tensile one, the $\Delta$ has a rapid increase, and then a slight decrease. With increasing compressive strain (1.00 to 0.90), the position of CBM (VBM) changes from K ($\Gamma$) point to one point along the K-$\Gamma$ direction (K point), when the compressive strain reaches about 0.94 (0.96). The compressive strain can also tune the numbers and relative positions of valence band extrema (VBE) or CBE. For example, at 0.96, the four CBE can be observed, and they energies are very close, which has very important effects on transport properties. To explore orbital contribution to the conduction bands in the case of 0.96 strain, we project the states to atomic orbitals at 0.96 strained and unstrained conditions, which are shown in Figure 4. At 0.96 strain, the composition of the low-energy states has little change with respect to unstrained one. At 0.98, the energy of two VBE are nearly the same. The compressive strain can make K point with spin splitting become VBM, which is very useful to allow spin manipulation for spin-valley physics. For example, at 0.94 strain, the VBM at K point is 0.49 eV higher than that at $\Gamma$ point. It is clearly seen that the increasing tensile strain can make valence band around the $\Gamma$ point near the Fermi level more flat.

Finally, the elastic constants $C_{ij}$ are calculated as a function of $a/a_{0}$ to study the mechanical stability of $\mathrm{MoSi_{2}N_{4}}$ monolayer with strain. For for 2D hexagonal crystals, the Born criteria of mechanical stability ela ($C_{11}>0$ and $C_{66}>0$) should be satisfied. The calculated $C_{11}$ and $C_{66}$ as a function of strain are plotted in Figure 5, and it is clearly seen that the $\mathrm{MoSi_{2}N_{4}}$ monolayer in considered strain range is mechanically stable, which is very important for farther experimental exploration.

Proposed by Hicks and Dresselhaus in 1993q2 ; q3 , the potential thermoelectric materials can be achieved in the low-dimensional systems or nanostructures. The dimensionless figure of merit, $ZT=S^{2}\sigma T/(\kappa_{e}+\kappa_{L})$, can be used to measure the efficiency of thermoelectric conversion of a thermoelectric material, where S, $\sigma$, T, $\kappa_{e}$ and $\kappa_{L}$ are the Seebeck coefficient, electrical conductivity, working temperature, electronic and lattice thermal conductivities, respectively. It is noted that, for the 2D material, the calculated $\sigma$, $\kappa_{e}$ and $\kappa_{L}$ depend on $Lz$ (here, $Lz$=40 $\mathrm{{\AA}}$), and the S and $ZT$ is independent of $Lz$. For 2D materials, we use electrons or holes per unit cell instead of doping concentration, which is described by N, and the N $<$($>$) 0 mean n- (p-) type doping. It is proved that the SOC has important effects on transport coefficients of TMD and Janus TMD monolayersm11 ; m13 ; m11-1 . However, the SOC has neglectful influences on transport properties of unstrained $\mathrm{MoSi_{2}N_{4}}$ monolayer, which can be observed from typical Seebeck coefficient S in Figure 6. This is because the energy bands near the Fermi level between GGA and GGA+SOC is nearly the same. However, the SOC has an important effect on p-type transport coefficients with the condition of compressive strain. For example at 0.96 strain, a detrimental effect on Seebeck coefficient S can be observed, when including SOC (See Figure 6). This is because the SOC can remove the band degeneracy near the VBM. So, the SOC is included to investigate the biaxial strain effects on transport coefficients of $\mathrm{MoSi_{2}N_{4}}$ monolayer.

Using GGA+SOC, the room temperature S, $\mathrm{\sigma/\tau}$ and $\mathrm{S^{2}\sigma/\tau}$ of $\mathrm{MoSi_{2}N_{4}}$ monolayer under different strain (0.90 to 1.10) are shown in Figure 7. It is clearly seen that the compressive strain has important effects on S, especially for n-type doping. However, the tensile strain produces small influences on S, especially for n-type S. These can be explained by strain-induced energy bands. When the strain is less than or equal to about 0.98, the n-type S (absolute value) can be observably improved, which is due to compressive strain-driven accidental conduction band degeneracies, namely bands convergence. With expanding compressive strain, in the low doping, the p-type S firstly increases, and has almost no change. This is because the valence bands convergence can be observed at about 0.98, and then is removed (At 0.98, the energy of two VBE are nearly the same, and only one VBE near the Fermi level can be observed with compressive strain from 0.96 to 0.90.). Foe considered tensile strain, the conduction bands near the Fermi level have little change, which leads to almost unchanged n-type S. When the strain changes from 1.00 to 1.10, the p-type S increases, which is due to tensile strain-induced more flat valence bands around $\Gamma$ point near the Fermi level. This can be understood by $S=\frac{8\pi^{2}K_{B}^{2}}{3eh^{2}}m^{*}T(\frac{\pi}{3n})^{2/3}$, in which $m^{*}$, T and $n$ is the effective mass of the carrier, temperature and carrier concentration, respectively. The flat bands can produce very large effective mass of the carrier, which will lead to improved S. It is found that the strain has nearly the opposite effects on $\mathrm{\sigma/\tau}$ with respect to S. It is found that the compressive strain can dramatically improve $\mathrm{S^{2}\sigma/\tau}$ due to the strain-enhanced S.

An upper limit of $ZT$ can be measured by $ZT_{e}=S^{2}\sigma T/\kappa_{e}$, neglecting the $\kappa_{L}$. The room temperature $ZT_{e}$ of $\mathrm{MoSi_{2}N_{4}}$ monolayer under different strain as a function of doping level are also shown in Figure 7. Calculated results show that the dependence of $ZT_{e}$ is very similar to one of S (absolute value), which can be explained by the Wiedemann-Franz law: $\kappa_{e}=L\sigma T$ ($L$ is the Lorenz number). And then the $ZT_{e}$ can be reformulated by $ZT_{e}=S^{2}/L$. Thus, the strain-induced bands convergence improves S, which is beneficial to better $ZT_{e}$.

In summary, we investigate the biaxial strain (0.90 to 1.10) effects on electronic structures and transport coefficients of monolayer $\mathrm{MoSi_{2}N_{4}}$ by the reliable first-principles calculations. With the strain from 0.90 to 1.10, the energy band gap of $\mathrm{MoSi_{2}N_{4}}$ monolayer shows a nonmonotonic behavior. It is found that the SOC has little effects on transport coefficients of unstrained $\mathrm{MoSi_{2}N_{4}}$ in considered doping range due to the hardly changed dispersion of bands near the Fermi level. However, the SOC has very important influences on transport properties of strained $\mathrm{MoSi_{2}N_{4}}$, for example 0.96 strain, which is due to the position change of VBM. Calculated results show that compressive strain can tune the numbers and relative positions of CBE, which can lead to enhanced n-type S, and then better n-type $ZT_{e}$. Our works may provide an idea to optimize the electronic structures and transport properties of monolayer $\mathrm{MoSi_{2}N_{4}}$.

The data that support the findings of this study are available from the corresponding author upon reasonable request.