Coexistence of intrinsic piezoelectricity, ferromagnetism and nontrivial band topology in Li-decorated Janus monolayer $\mathrm{Fe_{2}SSe}$ with high Curie temperature

The quantum Hall (QH) effect can be achieved in a 2D electron gas by a strong perpendicular external magnetic fieldqh1 , while the occurrence of the QAH effect aries from SOC and time-reversal (TR) symmetry broken in the presence of magnetic orderqh2 . The QAH effect is generally confirmed by a nonzero Chern number in accordance with the number of edge states , and only one spin species are allowed to flow unidirectionally, resulting in a quantized Hall conductancezhj1 ; zhj2 ; zhj3 . The discovery of TR invariant topological insulators promotes the experimental realization of QAH effectqh3 ; qh4 . Following the theoretical workqh5 , the QAH effect is achieved experimentally in thin films of Cr doped $\mathrm{(Bi,Sb)_{2}Te}$ below 30 mK, with quantized Hall conductance being observedqh6 . The QAH effect achieves the coexistence of magnetism and topological electronic band structure in a single compound. It’s a natural idea to combine more properties in a material, like the coexistence of QAH effect and piezoelectricity.

The piezoelectricity of 2D materials has been widely investigated in recent years z . The piezoelectricity of 2D materials has been observed experimentallyq5 ; q6 ; q8 ; q8-1 , and the density functional theory (DFT) calculations have also predicted the piezoelectric properties of many 2D materials q7-0 ; q7-1 ; q7-2 ; q7-4 ; q7-7 ; q7-8 ; q7-10 ; q9-0 ; q9-1 ; q9 , which lack centrosymmetry. Recently, some 2D multifunctional piezoelectric materials have been predicted by the first-principle calculations. In the 2D vanadium dichalcogenides, $\mathrm{VSi_{2}P_{4}}$, $\mathrm{CrBr_{1.5}I_{1.5}}$ and $\mathrm{InCrTe_{3}}$qt1 ; q15 ; q15-1 ; q15-2 , the piezoelectric ferromagnetism (PFM) has been predicted, which combines piezoelectricity and ferromagnetism. The combination of piezoelectricity with topological insulating phase, namely piezoelectric quantum spin Hall insulator (PQSHI), has also been realized in monolayer InXO (X=Se and Te)gsd1 and Janus monolayer $\mathrm{SrAlGaSe_{4}}$gsd2 . These discoveries provide possibility for using piezoelectric effect to control the quantum or spin transport process, which may induce novel device applications or scientific breakthroughs.

In fact, in our recent work, the PQAHI has also been achieved in Janus monolayer $\mathrm{Fe_{2}IX}$ (X=Cl and Br)gsd22 , based on QAH insulator $\mathrm{Fe_{2}I_{2}}$ fe . Recently, the lithium decoration of layered iron-based superconductor materials FeX (X=S, Se and Te), namely LiFeX, are predicted as room-temperature QAH insulators with high Chern number (C=2)pr , and the guidances on experimental realization have also been discussed. The LiFeX (X=S, Se and Te) have similar crystal structure with $\mathrm{Fe_{2}I_{2}}$, and no piezoelectricity can be observed due to inversion symmetry. A natural idea is to achieve PQAHI from LiFeX (X=S, Se and Te) monolayer by removing inversion symmetry. As an example, the monolayer FeSe has particular sandwiched structure (Se-Fe-Se), and Janus structure can be built by replacing the top Se atomic layer with S atoms ($\mathrm{Fe_{2}SSe}$), and then apply the gating techniques to inject a large amount of Li ions into monolayer $\mathrm{Fe_{2}SSe}$ ($\mathrm{Li_{2}Fe_{2}SSe}$). The related experimental techniques have been widely usedp1 ; p1-11 ; p1-12 ; p1-13 ; p1-14 .

In this work, it is found that, by first-principles calculations, Janus monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ is a 2D ferromagnetic semiconductor with out-of-plane magnetization, which can achieve the QAH effect at quite high temperature. Janus monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ is proved to be dynamically, mechanically and thermally stable. According to the results of Berry curvature and Chiral edge states, a high Chern number C=2 is obtained for $\mathrm{Li_{2}Fe_{2}SSe}$. It is found that QAH effect of $\mathrm{Li_{2}Fe_{2}SSe}$ is robust against biaxial strain and electronic correlations. A very high Curie temperature of about 1000 K is estimated by MC simulations using Heisenberg model. A particular symmetry leads to only out-of-plane piezoelectric response, and the predicted out-of-plane $d_{31}$ is -0.238 pm/V, which is comparable with ones of other 2D known materials. These results indicate that 2D Janus $\mathrm{Li_{2}Fe_{2}SSe}$ may be promising candidate for realizing the room-temperature PQAHI in experiments, which is very useful for developing 2D piezoelectric spin topological devices.

The rest of the paper is organized as follows. In the next section, we shall give our computational details and methods. In the next few sections, we shall present crystal structure, structural stabilities, topological properties, strain and correlation effects on topological properties, Curie temperature and piezoelectric properties of 2D Janus $\mathrm{Li_{2}Fe_{2}SSe}$. Finally, we shall give our discussion and conclusions.

All the calculations on the elastic, electronic, topological and piezoelectric properties are based on DFT1 , as implemented in the VASP codepv1 ; pv2 ; pv3 . The projector-augmented-wave (PAW) potential and the plane-wave basis with a kinetic energy cutoff of 500 eV are employed for Janus monolayer $\mathrm{Li_{2}Fe_{2}SSe}$. We use popular generalized gradient approximation (GGA) of Perdew, Burke and Ernzerhofpbe as the exchange-correlation functional, and the SOC is included by a second variational procedure on a fully self-consistent basis. The total energy convergence criterion is set for $10^{-8}$ eV with the Gaussian smearing method. The force convergence criterion is set for less than 0.0001 $\mathrm{eV.{\AA}^{-1}}$ for optimizing the lattice constants and atomic coordinates by the conjugate gradient (CG) scheme. To avoid artificial interactions caused by the periodic boundary condition, the vacuum layer is set to more than 15 $\mathrm{{\AA}}$. The Fe-3$d$ orbitals generally have important correlation effects, and the DFT+$U$ methodu is employed for the treatment of the strongly correlated $3d$ electrons with $U_{eff}$ = 2.5 eVfe ; fe1 .

With FM ground state, the interatomic force constants (IFCs) with the 5$\times$5$\times$1 supercell are calculated through the direct supercell method. Based on harmonic IFCs, the phonon dispersions are attained by using Phonopy codepv5 . An effective tight-binding Hamiltonian constructed from the maximally localized Wannier function (MLWF) is used to calculate the surface states and Berry curvature with the iterative Green function method, as implemented in the package WannierToolsw1 ; w2 . The Curie temperature is determined by MC simulations, as implemented by Mcsolver codemc .

The elastic stiffness tensor $C_{ij}$ and piezoelectric stress tensor $e_{ij}$ are carried out by using strain-stress relationship (SSR) with GGA and density functional perturbation theory (DFPT) methodpv6 using GGA+SOC. The Brillouin zone (BZ) integration is sampled by using a 18$\times$18$\times$1 Monkhorst-Pack grid for the self-consistent calculations and elastic coefficients $C_{ij}$. A very dense mesh of 26$\times$26$\times$1 k-points in the BZ is adopted to attain the accurate $e_{ij}$. The 2D elastic coefficients $C^{2D}_{ij}$ and piezoelectric stress coefficients $e^{2D}_{ij}$ have been renormalized by $C^{2D}_{ij}$=$Lz$$C^{3D}_{ij}$ and $e^{2D}_{ij}$=$Lz$$e^{3D}_{ij}$, where the $Lz$ is the length of unit cell along z direction.

The FeSe monolayer has Se-Fe-Se trilayers with a tetragonal lattice in the $P4/nmm$ space group, the unit cell of which contains four atoms with two co-planar Fe atoms. It is proved that the room-temperature QAH insulator can be achieved by Li decoration of FeSe monolayerpr . In fact, the $\mathrm{Fe_{2}I_{2}}$ monolayer, having the same crystal structure with FeSe, is predicted to be a QAH insulatorfe . The I element has one more valence electron than Se element, and the Li atom with an ultralow electronegativity easily loses one valence. So, the Li decoration of FeSe monolayer can become a QAH insulator. It is well known that Janus monolayer MoSSe (S-Mo-Se) can be constructed from $\mathrm{MoSe_{2}}$ with three atomic sublayers (Se-Mo-Se)p1 . In our previous work, the Janus monolayer $\mathrm{Fe_{2}IX}$ (X=Cl and Br) is predicted to be PQAHIgsd22 , which is built by replacing one of two I layers with X (X=Cl and Br) atoms in monolayer $\mathrm{Fe_{2}I_{2}}$. It is a natural idea to construct Janus $\mathrm{Fe_{2}SSe}$ monolayer (S-Fe-Se) by replacing one of two Se layers with S atoms in monolayer $\mathrm{FeSe}$, and then to achieve PQAHI by Li decoration of Janus monolayer $\mathrm{Fe_{2}SSe}$ ($\mathrm{Li_{2}Fe_{2}SSe}$). Compared with FeSe monolayer, the $\mathrm{Li_{2}Fe_{2}SSe}$ monolayer with $P4mm$ space group (No.99) lacks centrosymmetry, giving rise to piezoelectricity. The top and side views of schematic crystal structure of Janus monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ are shown in Figure 1.

The ground state of Janus monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ can be determined by comparing the energy difference between antiferromagnetic (AFM) and FM states. The magnetic ground state of $\mathrm{Li_{2}Fe_{2}SSe}$ monolayer is FM, which can be seen from the (c) in Figure 1. The optimized lattice constants $a_{0}$ is 3.636 $\mathrm{{\AA}}$ with FM order, which is between ones of LiFeS (3.542 $\mathrm{{\AA}}$) and LiFeSe (3.655 $\mathrm{{\AA}}$)pr . The thermal stability of magnetic ordering can be described by the magnetic anisotropy energy (MAE), stemming from the SOC effect. The (100) and (001) directions are used to obtain relative stabilities by using GGA+SOC. The energy difference of the magnetic moments constrained in the (100) and (001) direction is 172 $\mu eV$/Fe, indicating that the out-of-plane (001) direction is the easy one for magnetization in monolayer $\mathrm{Li_{2}Fe_{2}SSe}$. A built-in electric field can be produced by the the inequivalent bond lengths and bond angles (see Table 1) due to different atomic sizes and electronegativities of S and Se atoms. For LiFeSe monolayer, the key space-group symmetry operations contain space inversion $P$, $C_{4}$ rotation, $M_{x}$ and $M_{y}$ mirrors and glide mirror $G_{z}=\{M_{z}|\frac{1}{2},\frac{1}{2},0\}$pr . For $\mathrm{Li_{2}Fe_{2}SSe}$ monolayer, besides the missing $P$, the glide mirror $G_{z}$ is also removed.

The phonon calculation of $\mathrm{Li_{2}Fe_{2}SSe}$ (see (d) in Figure 1) reveals that there are no imaginary frequency modes, confirming its dynamic stability, which means that $\mathrm{Li_{2}Fe_{2}SSe}$ monolayer can exist as free-standing 2D crystal. A total of 18 branches due to 6 atoms per unitcell can be observed, including 15 optical and 3 acoustical phonon branches. The out-of-plane acoustic (ZA) branch (out-of-plane vibrations) deviates from linearity, in accord with the conclusion that the ZA phonon branch should have quadratic dispersion for the unstrained monolayerr1 ; r2 . The thermal stability of $\mathrm{Li_{2}Fe_{2}SSe}$ monolayer is further assessed by performing ab initio molecular dynamics (AIMD) simulations using NVT ensemble with a 4$\times$4$\times$1 supercell at 300 K. For $\mathrm{Li_{2}Fe_{2}SSe}$ monolayer, the temperature and total energy fluctuations as a function of the simulation time are shown in Figure 2. It is found that the total energy and temperature fluctuate smoothly with small amplitudes after the preheating process, which means a favorable thermal stability for $\mathrm{Li_{2}Fe_{2}SSe}$ monolayer at room tempearure.

The mechanical stability of monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ can be checked by Born criteria of mechanical stability:

where the $C_{11}$, $C_{12}$ and $C_{66}$ are three independent elastic constants. By using Voigt notation, the elastic tensor with $4mm$ point-group symmetry can be reduced into:

The calculated $C_{11}$, $C_{12}$ and $C_{66}$ are 91.26 $\mathrm{Nm^{-1}}$, 32.15 $\mathrm{Nm^{-1}}$ and 43.00 $\mathrm{Nm^{-1}}$, which satisfy the above Born criteria of mechanical stability, indicating mechanical stability of monolayer $\mathrm{Li_{2}Fe_{2}SSe}$.

Due to $C_{4}$ rotation symmetry, the mechanical properties of monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ have $C_{4}$ symmetry, and the direction-dependent in-plane Young’s moduli $C_{2D}(\theta)$ and Poisson’s ratios $\nu_{2D}(\theta)$ can be attained by ela ; ela1 :

in which $m=sin(\theta)$, $n=cos(\theta)$ and $B=(C_{11}C_{22}-C_{12}^{2})/C_{66}$. The $\theta$ is the angle of the direction with the x direction as $0^{\circ}$. The $C_{2D}(\theta)$ and $\nu_{2D}(\theta)$ as a function of the angle $\theta$ are plotted in Figure 2. It is clearly seen that they show $C_{4}$ symmetry, and we only consider the $0^{\circ}$-$90^{\circ}$ angle range. The softest direction is along the (100) direction, while the hardest direction is along the (110) direction, and the corresponding Young’s moduli is 79.94 $\mathrm{Nm^{-1}}$ and 101.36 $\mathrm{Nm^{-1}}$. The maximum value of Young’s moduli is less than that of many 2D materialsq7-0 ; q7-4 ; gra , which means that the monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ has extraordinary flexibility. For Poisson’s ratios, the minima is along the (110) direction (0.179), while the maxima is along the (100) direction (0.352).

The energy band structures of monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ with GGA is plotted in Figure 3, along with atom projected band structure in Figure 4. Without SOC, monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ shows a 2D half Dirac semimetal state with a large-gap insulator for spin up (1.41 eV) and a gapless Dirac semimetal for spin down. Due to $C_{4}$ symmetry, the four Dirac cones in the BZ can be observed for spin down along the mirror symmetry invariant lines $\Gamma$-X and $\Gamma$-Y. The important observation is that the states around the Fermi level for spin down are dominated by the Fe-$d$ orbitals, which means that the Fe-$d$ orbitals are partially occupied. However, the spin-up channel is fully occupied. These lead to the high-spin state for Fe atom, giving a spin magnetic moment of 3 $\mu_{B}$. To understand the composition of Dirac cone, we project the states to five Fe-$d$ orbitals for spin down, which are plotted in Figure 5. Calculated results show that the states around Dirac cone are dominated by the $d_{xy}$ and $d_{z^{2}}$ orbital. We then investigate the electronic band structures with SOC, and the energy bands are shown in Figure 3. When including SOC, the Dirac point splits apart with a noticeable energy gap opened, and the corresponding gap is 96.2 meV for $\mathrm{Li_{2}Fe_{2}SSe}$, which suggests nontrivial topology.

When the TR symmetry of a material breaks with a finite magnetic ordering, the topologically nontrivial properties can be verified by a non-zero Chern number in the valence bands. The Chern number of monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ can be attained by integrating the Berry curvature ($\Omega_{z}(k)$) of the occupied bands:

in which the $\mu_{n,k}$ is the lattice periodic part of the Bloch wave functions. The distributions of Berry curvature of monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ are shown in Figure 6. As can be observed, the nonzero Berry curvature is mainly distributed around four Dirac cones, and they have the same sign because of $C_{4}$ symmetry. Two Berry curvature peaks contribute to the nonzero Chern number 1, and the total Chern number from the 4 Berry curvature peaks equals 2. In other words, a quantized Berry phase of $\pi$ can be attained for each gapped Dirac cone, and the total Berry phase of 4$\pi$ is attained (four Dirac cones), giving rise to a Chern number C=2.

Based on the bulk-edge correspondence, the non-zero Chern number can be further confirmed by the number of nontrivial chiral edge states inside the bulk gap of a semiinfinite system. The local density of states vs momentum and energy at the edge can be obtained from the imaginary part of the surface Green’s function:

The topological edge states along the (100) direction are shown in Figure 6. It is clearly seen that the bulk states are connected by two chiral edge states, which indicates that Chern number equals to 2. These results show that monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ is a QAH insulator.

The strain generally can tune the SOC-induced bulk gap, MAE and magnetic order. It is important to investigate robustness of QAH properties of monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ against biaxial strain. Here, we use $a/a_{0}$ (0.96-1.04) to describe the biaxial strain, where $a$ ($a_{0}$) is the strained (equilibrium) lattice constants. The energy difference between FM and AFM orders and MAE as a function of $a/a_{0}$ are plotted in Figure 7. It is found that all strained monolayers are FM ground state in considered strain range, and the energy difference between FM and AFM order decreases with $a/a_{0}$ from 0.96 to 1.04. However, the MAE shows nonmonotonicity, which firstly increases, and then decreases. In considered strain range, the out-of-plane (001) direction is always the easy one. The energy band structures of some representative strained monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ using GGA+SOC are plotted in Figure 8, and the gap as a function of strain is shown in Figure 9. It is found that the gap increases almost linearly with increasing strain in considered strain range, and the gap changes from 49.7 meV to 135.8 meV. The topological edge states of strained monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ are calculated, and they all are QAH insulators with Chern number C=2. We show topological edge states at representative 1.04 strain in Figure 10, which shows clearly two chiral topologically protected gapless edge states. These results show that the QAH topological properties of monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ are robust against strain.

To check the Coulomb interaction $U$ effects on QAH properties, we calculate the energy difference between FM and AFM orders, MAE, energy bands and topological edge states with different values of $U$ (0-4 eV). The energy difference between FM and AFM order and MAE as a function of $U$ are plotted in Figure 7. It is found that the ground state always is FM order with different $U$. It is found that the MAE increases with increasing $U$. When the $U$ value is larger than 0 eV, the out-of-plane (001) direction is always the easy one in considered $U$ range. For $U$=0 eV, the MAE only is -0.5 $\mu eV$/Fe, and the spin orientation in the energy band calculations is chosen in the out-of-plane direction. The energy band structures of monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ with some representative $U$ value using GGA+SOC are plotted in Figure 8, and the gap as a function of $U$ is shown in Figure 9. With increasing $U$, it is clearly seen that the gap increases from -4.1 meV to 257.6 meV. For U=0 eV, the monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ becomes metal. In considered $U$ range except $U$=0 eV, the topological edge states of monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ are calculated, and they all are QAH insulators with Chern number C=2. The topological edge states at representative $U$=4 eV are shown in Figure 10, and two chiral topologically protected gapless edge states are present in the bulk gap. These results indicate the robustness of nontrivial topology against the correlation effect in the 3$d$ electrons of Fe atoms.

One of the important properties of ferromagnets for the practical application is the Curie temperatures ($T_{C}$). Using MC simulations based on the Heisenberg model, we have calculated the Curie temperature of monolayer $\mathrm{Li_{2}Fe_{2}SSe}$. For simplicity, only the nearest neighbor (NN) exchange interaction is considered, and the spin Heisenberg Hamiltonian is defined as:

where $J$, $S$ and $A$ are the exchange parameter, the spin vector of each atom and MAE, respectively. Based on the energy difference between AFM and FM, the magnetic coupling parameter is calculated as $J$=($E_{AFM}$-$E_{FM}$)/8 with normalized $S$ ($|S|$ = 1). The calculated $J$ value is 137.6 meV, and the A is 172 $\mu eV$/Fe.

A 50$\times$50 supercell with periodic boundary conditions is employed, and $10^{7}$ loops are adopted to perfotme the MC simulation. We show the normalized magnetic moment and auto-correlation of monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ as a function of temperature in Figure 11. It is found that $T_{C}$ is as high as 996 K for monolayer $\mathrm{Li_{2}Fe_{2}SSe}$, which is smaller than ones of Li-decorated monolayer FeX (X= S, Se and Te)pr . However, the $T_{C}$ of monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ is significantly higher than that of previously reported many 2D FM semiconductors, like $\mathrm{CrI_{3}}$ monolayer (about 45 K)m7-6 , CrOCl monolayer (about 160 K)tc1 , Janus $\mathrm{Fe_{2}IX}$ (X=Cl and Br) monolayer (about 400 K)gsd22 , $\mathrm{Fe_{2}I_{2}}$ monolayer (about 400 K)fe and $\mathrm{Cr_{2}Ge_{2}Te_{6}}$ monolayer (about 20 K)tc2 .

The piezoelectric effects of a material can be described by third-rank piezoelectric stress tensor $e_{ijk}$ and strain tensor $d_{ijk}$. The $e_{ijk}$ and $d_{ijk}$ can be expressed as:

and

They can be related by elastic tensor $C_{mnjk}$:

in which $P_{i}$, $\varepsilon_{jk}$ and $\sigma_{jk}$ are polarization vector, strain and stress, respectively. The $e_{ijk}^{elc}$/$d_{ijk}^{elc}$ is clamped-ion piezoelectric coefficients (only electronic contributions). The $e_{ijk}$/$d_{ijk}$ is relax-ion piezoelectric coefficients as a realistic result (the sum of ionic and electronic contributions).

The Li-decorated monolayer FeX (X= S, Se and Te) are centrosymmetric, which means that they have no piezoelectricity. However, the monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ lacks glide mirror $G_{z}$ symmetry, but has $M_{x}$ and $M_{y}$ mirrors symmetry, which gives only out-of-plane piezoelectricity, and the in-plane piezoelectricity will disappear. By using Voigt notation, the piezoelectric stress and strain tensors of monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ can be expressed as:

The $e_{31}$ can be directly calculated by VASP code, and the $d_{31}$ can be derived by Equation 11, Equation 12 and Equation 13.

Next, we use the primitive cell to calculate the $e_{31}$ of Janus monolayer $\mathrm{Li_{2}Fe_{2}SSe}$. Calculated results show that the $e_{31}$ is -0.294$\times$$10^{-10}$ C/m with the ionic contribution 0.133$\times$$10^{-10}$ C/m and electronic one -0.427 $\times$$10^{-10}$ C/m. It is found that the electronic and ionic parts have the opposite contributions, and the electronic contribution dominates the $e_{31}$. Based on Equation 14, the calculated $d_{31}$ is -0.238 pm/V. The $d_{31}$ of Janus monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ is higher than or comparable with ones of many 2D materialso1 ; o2 ; o3 ; o4 . The Coulomb interaction $U$ effects on piezoelectric properties of monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ are also considered. The elastic constants $C_{ij}$, the piezoelectric stress coefficients ($e_{31}$) and the piezoelectric strain coefficients ($d_{31}$) as a function of $U$ are plotted in Figure 12. A complex $U$ dependence for $e_{31}$ is observed, which leads to complicated $U$ effects on $d_{31}$. It is found that the smallest $d_{31}$ is -0.100 pm/V with $U$=3 eV.

In fact, Janus monolayer $\mathrm{Li_{2}Fe_{2}SeTe}$ is also a QAH insulator. The energy difference between AFM and FM orders is 0.875 eV per unitcell, and the out-of-plane (001) direction is the easy one with MAE of 45 $\mu eV$/Fe. The calculated $C_{11}$, $C_{12}$ and $C_{66}$ are 79.13 $\mathrm{Nm^{-1}}$, 19.06 $\mathrm{Nm^{-1}}$ and 31.36 $\mathrm{Nm^{-1}}$, which satisfy Born criteria of mechanical stability, indicating mechanical stability of monolayer $\mathrm{Li_{2}Fe_{2}SeTe}$. The phonon spectra of $\mathrm{Li_{2}Fe_{2}SeTe}$ is plotted in Figure 13 with no imaginary frequency modes, confirming its dynamic stability. The AIMD simulations also confirm the thermal stability of monolayer $\mathrm{LiFe_{2}SeTSe}$ at room temperature. The energy band structures and topological edge states of monolayer $\mathrm{Li_{2}Fe_{2}SeTe}$ are shown in Figure 13 and Figure 14, respectively. The energy band gap is 161 meV, and the bulk states are connected by two chiral edge states, indicating Chern number C=2. The calculated $d_{31}$ is 0.1 pm/V with $e_{31}$ of 0.099$\times$$10^{-10}$ C/m.

In summary, using DFT+$U$ calculations, we have performed a systematic investigation of the electronic, magnetic, topological and piezoelectric properties in Janus monolayer $\mathrm{Li_{2}Fe_{2}SSe}$, which is predicted as an intriguing 2D PQAHI. It is proved that monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ is mechanically, dynamically and thermally stable. Also, the nontrivial properties are confirmed by a nonzero Chern number (C=2) and two gapless chiral edge states. Moreover, the monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ possesses out-of-plane magnetic anisotropy and very high Curie temperature (about 1000 K). The emergence of QAH effect in monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ is robust against strain and Hubbard $U$ electronic correlation. The predicted $d_{31}$ is comparable with ones of other 2D known materials. The predicted PQAHI in monolayer $\mathrm{Li_{2}Fe_{2}SSe}$ is expected to work safely above room temperature, and provides a more promising platform for realizing low-dissipation topotronics devices, and provides possibility to use the piezotronic effect to control QAH effects.