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Compared to the ferromagnetic materials that realize the anomalous valley Hall effect by breaking time-reversal symmetry and spin-orbit coupling, the antiferromagnetic materials with the joint spatial inversion and time-reversal (PT) symmetry are rarely reported that achieve the anomalous valley Hall effect. Here, we predict that the antiferromagnetic monolayer MnBr possesses spontaneous valley polarization. The valley splitting of valence band maximum is 21.55 meV at K and K' points, which is originated from Mn-dx2-y2 orbital by analyzing the effective Hamiltonian. Importantly, monolayer MnBr has zero Berry curvature in the entire momentum space but non-zero spin-layer locked Berry curvature, which offers the condition for the anomalous valley Hall effect. In addition, the magnitude of valley splitting can be signally tuned by the onsite correlation, strain, magnetization rotation, electric field, and built-in electric field. The electric field and built-in electric field induce spin splitting due to breaking the P symmetry. Therefore, the spin-layer locked anomalous valley Hall effect can be observed in MnBr. More remarkably, the ferroelectric substrate Sc2CO2 can tune monolayer MnBr to realize the transition from metal to valley polarization semiconductor. Our findings not only extend the implementation of the anomalous valley Hall effect, but also provides a platform for designing low-power and non-volatile valleytronics devices., Comment: 9 pages, 10 figures

Superior to ferromagnetic (FM) valleytronics, antiferromagnetic (AFM) counterpart exhibits ultradense and ultrafast potential due to their intrinsic advantages of zero stray field, terahertz dynamics, and compensated moment of antiferromagnets. However, the physics of spontaneous valley polarization is mainly rooted in FM hexagonal lattices and is rarely used to explore the simultaneous spin and valley polarizations in AFM materials. Here, we propose a general stacking way to achieve valley polarization in AFM bilayer systems. The hexagonal ferrovalley material is used as the basic building unit, and then the space-inversion centrosymmetric bilayer system with interlayer AFM ordering is constructed by horizontal mirror and 2-fold rotational operations, which can exhibit spontaneous valley polarization. In this construction process, the rarely explored \textit{layer-locked hidden valley polarization}, hidden Berry curvature and layer Hall effect are involved, and an out-of-plane electric field can be used to detect hidden valley polarization and to realize layer-locked anomalous valley Hall effect. We use three examples to illustrate our proposal. Firstly, the Janus GdBrI is used to prove concepts and effects involved in our design process. Secondly, the $\mathrm{RuBr_2}$ is used to demonstrate other phenomena, including valley polarization transition and \textit{near-ideal quantum spin Hall insulator}. Finally, we use our design principles to understand the valley polarization of experimentally synthesized MnSe from a new perspective. Our works establish a robust general scheme to achieve valley polarization in AFM bilayer systems, thereby opening up new avenues for AFM valleytronics., Comment: 7 pages, 4 figures

The altermagnetism caused by alternating crystal environment provides a unique opportunity for designing new type of valley polarization. Here, we propose a way to realize valley polarization in two-dimensional (2D) tetragonal altermagnetism by regulating the direction of magnetization. The valley polarization along with spin polarization will arise when the orientation of magnetization breaks the $C_{4z}$ lattice rotational symmetry, particularly in the conventional in-plane $x$ or $y$ directions. When the direction of magnetization switches between the $x$ and $y$ direction, the valley and spin polarizations will be reversed. This is different from the widely studied valley polarization, which occurs in out-of-plane hexagonal magnetic materials with valley physics at -K/K point. Followed by first-principles calculations, our proposal is demonstrated in a 2D Janus tetragonal altermagnetic $\mathrm{Fe_2MoS_2Se_2}$ monolayer with good stability but very small valley splitting of 1.6 meV. To clearly see the feasibility of our proposal, an unrealistic material $\mathrm{Ru_2MoS_2Se_2}$ is used to show large valley splitting of 90 meV. In fact, our proposal can be readily extended to 2D tetragonal ferromagnetic (FM) materials, for example FM $\mathrm{Fe_2I_2}$ monolayer. Our findings can enrich the valley physics, and provide new type of valley materials., Comment: 7 pages, 6 figures

The antiferromagnetic (AFM) valleytronics can be intrinsically more energy-saving and fast-operating in device applications. In general, the lacking spontaneous spin-splitting hinders the implementation and detection of anomalous valley Hall effect (AVHE). Here, we propose to implement AVHE in electric-potential-difference antiferromagnetic $\mathrm{Cr_2CHCl}$ monolayer with excellent stability, where the spontaneous spin-splitting can be induced due to layer-dependent electrostatic potential caused by out-of-plane built-in electric field. From a symmetry perspective, the introduction of Janus structure breaks the combined symmetry ($PT$ symmetry) of spatial inversion ($P$) and time reversal ($T$), which gives rise to spin-splitting. Both unstarined and strained monolayer $\mathrm{Cr_2CHCl}$ possess valley splitting of larger than 51 meV, which is higher than the thermal energy of room temperature (25 meV). The layer-locked Berry curvature gives rise to layer-locked AVHE. Our work reveals a route to achieve AVHE in AFM monolayer with spontaneous spin-splitting., Comment: 6 pages, 6 figures. arXiv admin note: text overlap with arXiv:2404.17596, arXiv:2312.07202, arXiv:2405.18826

The combination of altermagnetism, twistronics and valleytronics is of great significance for potential applications in advanced electronic devices. Twisted magnetic van der Waals bilayers have been identified as an ideal platform for altermagnetism of any type, such as $d$-wave, $g$-wave, and $i$-wave, by choosing the constituent monolayer with specific symmetry [arXiv:2404.17146 (2024)]. Here, we propose a way for achieving valley polarization in twisted altermagnetism by applying out-of-plane external electric field. Since the out-of-plane electric field creates a layer-dependent electrostatic potential, the valleys form different layers will stagger, producing valley polarization. We also demonstrate the effectiveness of our proposed way using the twisted tight-binding model. It is found that the applied electric field can also induce valley/spin-gapless semiconductor and half metal besides valley polarization. Based on first-principles calculations, our proposed way to achieve valley polarization can be verified in twisted bilayer VOBr and monolayer $\mathrm{Ca(CoN)_2}$ as a special twisted altermagnet. These findings provide new opportunities for innovative spintronics, twistronics and valleytronics applications., Comment: 5 pages, 5 figures

Exploring combination of antiferromagnetic (AFM) spintronics and anomalous valley Hall effect (AVHE) is one of the most important questions for valleytronic applications. The key to address this issue is to achieve spin splitting around the valleys in AFM systems. Here, we propose a possible way for achieving AVHE in hexagonal AFM monolayer, which involves the isovalent alloying. This can break the combined symmetry ($PT$ symmetry) of spatial inversion ($P$) and time reversal ($T$), giving rise to spin splitting. More specifically, the large spin splitting around the Fermi energy level owes to $d$ orbital mismatch among these different transition metal ions. Based on first-principles calculations, the proposed way can be verified in out-of-plane AFM $\mathrm{CrMoC_2S_6}$ monolayer, which possesses spontaneous valley polarization and spitting splitting, providing possibility to realize AVHE. It is also proved that tensile strain can strengthen the valley splitting and maintain the out-of-plane AFM ordering. Our works provide an experimentally feasible way for developing AFM valleytronic devices., Comment: 6 pages, 7 figures

Superior to ferromagnetic (FM) materials, antiferromagnetic (AFM) materials do not have any net magnetic moment and are robust to external magnetic perturbation with ultra-high dynamic speed. To achieve spontaneous valley polarization and anomalous valley Hall effect (AVHE) in AFM materials is of great significance for potential applications in spintronics and valleytronics. Here, we predict an A-type AFM monolayer $\mathrm{Fe_2CF_2}$ with large spontaneous valley polarization. Monolayer $\mathrm{Fe_2CF_2}$ has zero Berry curvature in momentum space but non-zero layer-locked hidden Berry curvature in real space, which provides the basic conditions for the realization of AVHE. Because $\mathrm{Fe_2CF_2}$ possesses the combined symmetry ($PT$ symmetry) of spatial inversion ($P$) and time reversal ($T$) symmetry, the spin is degenerate, which prevents the AVHE. An out-of-plane external electric field can be used to produce spin splitting due to the introduction of layer-dependent electrostatic potential, and then layer-locked AVHE can be realized in $\mathrm{Fe_2CF_2}$. Moreover, the spin order of spin splitting can be reversed, when the direction of electric field is reversed. It is proved that the AVHE can be achieved in Janus $\mathrm{Fe_2CFCl}$ without external electric field due to intrinsic built-in electric field. Our works provide an AFM monolayer with excellent properties to realize AVHE.

The combination of antiferromagnetic (AFM) spintronics and anomalous valley Hall effect (AVHE) is of great significance for potential applications in valleytronics. Here, we propose a design principle for achieving AVHE in A-type hexagonal AFM monolayer. The design principle involves the introduction of layer-dependent electrostatic potential caused by out-of-plane external electric field, which can break the combined symmetry ($PT$ symmetry) of spatial inversion ($P$) and time reversal ($T$), producing spin splitting. The spin order of spin splitting can be reversed by regulating the direction of electric field. Based on first-principles calculations, the design principle can be verified in AFM $\mathrm{Cr_2CH_2}$. The layer-locked hidden Berry curvature can give rise to layer-Hall effect, including valley layer-spin Hall effect and layer-locked AVHE. Our works provide an experimentally feasible way to realize AVHE in AFM monolayer., Comment: 5 pages,6 figures

Antiferromagnetic (AFM) spintronics provides a route towards energy-efficient and ultrafast device applications. Achieving anomalous valley Hall effect (AVHE) in AFM monolayers is thus of considerable interest for both fundamental condensed matter physics and device enginering. Here we propose a route to achieve AVHE in A-type AFM insulator composed of vertically-stacked monolayer quantum anomalous Hall insulator (QAHI) with strain and electric field modulations. Uniaxial strain and electric field generate valley polarization and spin splitting, respectively. Using first-principles calculations, $\mathrm{Fe_2BrMgP}$ monolayer is predicted to be a prototype A-type AFM hosting \emph{valley-polarized quantum spin Hall insulator} (VQSHI) in which AVHE and quantum spin Hall effect (QSHE) are synergized in a single system. Our findings reveal a route to achieve multiple Hall effects in 2D tetragonal AFM monolayers., Comment: 6 pages, 6 figures

Recent discovery of ultrathick $\mathrm{MoSi_2N_4(MoN)_n}$ monolayers open up an exciting platform to engineer 2D material properties via intercalation architecture. Here we computationally investigate a series of ultrathick MA$_2$N$_4$(M'N) monolayers (M, M' = Mo, W; A = Si, Ge) under both homolayer and heterolayer intercalation architectures in which the same and different species of transition metal nitride inner core layers are intercalated by outer passivating nitride sublayers, respectively. The MA$_2$N$_4$(M'N) monolayers are thermally, dynamically and mechanically stable with excellent mechanical strength and metallic properties. Intriguingly, the metallic states around Fermi level are localized within the inner core layers. Carrier conduction mediated by electronic states around the Fermi level is thus spatially insulated from the external environment by the native outer nitride sublayers, suggesting the potential of MA$_2$N$_4$(M'N) in back-end-of-line (BEOL) metal interconnect applications. Nitrogen vacancy defect at the outer sublayers creates `punch through' states around the Fermi level that bridges the carrier conduction in the inner core layers and the outer environment, forming a electrical contact akin to the `vias' structures of metal interconnects. We further show that MoSi$_2$N$_4$(MoN) can serve as a quasi-Ohmic contact to 2D WSe$_2$. These findings reveal the promising potential of ultrathick MA$_2$N$_4$(MN) monolayers as metal electrodes and BEOL interconnect applications., Comment: 13 pages, 7 figures, 3 tables

Antiferromagnetic (AFM) materials have potential advantages for spintronics due to their robustness, ultrafast dynamics, and magnetotransport effects. However, the missing spontaneous polarization and magnetization hinders the efficient utilization of electronic spin in these AFM materials. Here, we propose a simple way to produce spin-splitting in AFM materials by making the magnetic atoms with opposite spin polarization locating in the different environment (surrounding atomic arrangement), which does not necessarily require the presence of spin-orbital coupling (SOC). We confirm our proposal by four different types of two-dimensional (2D) AFM materials within the first-principles calculations. Our works provide a intuitional design principle to find or produce spin-splitting in AFM materials., Comment: 6 pages, 9 figures

Two-dimensional (2D) half-metallic materials are highly desirable for nanoscale spintronic applications. Here, we propose a new mechanism that can achieve half-metallicity in 2D ferromagnetic (FM) material with two-layer magnetic atoms by electric field tuning. We use a concrete example of experimentally synthesized CrSBr monolayer to illustrate our proposal through the first-principle calculations. It is found that the half-metal can be achieved in CrSBr within appropriate electric field range, and the corresponding amplitude of electric field intensity is available in experiment. Janus monolayer $\mathrm{Cr_2S_2BrI}$ is constructed, which possesses built-in electric field due to broken horizontal mirror symmetry. However, $\mathrm{Cr_2S_2BrI}$ without and with applied external electric field is always a FM semiconductor. A possible memory device is also proposed based on CrSBr monolayer. Our works will stimulate the application of 2D FM CrSBr in future spintronic nanodevices., Comment: 6 pages, 7 figures

The antiferromagnetic (AFM) materials are robust to external magnetic perturbation due to missing any net magnetic moment. In general, the spin splitting in the band structures disappears in these antiferromagnets. However, the altermagnetism can achieve spin-split bands in collinear symmetry-compensated antiferromagnet with special magnetic space group. Here, we propose a new mechanism that can achieve spin splitting in two-dimensional (2D) Janus A-type AFM materials. Since the built-in electric field caused by Janus structure creates a layer-dependent electrostatic potential, the electronic bands in different layers will stagger, producing the spin splitting, which can be called electric-potential-difference antiferromagnetism (EPD-AFM). We demonstrate that Janus monolayer $\mathrm{Mn_2ClF}$ is a possible candidate to achieve the EPD-AFM by the first-principles calculations. It is proposed that the spin splitting can be tuned in EPD-AFM by piezoelectric effect. Our works provide a new design principle for generating spin polarization in 2D AFM materials., Comment: 6 pages, 6 figures

The altermagnetism can achieve spin-split bands in collinear symmetry-compensated antiferromagnets. Here, we predict altermagnetic order in Janus monolayer $\mathrm{Cr_2SO}$ with eliminated inversion symmetry, which can realize the combination of piezoelectricity and altermagnetism in a two-dimensional material, namely 2D piezoelectric altermagnetism. It is found that $\mathrm{Cr_2SO}$ is an altermagnetic semiconductor, and the spin-split bands of both valence and conduction bands are near the Fermi level. The $\mathrm{Cr_2SO}$ has large out-of-plane piezoelectricity ($|d_{31}|$$=$0.97 pm/V), which is highly desirable for ultrathin piezoelectric device application. Due to spin-valley locking, both spin and valley can be polarized by simply breaking the corresponding crystal symmetry with uniaxial strain. Our findings provide a platform to integrate spin, piezoelectricity and valley in a single material, which is useful for multi-functional device applications., Comment: 6 pages, 5 figures

The persistent spin helix (PSH) is robust against spin-independent scattering and renders an extremely long spin lifetime, which can improve the performance of potential spintronic devices. To achieve the PSH, a unidirectional spin configuration is required in the momentum space. Here, T-XY (X$\neq$Y=P, As, Sb and Bi) monolayers with dynamical, mechanical and thermal stabilities are predicted to intrinsically possess PSH. Due to the $C_{2\upsilon}$ point-group symmetry, a unidirectional spin configuration is preserved in the out-of-plane direction for both conduction and valence bands around the high-symmetry $\Gamma$ point. That is, the expectation value of the spin $S$ only has the out-of-plane component $S_z$. The application of an out-of-plane external electric field can induce in-plane components $S_x$ and $S_y$, thus offering a promising platform for the on-off logical functionality of spin devices. Our work reveals a new family of T-phase two-dimensional (2D) materials, which could provide promising applications in spintronic devices., Comment: 7 pages, 7 figures

Recent experimental synthesis of ambient-stable MoSi2N4 monolayer have garnered enormous research interests. The intercalation morphology of MoSi2N4 - composed of a transition metal nitride (Mo-N) inner sub-monolayer sandwiched by two silicon nitride (Si-N) outer sub-monolayers - have motivated the computational discovery of an expansive family of synthetic MA2Z4 monolayers with no bulk (3D) material counterpart (where M = transition metals or alkaline earth metals; A = Si, Ge; and N = N, P, As). MA2Z4 monolayers exhibit interesting electronic, magnetic, optical, spintronic, valleytronic and topological properties, making them a compelling material platform for next-generation device technologies. Furthermore, heterostructure engineering enormously expands the opportunities of MA2Z4. In this review, we summarize the recent rapid progress in the computational design of MA2Z4-based heterostructures based on first-principle density functional theory (DFT) simulations - a central \emph{work horse} widely used to understand the physics, chemistry and general design rules for specific targeted functions. We systematically classify the MA2Z4-based heterostructures based on their contact types, and review their physical properties, with a focus on their performances in electronics, optoelectronics and energy conversion applications. We review the performance and promises of MA2Z4-based heterostructures for device applications that include electrical contacts, transistors, spintronic devices, photodetectors, solar cells, and photocatalytic water splitting. This review unveils the vast device application potential of MA2Z4-based heterostructures, and paves a roadmap for the future experimental and theoretical development of MA2Z4-based functional heterostructures and devices., Comment: 32 pages, 15 figures

Coexistence of ferromagnetism, piezoelectricity and valley in two-dimensional (2D) materials is crucial to advance multifunctional electronic technologies. Here, Janus ScXY (X$\neq$Y=Cl, Br and I) monolayers are predicted to be in-plane piezoelectric ferromagnetic (FM) semiconductors with dynamical, mechanical and thermal stabilities. The predicted piezoelectric strain coefficients $d_{11}$ and $d_{31}$ (absolute values) are higher than ones of most 2D materials. Moreover, the $d_{31}$ (absolute value) of ScClI reaches up to 1.14 pm/V, which is highly desirable for ultrathin piezoelectric device application. To obtain spontaneous valley polarization, charge doping are explored to tune the direction of magnetization of ScXY. By appropriate hole doping, their easy magnetization axis can change from in-plane to out-of-plane, resulting in spontaneous valley polarization. Taking ScBrI with 0.20 holes per f.u. as a example, under the action of an in-plane electric field, the hole carriers of K valley turn towards one edge of the sample, which will produce anomalous valley Hall effect (AVHE), and the hole carriers of $\Gamma$ valley move in a straight line. These findings could pave the way for designing piezoelectric and valleytronic devices., Comment: 7 pages,7 figures

In this paper, the electrical transport, thermal transport, and thermoelectric properties of three new Janus STe$_{2}$, SeTe$_{2}$, and Se$_{2}$Te monolayers are systematically studied by first-principles calculations, as well as the comparative with available literature's results using different methods. It is found that the Seebeck coefficient and conductivity have opposite dependence on temperature, and we illustrate this phenomenon in detail. The decrease of the thermoelectric power factor (PF) with temperature originates from the decrease in conductivity. To obtain accurate and convergent lattice thermal conductivity, the root mean square (RMS) is calculated to obtain a reasonable cutoff radius for the calculation of third-order forces. Janus STe$_{2}$, SeTe$_{2}$, and Se$_{2}$Te monolayers exhibit ultra-low lattice thermal conductivity of 0.2, 0.133, and 4.81$\times10^{-4}$ W/mK at 300 K, which result from the strong coupling effect between the acoustic mode and the low-frequency optical branch, low phonon group velocity, small phonon lifetime, and large anharmonicity. Consequently, ultra-high $\textit{ZT}$ values of 2.11 (2.09), 3.28 (4.24), and 3.40 (6.51) for n-type(p-type) carrier doping of STe$_{2}$, SeTe$_{2}$, and Se$_{2}$Te are obtained, indicating that they are promising thermoelectric materials., Comment: 7 pages, 7 figures

Valleytronic materials can provide new degrees of freedom to future electronic devices. In this work, the concepts of the ferrovalley metal (FVM) and valley gapless semiconductor (VGS) are proposed, which can be achieved in valleytronic bilayer systems by electric-field tuning, where the interaction between out-of-plane ferroelectricity and A-type antiferromagnetism can induce layer-polarized anomalous valley Hall (LP-AVH) effect. The K and -K valleys of FVM are both metallic, and electron and hole carriers simultaneously exist. In the extreme case, the FVM can become VGS by analogizing spin gapless semiconductor (SGS). Moreover, it is proposed that the valley splitting enhancement and valley polarization reversal can be achieved by electric field in valleytronic bilayer systems. Taking the bilayer $\mathrm{RuBr_2}$ as an example, our proposal is confirmed by the first-principle calculations. The FVM and VGS can be achieved in bilayer $\mathrm{RuBr_2}$ by applying electric field. With appropriate electric field range, increasing electric field can enhance valley splitting, and the valley polarization can be reversed by flipping electric field direction. To effectively tune valley properties by electric field in bilayer systems, the parent monolayer should possess out-of-plane magnetization, and have large valley splitting. Our results shed light on the possible role of electric field in tuning valleytronic bilayer systems, and provide a way to design the ferrovalley-related material by electric field., Comment: 7 pages, 7 figures

Two-dimensional (2D) niobium oxydihalide NbOI$_2$ has been recently demonstrated as an excellent in-plane piezoelectric and nonlinear optical materials. Here we show that Janus niobium oxydihalide, NbO$XY$ (X, Y = Cl, Br, I and X$\neq$Y), is a multifunctional anisotropic semiconductor family with exceptional piezoelectric, electronic, photocatalytic and optical properties. NbO$XY$ are stable and mechancially flexible monolayers with band gap around the visible light regime of $\sim 1.9$ eV. The anisotropic carrier mobility of NbO$XY$ lies in the range of $10^3 \sim 10^4$ cm$^2$V$^{-1}$s$^{-1}$, which represents some of the highest among 2D semiconductors of bandgap $\gtrsim 2$ eV. Inversion symmetry breaking in Janus NbO$XY$ generates sizable out-of-plane $d_{31}$ piezoelectric response while still retaining a strong in-plane piezoelectricity. Remarkably, NbO$XY$ exhibits an additional out-of-plane piezoelectric response, $d_{32}$ as large as 0.55 pm/V. G$_0$W$_0$-BSE calculation further reveals the strong linear optical dichroism of NbO$XY$ in the visible-to-ultraviolet regime. The optical absorption peaks with $14\sim18$ \% in the deep UV regime ($5\sim6$ eV), outperforming the vast majority of other 2D materials. The high carrier mobility, strong optical absorption, sizable built-in electric field and band alignment compatible with overall water splitting further suggest the strengths of NbO$XY$ in energy conversion application. We further propose a directional stress sensing device to demonstrate how the out-of-plane piezoelectricity can be harnessed for functional device applications. Our findings unveil NbO$XY$ as an exceptional multifunctional 2D semiconductor for flexible electronics, optoelectronics, UV photonics, piezoelectric and sustainable energy applications., Comment: 16 Pages, 7 Figures, 3 Tables

One of the key problems in valleytronics is to realize valley polarization. Ferrovalley (FV) semiconductor and half-valley-metal (HVM) have been proposed, which possess intrinsic spontaneous valley polarization. Here, we propose the concept of quasi-half-valley-metal (QHVM), including electron and hole carriers with only a type of carriers being valley polarized. The QHVM may realize separation function of electron and hole. A concrete example of $\mathrm{VGe_2P_4}$ monolayer is used to illustrate our proposal through the first-principle calculations. To better realize QHVM, the electric field is applied to tune related valley properties of $\mathrm{VGe_2P_4}$. Within considered electric field range, $\mathrm{VGe_2P_4}$ is always ferromagnetic (FM) ground state, which possesses out-of-plane magnetization by calculating magnetic anisotropy energy (MAE) including magnetic shape anisotropy (MSA) and magnetocrystalline anisotropy (MCA) energies. These out-of-plane FM properties guarantee intrinsic spontaneous valley polarization in $\mathrm{VGe_2P_4}$. Within a certain range of electric field, the QHVM can be maintained, and the related polarization properties can be effectively tuned. Our works pave the way toward two-dimensional (2D) functional materials design of valleytronics., Comment: 8 pages, 10 figures. arXiv admin note: text overlap with arXiv:2208.02425

The manipulation of the Rashba spin-splitting is crucial for the development of nanospintronic technology. Here, it is proposed that the Rashba spin-splitting can be turned on and off by reversing electric-field direction. By the first-principle calculations, our proposal is illustrated by a concrete example of Janus monolayer RbKNaBi. The designed RbKNaBi possesses dynamical, thermal and mechanical stability, and is a large-gap quantum spin Hall insulator (QSHI) with Rashba spin-splitting near the Fermi level. A small built-in electric field is predicted due to very small electronegativity difference between the bottom and top atoms, which is very key to switch Rashba spin-splitting through the experimentally available electric field intensity. Due to out-of-plane structural asymmetry, the Janus monolayer has distinctive behaviors by applying external electric field $E$ with the same magnitude but different directions ($z$ or $-z$). Our results reveal that the Rashba energy ($E_R$) and Rashba constant ($\alpha_R$) are increased by the positive $E$, while a negative $E$ suppresses the Rashba splitting to disappear, and then appears again. In a certain $E$ region (0.15 $\mathrm{V/\AA}$ to 0.25 $\mathrm{V/\AA}$), switching Rashba spin-splitting can be achieved by only reversing electric-field direction. Besides, the piezoelectric strain coefficients $d_{11}$ and $d_{31}$ (5.52 pm/V and -0.41 pm/V) are predicted, which are higher than or compared with those of many 2D materials. By piezoelectric effect, the strain can also be used to tune Rashba spin-splitting of RbKNaBi. Moreover, a possible spintronic device is proposed to realize the function of spintronic switch., Comment: 9 pages,10 figures

Strong structural asymmetry is actively explored in two-dimensional (2D) materials, because it can give rise to many interesting physical properties. Motivated by the recent synthesis of monolayer $\mathrm{Si_2Te_2}$, we explore a family of 2D materials, termed as the Janus Si dichalcogenides (JSD), which parallel the Janus transition metal dichalcogenides and exhibit even stronger inversion asymmetry. Using first-principles calculations, we demonstrate excellent stability of these materials. We show that their strong structural asymmetry leads to pronounced intrinsic polar field, sizable spin splitting due to spin-orbit coupling, and large piezoelectric response. The spin splitting involves an out-of-plane component, which is beyond the linear Rashba model. The piezoelectric tensor has large value in both in-plane $d_{11}$ coefficient and out-of-plane $d_{31}$ coefficient, making the monolayer JSDs distinct among the existing 2D piezoelectrics. In addition, we find interesting strain-induced phase transitions in these materials. Particularly, there are multiple valleys in the conduction band that compete for the conduction band minimum, which will lead to notable changes in optical and transport properties under strain. Our work reveals a new family of Si based 2D materials, which could find promising applications in spintronic and piezoelectric devices., Comment: 8 pages, 8 figures

Valleytronics has been widely investigated for providing new degrees of freedom to future information coding and processing. Here, it is proposed that valley polarization can be achieved by electric field induced magnetic anisotropy (MA) transformation. Through the first-principle calculations, our idea is illustrated by a concrete example of $\mathrm{VSi_2P_4}$ monolayer. The increasing electric field can induce a transition of MA from in-plane to out-of-plane by changing magnetic anisotropy energy (MAE) from negative to positive value, which is mainly due to increasing magnetocrystalline anisotropy (MCA) energy. The out-of-plane magnetization is in favour of spontaneous valley polarization in $\mathrm{VSi_2P_4}$. Within considered electric field range, $\mathrm{VSi_2P_4}$ is always ferromagnetic (FM) ground state. In a certain range of electric field, the coexistence of semiconductor and out-of-plane magnetization makes $\mathrm{VSi_2P_4}$ become a true ferrovalley (FV) material. The anomalous valley Hall effect (AVHE) can be observed under in-plane and out-of-plane electrical field in $\mathrm{VSi_2P_4}$. Our works pave the way to design the ferrovalley material by electric field., Comment: 6 pages, 6 figures. arXiv admin note: text overlap with arXiv:2207.13420

Two-dimensional (2D) ferromagnets have been a fascinating subject of research, and magnetic anisotropy (MA) is indispensable for stabilizing the 2D magnetic order. Here, we investigate magnetic anisotropy energy (MAE), magnetic and electronic properties of $\mathrm{VSi_2P_4}$ by using the generalized gradient approximation plus $U$ (GGA+$U$) approach. For large $U$, the magnetic shape anisotropy (MSA) energy has a more pronounced contribution to the MAE, which can overcome the magnetocrystalline anisotropy (MCA) energy to evince an easy-plane. For fixed out-of-plane MA, monolayer $\mathrm{VSi_2P_4}$ undergoes ferrovalley (FV), half-valley-metal (HVM), valley-polarized quantum anomalous Hall insulator (VQAHI), HVM and FV states with increasing $U$. However, for assumptive in-plane MA, there is no special quantum anomalous Hall (QAH) state and spontaneous valley polarization within considered $U$ range. According to the MAE and electronic structure with fixed out-of-plane or in-plane MA, the intrinsic phase diagram shows common magnetic semiconductor (CMS), FV and VQAHI in monolayer $\mathrm{VSi_2P_4}$. At representative $U$$=$3 eV widely used in references, $\mathrm{VSi_2P_4}$ can be regarded as a 2D-$XY$ magnet, not Ising-like 2D long-range order magnets predicted in previous works with only considering MCA energy. Our findings shed light on importance of MSA in determining magnetic and electronic properties of monolayer $\mathrm{VSi_2P_4}$., Comment: 7 pages,7 figures. arXiv admin note: text overlap with arXiv:2205.07012

For two-dimensional (2D) material, piezoelectric ferromagnetism (PFM) with large out-of-plane piezoresponse is highly desirable for multifunctional ultrathin piezoelectric device application. Here, we predict that Janus monolayer CrSCl is an out-of-plane ferromagnetic (FM) semiconductor with large vertical piezoelectric response and high Curie temperature. The predicted out-of-plane piezoelectric strain coefficient $d_{31}$ is -1.58 pm/V, which is higher than ones of most 2D materials (compare absolute values of $d_{31}$). The large out-of-plane piezoelectricity is robust against electronic correlation and biaxial strain, confirming reliability of large $d_{31}$. Calculated results show that tensile strain is conducive to high Curie temperature, large magnetic anisotropy energy (MAE) and large $d_{31}$. Finally, by comparing $d_{31}$ of CrYX (Y=S; X=Cl, Br I) and CrYX (Y=O; X=F, Cl, Br), we conclude that the size of $d_{31}$ is positively related to electronegativity difference of X and Y atoms. Such findings can provide valuable guidelines for designing 2D piezoelectric materials with large vertical piezoelectric response., Comment: 6 pages, 5 figures

Two-dimensional (2D) materials van der Waals heterostructures (vdWHs) provides a revolutionary route towards high-performance solar energy conversion devices beyond the conventional silicon-based pn junction solar cells. Despite tremendous research progress accomplished in recent years, the searches of vdWHs with exceptional excitonic solar cell conversion efficiency and optical properties remain an open theoretical and experimental quest. Here we show that the vdWH family composed of MoSi$_2$N$_4$ and WSi$_2$N$_4$ monolayers provides a compelling material platform for developing high-performance ultrathin excitonic solar cells and photonics devices. Using first-principle calculations, we construct and classify 51 types of MoSi$_2$N$_4$ and WSi$_2$N$_4$-based [(Mo,W)Si$_2$N$_4$] vdWHs composed of various metallic, semimetallic, semiconducting, insulating and topological 2D materials. Intriguingly, MoSi$_2$N$_4$/(InSe, WSe$_2$) are identified as Type-II vdWHs with exceptional excitonic solar cell power conversion efficiency reaching well over 20%, which are competitive to state-of-art silicon solar cells. The (Mo,W)Si$_2$N$_4$ vdWH family exhibits strong optical absorption in both the visible and ultraviolet regimes. Exceedingly large peak ultraviolet absorptions over 40%, approaching the maximum absorption limit of a free-standing 2D material, can be achieved in (Mo,W)Si$_2$N$_4$/$\alpha_2$-(Mo,W)Ge$_2$P$_4$ vdWHs. Our findings unravel the enormous potential of (Mo,W)Si$_2$N$_4$ vdWHs in designing ultimately compact excitonic solar cell device technology., Comment: 30 pages, 7 figures

The combination of piezoelectricity with nontrivial topological insulating phase in two-dimensional (2D) systems, namely piezoelectric quantum spin Hall insulator (PQSHI), is intriguing for exploring novel topological states toward the development of high-speed and dissipationless electronic devices. In this work, we predict a PQSHI Janus monolayer VCClBr constructed from $\mathrm{VCCl_2}$, which is dynamically, mechanically and thermally stable. In the absence of spin orbital coupling (SOC), VCClBr is a narrow gap semiconductor with gap value of 57 meV, which is different from Dirac semimetal $\mathrm{VCCl_2}$. The gap of VCClBr is due to built-in electric field caused by asymmetrical upper and lower atomic layers, which is further confirmed by external-electric-field induced gap in $\mathrm{VCCl_2}$. When including SOC, the gap of VCClBr is improved to 76 meV, which is larger than the thermal energy of room temperature (25 meV). The VCClBr is a 2D topological insulator (TI), which is confirmed by $Z_2$ topological invariant and nontrivial one-dimensional edge states. It is proved that the nontrivial topological properties of VCClBr are robust against strain (biaxial and uniaxial cases) and external electric field. Due to broken horizontal mirror symmetry, only out-of-plane piezoelectric response can be observed, when biaxial or uniaxial in-plane strain is applied. The predicted piezoelectric strain coefficients $d_{31}$ and $d_{32}$ are -0.425 pm/V and -0.219 pm/V, which are higher than or compared with ones of many 2D materials. Finally, another two Janus monolayer VCFBr and VCFCl (dynamically unstable) are constructed, and they are still PQSHIs. Moreover, their $d_{31}$ and $d_{32}$ are higher than ones of VCClBr, and the $d_{31}$ (absolute value) of VCFBr is larger than one., Comment: 10 pages, 12 figures

Strain is an effective method to tune the electronic properties of two-dimension (2D) materials, and can induce novel phase transition. Recently, 2D $\mathrm{MA_2Z_4}$ family materials are of interest because of their emerging topological, magnetic and superconducting properties. Here, we investigate the impact of strain effects ($a/a_0$:0.96$\sim$1.04) on the physical properties of Janus monolayer $\mathrm{VSiGeN_4}$ as a derivative of $\mathrm{VSi_2N_4}$ or $\mathrm{VGe_2N_4}$, which possesses dynamical, mechanical and thermal stabilities. For out-of-plane magnetic anisotropy, with increasing strain, $\mathrm{VSiGeN_4}$ undergoes transition between ferrovalley semiconductor (FVS), half-valley-metal (HVM), valley-polarized quantum anomalous Hall insulator (VQAHI), HVM and FVS. These imply twice topological phase transitions, which are related with sign-reversible Berry curvature and band inversion between $d_{xy}$+$d_{x^2-y^2}$ and $d_{z^2}$ orbitals for K or -K valley. The band inversion also leads to transformation of valley splitting strength between valence and conduction bands. However, for in-plane magnetic anisotropy, no special quantum anomalous Hall (QAH) states and valley polarization exist within the considered strain range. The actual magnetic anisotropy energy (MAE) shows no special QAH and HVM states in monolayer $\mathrm{VSiGeN_4}$. Fortunately, these can be easily achieved by external magnetic field, which adjusts the easy magnetization axis of $\mathrm{VSiGeN_4}$ from in-plane one to out-of-plane one. Our findings shed light on how strain can be employed to engineer the electronic states of $\mathrm{VSiGeN_4}$, which may open new perspectives for multifunctional quantum devices in valleytronics and spintronics., Comment: 9 pages, 9 figures

The combination of piezoelectricity and ferromagnetic (FM) order in a two-dimensional (2D) material, namely 2D piezoelectric ferromagnetism (PFM), may open up unprecedented opportunities for novel device applications. Here, we predict an in-plane FM semiconductor Janus monolayer NiClI with considerably large magnetic anisotropy energy (MAE) of 1.439 meV, exhibiting dynamic, mechanical and thermal stabilities. The NiClI monolayer possesses larger in-plane piezoelectricity ($d_{11}$$=$5.21 pm/V) comparable to that of $\mathrm{MoS_2}$. Furthermore, NiClI has huge out-of-plane piezoelectricity ($d_{31}$$=$1.89 pm/V), which is highly desirable for ultrathin piezoelectric device application. It is proved that huge out-of-plane piezoelectricity is robust against electronic correlation, which confirms reliability of huge $d_{31}$. Finally, being analogous to NiClI, PFM with large out-of-plane piezoelectricity can also be achieved in the Janus monolayers of NiClBr and NiBrI, with the predicted $d_{31}$ of 0.73 pm/V and 1.15 pm/V, respectively. The predicted huge out-of-plane piezoelectric response makes Janus monolayer NiClI a good platform for multifunctional semiconductor spintronic applications, which is also compatible with the bottom/top gate technologies of conventional semiconductor nanoelectronic devices., Comment: 6 pages, 4 figures

Coexistence of intrinsic ferromagnetism and piezoelectricity, namely piezoelectric ferromagnetism (PFM), is crucial to advance multifunctional spintronic technologies. In this work, we demonstrate that Janus monolayer YBrI is a PFM, which is dynamically, mechanically and thermally stable. Electronic correlation effects on physical properties of YBrI are investigated by using generalized gradient approximation plus $U$ (GGA+$U$) approach. For out-of-plane magnetic anisotropy, YBrI is a ferrovalley (FV) material, and the valley splitting is larger than 82 meV in considered $U$ range. The anomalous valley Hall effect (AVHE) can be achieved under an in-plane electric field. However, for in-plane magnetic anisotropy, YBrI is a common ferromagnetic (FM) semiconductor. When considering intrinsic magnetic anisotropy, the easy axis of YBrI is always in-plane with magnetic anisotropy energy (MAE) from 0.309 meV to 0.237 meV ($U$=0.0 eV to 3.0 eV). However, the magnetization can be adjusted from the in-plane to off-plane direction by external magnetic field, and then lead to the occurrence of valley polarization. Moreover, missing centrosymmetry along with mirror symmetry breaking results in both in-plane and out-of-plane piezoelectricity in YBrI monolayer. At a typical $U$=2.0 eV, the $d_{11}$ is predicted to be -5.61 pm/V, which is higher than or compared with ones of other two-dimensional (2D) known materials. The electronic and piezoelectric properties of YBrI can be effectively tuned by applying a biaxial strain. For example, tensile strain can enhance valley splitting and $d_{11}$ (absolute value). The predicted Curie temperature of YBrI is higher than those of experimentally synthesized 2D ferromagnetic materials $\mathrm{CrI_3}$ and $\mathrm{Cr_2Ge_2Te_6}$., Comment: 10 pages, 12 figures

Spin-orbit coupling (SOC) combined with electronic correlation can induce topological phase transition, producing novel electronic states. Here, we investigate the impact of SOC combined with correlation effects on physical properties of monolayer $\mathrm{OsBr_2}$, based on first-principles calculations with generalized gradient approximation plus $U$ (GGA+$U$) approach. With intrinsic out-of-plane magnetic anisotropy, $\mathrm{OsBr_2}$ undergoes threefold topological phase transition with increasing $U$, and valley-polarized quantum anomalous Hall insulator (VQAHI) to half-valley-metal (HVM) to ferrovalley insulator (FVI) to HVM to VQAHI to HVM to FVI transitions can be induced. These topological phase transitions are connected with sign-reversible Berry curvature and band inversion between $d_{xy}$/$d_{x^2-y^2}$ and $d_{z^2}$ orbitals. Due to $\bar{6}m2$ symmetry, piezoelectric polarization of $\mathrm{OsBr_2}$ is confined along the in-plane armchair direction, and only one $d_{11}$ is independent. For a given material, the correlation strength should be fixed, and $\mathrm{OsBr_2}$ may be a piezoelectric VQAHI (PVQAHI), piezoelectric HVM (PHVM) or piezoelectric FVI (PFVI). The valley polarization can be flipped by reversing the magnetization of Os atoms, and the ferrovalley (FV) and nontrivial topological properties will be suppressed by manipulating out-of-plane magnetization to in-plane one. In considered reasonable $U$ range, the estimated Curie temperatures all are higher than room temperature. Our findings provide a comprehensive understanding on possible electronic states of $\mathrm{OsBr_2}$, and confirm that strong SOC combined with electronic correlation can induce multiple quantum phase transition., Comment: 9 pages, 12 figures

Nontrivial band topology combined with magnetic ordering can produce quantum anomalous Hall insulator (QAHI), which may lead to advances in device concepts. Here, through first-principles calculations, stable monolayer $\mathrm{Fe_2Br_2}$ is predicted as a room-temperature large-gap high-Chern-number QAHI by using generalized gradient approximation plus $U$ (GGA+$U$) approach. The large gap is due to correlation-enhanced spin-orbit coupling (SOC) effect of Fe atoms, which equates with artificially increasing the strength of SOC without electronic correlation. Out-of-plane magnetic anisotropy is very key to produce quantum anomalous Hall (QAH) state because in-plane magneitization will destroy nontrivial band topology. In the absence of SOC, $\mathrm{Fe_2Br_2}$ is a half Dirac semimetal state protected by mirror symmetry, and the electronic correlation along with SOC effect creates QAH state with a sizable gap and two chiral edge modes. It is found that the QAH state is robust against biaxial strain ($a/a_0$: 0.96 to 1.04) in monolayer $\mathrm{Fe_2Br_2}$ with stable ferromagnetic (FM) ordering and out-of-plane magnetic anisotropy. Calculated results show that Curie temperature is sensitive to correlation strength and strain. The reduced correlation and compressive strain are in favour of high Curie temperature. These analysis and results can be readily extended to other monolayer $\mathrm{Fe_2XY}$ (X/Y=Cl, Br and I), which possesses the same Fe-dominated low-energy states with a $\mathrm{Fe_2Br_2}$ monolayer. These findings open new opportunities to design new high-temperature topological quantum devices., Comment: 9 pages, 11 figures

Coexistence of intrinsic ferrovalley (FV) and nontrivial band topology attracts intensive interest both for its fundamental physics and for its potential applications, namely valley-polarized quantum anomalous Hall insulator (VQAHI). Here, based on first-principles calculations by using generalized gradient approximation plus $U$ (GGA+$U$) approach, the VQAHI induced by electronic correlation or strain can occur in monolayer $\mathrm{RuBr_2}$. For perpendicular magnetic anisotropy (PMA), the ferrovalley (FV) to half-valley-metal (HVM) to quantum anomalous Hall (QAH) to HVM to FV transitions can be driven by increasing electron correlation $U$. However, there are no special QAH states and valley polarization for in-plane magnetic anisotropy. By calculating actual magnetic anisotropy energy (MAE), the VQAHI indeed can exist between two HVM states due to PMA, a unit Chern number/a chiral edge state and spontaneous valley polarization. The increasing $U$ can induce VQAHI, which can be explained by sign-reversible Berry curvature or band inversion between $d_{xy}$/$d_{x^2-y^2}$ and $d_{z^2}$ orbitals. Even though the real $U$ falls outside the range, the VQAHI can be achieved by strain. Taking $U$$=$2.25 eV as a concrete case, the monolayer $\mathrm{RuBr_2}$ can change from a common ferromagentic (FM) semiconductor to VQAHI under about 0.985 compressive strain. It is noted that the edge states of VQAHI are chiral-spin-valley locking, which can achieve complete spin and valley polarizations for low-dissipation electronics devices. Both energy band gap and valley splitting of VQAHI in monolayer $\mathrm{RuBr_2}$ are higher than the thermal energy of room temperature (25 meV), which is key at room temperature for device applications., Comment: 11 pages,14 figures

The electronic correlation may have essential influence on electronic structures in some materials with special structure and localized orbital distribution. In this work, taking Janus monolayer FeClF as a concrete example, the correlation effects on its electronic structures are investigated by using generalized gradient approximation plus $U$ (GGA+$U$) approach. For perpendicular magnetic anisotropy (PMA), the increasing electron correlation effect can induce the ferrovalley (FV) to half-valley-metal (HVM) to quantum anomalous Hall (QAH) to HVM to FV transitions. For QAH state, there are a unit Chern number and a chiral edge state connecting the conduction and valence bands. The HVM state is at the boundary of the QAH phase, whose carriers are intrinsically 100\% valley polarized. With the in-plane magnetic anisotropy, no special QAH states and prominent valley polarization are observed. However, for both out-of-plane and in-plane magnetic anisotropy, sign-reversible Berry curvature can be observed with increasing $U$. It is found that these phenomenons are related with the change of $d_{xy}$/$d_{x^2-y^2}$ and $d_{z^2}$ orbital distributions and different magnetocrystalline directions. It is also found that the magnetic anisotropy energy (MAE) and Curie temperature strongly depend on the $U$. With PMA, taking typical $U=$2.5 eV, the electron valley polarization can be observed with valley splitting of 109 meV, which can be switched by reversing the magnetization direction. The analysis and results can be readily extended to other nine members of monolayer FeXY (X/Y=F, Cl, Br and I) due to sharing the same Fe-dominated low-energy states and electronic correlations with FeClF monolayer., Comment: 10 pages, 12 figures

The valley degrees of freedom of carriers in crystals is useful to process information and perform logic operations, and it is a key factor for valley application to realize the valley polarization. Here, we propose a model that the valley polarization transition at different valley points (-K and K points) is produced by biaxial strain. By the first-principle calculations, we illustrate our idea with a concrete example of Janus $\mathrm{GdClF}$ monolayer. The predicted $\mathrm{GdClF}$ monolayer is dynamically, mechanically and thermally stable, and is a ferromagnetic (FM) semiconductor with perpendicular magnetic anisotropy (PMA), valence band maximum (VBM) at valley points and high Curie temperature ($T_C$). Due to its intrinsic ferromagnetism and spin orbital coupling (SOC), a spontaneous valley polarization will be induced, but the valley splitting is only -3.1 meV, which provides an opportunity to achieve valley polarization transition at different valley points by strain. In considered strain range ($a/a_0$: 0.94$\sim$1.06), the strained GdClF monolayer has always energy bandgap, strong FM coupling and PMA. The compressive strain is in favour of -K valley polarization, while the tensile strain makes for K valley polarization. The corresponding valley splitting at 0.96 and 1.04 strain are -44.5 meV and 29.4 meV, which are higher than the thermal energy of room temperature (25 meV). Due to special Janus structure, both in-plane and out-of-plane piezoelectric polarizations can be observed. It is found that the direction of in-plane piezoelectric polarizations can be overturned by strain, and the $d_{11}$ at 0.96 and 1.04 strain are -1.37 pm/V and 2.05 pm/V. Our works pave the way to design the ferrovalley material as multifunctional valleytronics and piezoelectric devices by strain., Comment: 9 pages, 10 figures. arXiv admin note: text overlap with arXiv:2109.13534

Achieving combination of spin and valley polarized states with topological insulating phase is pregnant to promote the fantastic integration of topological physics, spintronics and valleytronics. In this work, a spin-valley-coupled quantum spin Hall insulator (svc-QSHI) is predicted in Janus monolayer $\mathrm{CSb_{1.5}Bi_{1.5}}$ with dynamic, mechanical and thermal stabilities. The inequivalent valleys have opposite Berry curvature and spin moment, which can produce a spin-valley Hall effect. In the center of Brillouin zone, a Rashba-type spin splitting can be observed due to missing horizontal mirror symmetry. Moreover, monolayer $\mathrm{CSb_{1.5}Bi_{1.5}}$ shows unique Rashba-splitting edge states. Both energy band gap and spin-splitting at the valley point are larger than the thermal energy of room temperature (25 meV) with generalized gradient approximation (GGA) level, which is very important at room temperature for device applications. It is proved that the spin-valley-coupling and nontrivial quantum spin Hall (QSH) state are robust again biaxial strain. Our work may provide a new platform to achieve integration of topological physics, spintronics and valleytronics., Comment: 6 pages, 7 figures

Ferrovalley materials can achieve manipulation of the valley degree of freedom with intrinsic spontaneous valley polarization introduced by their intrinsic ferromagnetism. A good ferrovalley material should possess perpendicular magnetic anisotropy (PMA), valence band maximum (VBM)/conduction band minimum (CBM) at valley points, strong ferromagnetic (FM) coupling and proper valley splitting. In this work, the monolayer $\mathrm{GdCl_2}$ is proposed as a potential candidate material for valleytronic applications by the first-principles calculations. It is proved that monolayer $\mathrm{GdCl_2}$ is a FM semiconductor with the easy axis along out of plane direction and strong FM coupling. A spontaneous valley polarization with a valley splitting of 42.3 meV is produced due to its intrinsic ferromagnetism and spin orbital coupling (SOC). Although the VBM of unstrained monolayer $\mathrm{GdCl_2}$ is away from valley points, a very small compressive strain (about 1\%) can make VBM move to valley points. We propose a possible way to realize anomalous valley Hall effect in monolayer $\mathrm{GdCl_2}$ by piezoelectric effect, not an external electric field, namely piezoelectric anomalous valley Hall effect (PAVHE). This phenomenon could be classified as piezo-valleytronics, being similar to piezotronics and piezophototronics. The only independent piezoelectric strain coefficient $d_{11}$ is -2.708 pm/V, which is comparable to one of classical bulk piezoelectric material $\alpha$-quartz ($d_{11}$=2.3 pm/V). The biaxial in-plane strain and electronic correlation effects are considered to confirm the reliability of our results. Finally, the monolayer $\mathrm{GdF_2}$ is predicted to be a ferrovalley material with dynamic and mechanical stabilities, PMA, VBM at valley points, strong FM coupling, valley splitting of 47.6 meV, and $d_{11}$ of 0.584 pm/V., Comment: 10 pages, 12 figures

Recently, the quantum anomalous Hall (QAH) insulators are predicted in Lithium-decorated iron-based superconductor monolayer materials (LiFeX (X=S, Se and Te)) with very high Curie temperature (\textcolor[rgb]{0.00,0.00,1.00}{PRL 125, 086401 (2020)}), which combines the topological and ferromagnetic (FM) orders. It is interesting and useful to achieve coexistence of intrinsic piezoelectricity, ferromagnetism and nontrivial band topology in single two-dimensional (2D) material, namely 2D piezoelectric quantum anomalous hall insulator (PQAHI). In this work, 2D Janus monolayer $\mathrm{Li_2Fe_2SSe}$ is predict to be a room-temperature PQAHI, which possesses dynamic, mechanical and thermal stabilities. It is predicted to be a half Dirac semimetal without spin-orbit coupling (SOC). It is found that the inclusion of SOC opens up a large nontrivial gap, which means the nontrivial bulk topology (QAH insulator), confirmed by the calculation of Berry curvature and the presence of two chiral edge states (Chern number C=2). Calculated results show that monolayer $\mathrm{Li_2Fe_2SSe}$ possesses robust QAH states against biaxial strain and electronic correlations. Compared to LiFeX, the glide mirror $G_z$ of $\mathrm{Li_2Fe_2SSe}$ disappears, which will induce only out-of-plane piezoelectric response. The calculated out-of-plane $d_{31}$ of monolayer $\mathrm{Li_2Fe_2SSe}$ is -0.238 pm/V comparable with ones of other 2D known materials. Moreover, very high Curie temperature (about 1000 K) is predicted by using Monte Carlo (MC) simulations, which means that the QAH effect can be achieved at room temperature in Janus monolayer $\mathrm{Li_2Fe_2SSe}$. Similar to monolayer $\mathrm{Li_2Fe_2SSe}$, the PQAHI can also be realized in the Janus monolayer $\mathrm{Li_2Fe_2SeTe}$., Comment: 10 pages, 14 figures. arXiv admin note: text overlap with arXiv:2105.03004

A two-dimensional (2D) material with piezoelectricity, topological and ferromagnetic (FM) orders, namely 2D piezoelectric quantum anomalous hall insulator (PQAHI), may open new opportunities to realize novel physics and applications. Here, by first-principles calculations, a family of 2D Janus monolayer $\mathrm{Fe_2IX}$ (X=Cl and Br) with dynamic, mechanical and thermal stabilities is predict to be room-temperature PQAHI. At the absence of spin-orbit coupling (SOC), monolayer $\mathrm{Fe_2IX}$ (X=Cl and Br) is a half Dirac semimetal state. When the SOC is included, these monolayers become quantum anomalous hall (QAH) states with sizable gaps (more than two hundred meV) and two chiral edge modes (Chern number C=2). It is also found that monolayer $\mathrm{Fe_2IX}$ (X=Cl and Br) possesses robust QAH states against biaxial strain. By symmetry analysis, it is found that only out-of-plane piezoelectric response can be induced by a uniaxial strain in the basal plane. The calculated out-of-plane $d_{31}$ of $\mathrm{Fe_2ICl}$ ($\mathrm{Fe_2IBr}$) is 0.467 pm/V (0.384 pm/V), which is higher than or comparable with ones of other 2D known materials. Meanwhile, using Monte Carlo (MC) simulations, the Curie temperature $T_C$ is estimated to be 429/403 K for monolayer $\mathrm{Fe_2ICl}$/$\mathrm{Fe_2IBr}$ at FM ground state, which is above room temperature. Finally, the interplay of electronic correlations with nontrivial band topology is studied to confirm the robustness of QAH state. The combination of piezoelectricity, topological and FM orders makes monolayer $\mathrm{Fe_2IX}$ (X=Cl and Br) become a potential platform for multi-functional spintronic applications with large gap and high $T_C$. Our works provide possibility to use the piezotronic effect to control QAH effects, and can stimulate further experimental works., Comment: 10 pages, 10 figures

A two-dimensional (2D) material system with both piezoelectricity and ferromagnetic (FM) order, referred to as a 2D piezoelectric ferromagnetism (PFM), may open up unprecedented opportunities for intriguing physics. Inspired by experimentally synthesized Janus monolayer MoSSe from $\mathrm{MoS_2}$, in this work, the Janus monolayer $\mathrm{CrBr_{1.5}I_{1.5}}$ with dynamic, mechanical and thermal stabilities is predicted, which is constructed from synthesized ferromagnetic $\mathrm{CrI_3}$ monolayer by replacing the top I atomic layer with Br atoms. Calculated results show that monolayer $\mathrm{CrBr_{1.5}I_{1.5}}$ is an intrinsic FM half semiconductor with valence and conduction bands being fully spin-polarized in the same spin direction. Furthermore, monolayer $\mathrm{CrBr_{1.5}I_{1.5}}$ possesses a sizable magnetic anisotropy energy (MAE). By symmetry analysis, it is found that both in-plane and out-of-plane piezoelectric polarizations can be induced by a uniaxial strain in the basal plane. The calculated in-plane $d_{22}$ value of 0.557 pm/V is small. However, more excitingly, the out-of-plane $d_{31}$ is as high as 1.138 pm/V, which is obviously higher compared with ones of other 2D known materials. The strong out of-plane piezoelectricity is highly desirable for ultrathin piezoelectric devices. Moreover, strain engineering is used to tune piezoelectricity of monolayer $\mathrm{CrBr_{1.5}I_{1.5}}$. It is found that compressive strain can improve the $d_{22}$, and tensile strain can enhance the $d_{31}$. A FM order to antiferromagnetic (AFM) order phase transition can be induced by compressive strain, and the critical point is about 0.95 strain. That is to say that a 2D piezoelectric antiferromagnetism (PAFM) can be achieved by compressive strain, and the corresponding $d_{22}$ and $d_{31}$ are 0.677 pm/V and 0.999 pm/V at 0.94 strain, respectively., Comment: 10 pages, 15 figures

The realization of multifunctional two-dimensional (2D) materials is fundamentally intriguing, such as combination of piezoelectricity with topological insulating phase or ferromagnetism. In this work, a Janus monolayer $\mathrm{SrAlGaSe_4}$ is built from 2D $\mathrm{MA_2Z_4}$ family with dynamic, mechanical and thermal stabilities, which is piezoelectric due to lacking inversion symmetry. The unstrained $\mathrm{SrAlGaSe_4}$ monolayer is a narrow gap normal insulator (NI) with spin orbital coupling (SOC). However, the NI to topological insulator (TI) phase transition can be induced by the biaxial strain, and a piezoelectric quantum spin Hall insulator (PQSHI) can be achieved. More excitingly, the phase transformation point is only about 1.01 tensile strain, and nontrivial band topology can hold until considered 1.16 tensile strain. Moreover, a Rashba spin splitting in the conduction bands can exit in PQSHI due to the absence of a horizontal mirror symmetry and the presence of SOC. For monolayer $\mathrm{SrAlGaSe_4}$, both in-plane and much weak out-of-plane piezoelectric polarizations can be induced with a uniaxial strain applied. The calculated piezoelectric strain coefficients $d_{11}$ and $d_{31}$ of monolayer $\mathrm{SrAlGaSe_4}$ are -1.865 pm/V and -0.068 pm/V at 1.06 tensile strain as a representative TI. In fact, many PQSHIs can be realized from 2D $\mathrm{MA_2Z_4}$ family. To confirm that, similar to $\mathrm{SrAlGaSe_4}$, the coexistence of piezoelectricity and topological orders can be realized by strain (about 1.04 tensile strain) in the $\mathrm{CaAlGaSe_4}$ monolayer. Our works suggest that Janus monolayer $\mathrm{SrAlGaSe_4}$ is a pure 2D system for PQSHI, enabling future studies exploring the interplay between piezoelectricity and topological orders, which can lead to novel applications in electronics and spintronics., Comment: 9 pages,10 figures

The combination of piezoelectricity with other unique properties (like topological insulating phase and intrinsic ferromagnetism) in two-dimensional (2D) materials is much worthy of intensive study. In this work, the piezoelectric properties of 2D topological insulators InXO (X=Se and Te) from monolayer InX (X=Se and Te) with double-side oxygen functionalization are studied by density functional theory (DFT). The large piezoelectric strain coefficients (e.g. $d_{11}$=-13.02 pm/V for InSeO and $d_{11}$=-9.64 pm/V for InTeO) are predicted, which are comparable and even higher than ones of many other familiar 2D materials. Moreover, we propose two strategies to enhance piezoelectric response of monolayer InXO (X=Se and Te). Firstly, the biaxial strain (0.94-1.06) is applied, and the $d_{11}$ (absolute value) is increased by 53\%/56\% for monolayer InSeO/InTeO at 1.06 strain, which is due to increased $e_{11}$ (absolute value) and reduced $C_{11}-C_{12}$. In considered strain range, InXO (X=Se and Te) monolayers are always 2D topological insulators, which confirm the coexistence of piezoelectricity and nontrivial band topology. Secondly, a Janus monolayer $\mathrm{In_2SeTeO_2}$ is designed by replacing the top Se/Te atomic layer in monolayer InSeO/InTeO with Te/Se atoms, which is dynamically and mechanically stable. More excitingly, Janus monolayer $\mathrm{In_2SeTeO_2}$ is also a 2D topological insulator with sizeable bulk gap up to 0.158 eV, confirming the coexistence of intrinsic piezoelectricity and topological nature. The calculated $d_{11}$ is -9.9 pm/V, which is in the middle of ones of InSeO and InTeO monolayers., Comment: 10 pages, 11 figures

The recently experimentally synthesized monolayer $\mathrm{MoSi_2N_4}$ and $\mathrm{WSi_2N_4}$ (\textcolor[rgb]{0.00,0.00,1.00}{Science 369, 670-674 (2020})) lack inversion symmetry, which allows them to become piezoelectric. In this work, based on ab initio calculations, we report structure effect on intrinsic piezoelectricity in septuple-atomic-layer $\mathrm{MSi_2N_4}$ (M=Mo and W), and six structures ($\alpha_i$ ($i$=1 to 6)) are considered with the same space group.It is found that $\mathrm{MSi_2N_4}$ (M=Mo and W) with $\alpha_i$ ($i$=1 to 6) all are indirect band gap semiconductors. Calculated results show that $\mathrm{MoSi_2N_4}$ and $\mathrm{WSi_2N_4}$ monolayers have the same structural dependence on piezoelectric strain and stress coefficients ($d_{11}$ and $e_{11}$), together with the ionic and electronic contributions to $e_{11}$.Finally, we investigate the intrinsic piezoelectricity of monolayer $\mathrm{MA_2Z_4}$ (M=Cr, Mo and W; A=Si and Ge; Z=N and P) with $\alpha_1$ and $\alpha_2$ phases expect $\mathrm{CrGe_2N_4}$, because they all are semiconductors and their enthalpies of formation between $\alpha_1$ and $\alpha_2$ phases are very close. The most important result is that monolayer $\mathrm{MA_2Z_4}$ containing P atom have more stronger piezoelectric polarization than one including N atom. The largest $d_{11}$ among $\mathrm{MA_2N_4}$ materials is 1.85 pm/V, which is close to the smallest $d_{11}$ of 1.65 pm/V in $\mathrm{MA_2P_4}$ monolayers. For $\mathrm{MA_2P_4}$, the largest $d_{11}$ is up to 6.12 pm/V. Among the 22 monolayers, $\alpha_1$-$\mathrm{CrSi_2P_4}$, $\alpha_1$-$\mathrm{MoSi_2P_4}$, $\alpha_1$-$\mathrm{CrGe_2P_4}$, $\alpha_1$-$\mathrm{MoGe_2P_4}$ and $\alpha_2$-$\mathrm{CrGe_2P_4}$ have large $d_{11}$, which are greater than or close to 5 pm/V, a typical value for bulk piezoelectric materials., Comment: 8 pages, 11 figures

Janus two-dimensional (2D) materials have attracted much attention due to possessing unique properties caused by their out-of-plane asymmetry, which have been achieved in many 2D families. In this work, the Janus monolayers are predicted in new 2D $\mathrm{MA_2Z_4}$ family by means of first-principles calculations, $\mathrm{MoSi_2N_4}$ and $\mathrm{WSi_2N_4}$ of which have been synthesized in experiment(\textcolor[rgb]{0.00,0.00,1.00}{Science 369, 670-674 (2020)}). The predicted $\mathrm{MSiGeN_4}$ (M=Mo and W) monolayers exhibit dynamic, thermodynamical and mechanical stability, and they are indirect band-gap semiconductors. The inclusion of spin-orbit coupling (SOC) gives rise to the Rashba-type spin splitting, which is observed in the valence bands, being different from common conduction bands. Calculated results show valley polarization at the edge of the conduction bands due to SOC together with inversion symmetry breaking. It is found that $\mathrm{MSiGeN_4}$ (M=Mo and W) monolayers have high electron mobilities. Both in-plane and much weak out-of-plane piezoelectric polarizations can be observed, when a uniaxial strain in the basal plane is applied. The values of piezoelectric strain coefficient $d_{11}$ of the Janus $\mathrm{MSiGeN_4}$ (M=Mo and W) monolayers fall between those of the $\mathrm{MSi_2N_4}$ (M=Mo and W) and $\mathrm{MGe_2N_4}$ (M=Mo and W) monolayers, as expected. It is proved that strain can tune the positions of valence band maximum (VBM) and conduction band minimum (CBM), and enhance the the strength of conduction bands convergence caused by compressive strain. It is also found that tensile biaxial strain can enhance $d_{11}$ of $\mathrm{MSiGeN_4}$ (M=Mo and W) monolayers, and the compressive strain can improve the $d_{31}$ (absolute values)., Comment: 10 pages, 11 figures