Twodimensional hexagonal chromium chalcohalides with large vertical piezoelectricity, hightemperature ferromagnetism, and high magnetic anisotropy†
Received
29th April 2020
, Accepted 16th June 2020
First published on 17th June 2020
On the basis of density functional theory, we predicted that Janus CrTeI and CrSeBr monolayers possess highly energetic, dynamical, and mechanical stability. Due to noncentral symmetry, the two monolayers exhibit vertical piezoelectricity with large piezoelectric coefficients d_{31} (1.745 and 1.716 pm V^{−1} for CrBSe and CrTeI, respectively), which are larger than those of most materials in existence. Both systems are also ferromagnetic (FM) semiconductors, with Curie temperature (T_{C}) higher than 550 K and large inplane magnetic anisotropy energy. Superexchange interactions are responsible for hightemperature FM order. A semiconductor to half metal transition can be regulated by carrier doping, which can be carried out by gate voltages. Doped systems still retain the same FM order as pristine ones; in particular, hole doping enhances exchange coupling, thereby increasing T_{C}. The combination of piezoelectricity, high T_{C}, and controllable electronic structures and magnetic properties makes magnetic 2D Janus CrSeBr and CrTeI attractive materials for potential applications in nanoelectronics, electromechanics, and spintronics.
1. Introduction
Since the discovery of graphene,^{1} considerable efforts have been made in exploring many other twodimensional (2D) materials that exhibit novel properties in electronics, spintronics, optics, and magnetism due in part to quantum confinement and dimensionality reduction. The existing 2D materials include graphene,^{1} BN,^{2} black phosphorus,^{3} borophene,^{4} transition metal dichalcogenides (TMX_{2}),^{5} and MXenes.^{6,7} Among them, 2D materials with robust ferromagnetic (FM) order have become a highly desirable target for nanoscale spintronic applications,^{7–11} but suitable candidates are unavailable because of the lack of intrinsically robust magnetic order or sufficiently high Curie temperature (T_{C}). To date, few 2D magnetic materials have been demonstrated experimentally.^{12–15} The wellknown chromium triiodide (CrI_{3}) monolayer was experimentally found to exhibit Ising ferromagnetism with T_{C} of 45K,^{12} which is slightly lower than that of its bulk counterpart (61K). Closetoideal 2D intrinsic Heisenberg ferromagnetism was observed in the atomic layers of the Cr_{2}Ge_{2}Te_{6} system in experimental investigation^{13} and theoretic prediction,^{16} and the magnetic phase transition temperature was 67.3 K.^{17–19} The monolayer VSe_{2} grown on MoS_{2} exhibits room temperature FM ordering with a large magnetic moment,^{14,15} which is in accordance with theoretic reports.^{16,17} FM materials with T_{C} above room temperature are rare in nature, but these discoveries challenge the Mermin–Wagner theorem,^{20} which predicts that theoretically longrange magnetic order is prohibited in 2D materials at finite temperatures. This phenomenon intrigues scientists into further understanding the underlying physics of 2D magnetic order, and the findings may help in obtaining intrinsic longrange 2D FM materials.
Except for magnetic behavior, piezoelectricity is another particularly interesting and useful property that has attracted tremendous interest.^{21–26} Piezoelectricity reflects the coupling between mechanical stimulation and electronic output, thereby converting mechanical energy into electricity or vice versa.^{21,22} Therefore, nanoscale piezoelectric materials are promising candidates for applications in electromechanical sensors, actuators, transducers, and energy harvesters in the fields of nanorobotics, piezotronics, and nanoelectromechanical systems.^{21–29} From a physical standpoint, piezoelectric polarization can be realized spontaneously in 2D materials by designing materials without central symmetry, indicating that intrinsic piezoelectricity, ferroelasticity, and ferroelectricity exist in such 2D materials. Theoretic advances predict piezoelectric polarizations in 2D Janus transition metal dichalcogenide monolayers (TMXY)^{23,27} because such materials directly break the inversion and outofplane mirror symmetries.^{27} Experimental observation reported that Janus TMXY, that is, Janus MoSSe, has been successfully realized,^{28,29} which encourages researchers’ interest in exploring piezoelectric polarizations in Janus TMXY.^{21–27}
Materials with versatile properties offer an avenue for the design of multifunctional nanodevices, thereby extending their applications in various fields. However, at present, there are few reported 2D multifunctional materials that exhibit intrinsic piezoelectricity and ferromagnetism at room temperature. Theoretical calculations predict a kind of 2D Crbased chalcohalide with lepidocrocitetype structure as FM materials with T_{C} below temperature.^{30,31} However, outofplane piezoelectricity is still lacking because of mirror symmetry. In this study, we designed and investigated hexagonal chromium chalcohalide (CrSeBr and CrTeI) monolayers with Janus MoSSe type structure on the basis of density functional theory (DFT) calculations. The systems under study were magnetic semiconductors, demonstrating large vertical piezoelectricity, hightemperature FM ordering and magnetic anisotropy, and tunable halfmetallicity. FM 2D materials in conjunction with their piezoelectricity may result in new physics and innovative device designs for novel applications.
2. Computational details
DFT calculations were performed using the Vienna Ab initio Simulation Package,^{32,33} with projected augmented wave method. We adopted the generalizedgradient exchange–correlation functional within the Perdew–Burke–Ernzerhof (PBE) scheme^{34} to treat the exchange–correlation energy in all calculations. The energy cutoff for plane wave basis was expanded to 500 eV for the calculation of structural relaxation and electronic structures. The first Brillouin zone (BZ) integration was sampled by 13 × 13 × 1 Monkhorst–Pack k mesh for unit cell. The structures are fully relaxed until the Hellmann–Feynman forces on each atom were less than 0.005 eV Å^{−1}. The convergence criteria for the total energy was set to be 10^{−6} eV. To eliminate spurious interactions with its periodic images, a vacuum region of approximately 15 Å was applied in the direction perpendicular to the CrXY monolayer. To cope with the correlation effects of the localized Cr3d electrons, we adopted the GGA + U scheme with the onsite Coulomb U = 3.5 eV and exchange parameters J = 0.7 eV, respectively.^{35} Phonon dispersion calculations were obtained by the PHONOPY code on the basis of force constant method.^{36} To ensure the accuracy for magnetic anisotropy energy (MAE) calculation, the BZ of a rectangular supercell was sampled by Γcentered Monkhorst–Pack kpoint of 10 × 17 × 1, and the convergence criteria for energy increased to 10^{−8} eV. When evaluating the MAE of monolayers, we considered spin–orbit coupling (SOC).
3. Results and discussion
3.1 Geometry and stability
The singlelayer CrS_{2} in 1T phase exhibits a hexagonal structure with space group Pm1 (164#),^{37} whereas CrXY monolayers under study have similar structures with P3m1 space group (156#) due to the absence of outofplane mirror symmetry, as shown in Fig. 1(a). Cr atoms are surrounded by six anions to form a distorted octahedral structure. The predicted lattice constants (a) are 3.656 and 3.959 Å at the PBE level for CrSeBr and CrTeI, respectively. The corresponding bond lengths are 2.521, 2.655, 2.755, and 2.861 Å for Cr–Se, Cr–Br, Cr–Te, and Cr–I, respectively. The layer thicknesses (h) are 2.991 and 3.256 Å for CrSeBr and CrTeI, respectively.

 Fig. 1 (a) Top and side views of the schematic structures of CrXY monolayer, (b) phonon dispersion spectra of CrSeBr and CrTeI, (c) electron localization function plot of the plane determined by Cr, Se, and Br atoms for CrSeBr monolayer, and (d) spin charge densities of CrSeBr monolayer. The yellow and green correspond to the spin up and spin down, respectively. The isovalue surface level is at 0.01 e Å^{−3}.  
To ensure the stability of CrXY, we first considered the binding energy (E_{b}), which is formulated as E_{b} = (E_{CrXY} − E_{Cr} − E_{X} − E_{Y}), where E_{CrXY} is the total energy of CrXY and E_{Cr}, E_{X}, and E_{Y} denote the energy of an isolated atom of Cr, X, and Y in a large enough box, respectively. The calculated E_{b} at PBE level are −4.721 and −4.01 eV per atom for CrSeBr and CrTeI, respectively. Such large binding energies indicate strong interactions between atoms in CrXY monolayers. To further verify the dynamical stability of the CrXY monolayers, their phonon spectra were investigated, as shown in Fig. 1(b). No negative frequency phonons were observed in the whole BZ for CrXY monolayers. Thus, they should be dynamically stable structures. For CrSeBr, the longitudinal optical (LO) and transverse optical (TO) modes are well separated by a large phonon bandgap of 1.27 THz at the Γ point. A wide phonon bandgap between LO and TO branches indicates a strong ionic nature. This is because the optical mode in ionic crystal induces electric polarization, which in turn increases resilience for the LO branch, thereby resulting in high frequency for the LO branches. Longitudinal acoustic and in and outofplane transverse acoustic (TA and ZA) branches show linear dispersion in the vicinity of the center of the BZ.
To determine the mechanical stability and properties, we investigated the linear elastic constants of all monolayers. For the 2D crystallographic system with hexagonal symmetry, only two independent elastic constants were observed: C_{11} and C_{12}. C_{11} is equal to C_{22}, and 2C_{66} = (C_{11} − C_{12}). The corresponding mechanical stability was determined by the Born–Huang criteria:^{38,39}C_{11} > 0, C_{66} > 0, and C_{11} > C_{12}. The elastic energy for 2D hexagonal structure can be expressed as follows: .^{40} The energyversusstrain plot can be obtained by applying small strains in the range of [−0.02, 0.02] to the unstrained lattice configuration (Fig. S1, ESI†). Thus, the elastic constants can be obtained by the polynomial fitting of the strain–energy curves. The calculated elastic constants C_{ij} listed in Table 1 satisfy the mechanical stability criteria for the two monolayers, thereby verifying their mechanical stability. The corresponding Young's moduli are 50.37 and 40.10 N m^{−1} according to the formula Y = (C_{11}^{2} − C_{12}^{2})/C_{11} for CrSeBr and CrTeI, respectively. CrXY systems demonstrate larger mechanical flexibility than those of other wellknown 2D materials, such as graphene (340 ± 40 N m^{−1})^{41} and MoS_{2} (126.2 N m^{−1}).^{42} The significantly low value of Young's moduli in CrXY systems may be ascribed to long and weak bond lengths.
Table 1 Lattice parameters a, elastic coefficients C_{ij}, Young's modulus Y, piezoelectric coefficients e_{ij} and d_{ij} of CrXY. The units of C_{ij}, e_{ij}, and d_{ij} for these 2D materials are N m^{−1}, pC m^{−1}, and pm V^{−1}, respectively
Material 
a (Å) 
C
_{11}

C
_{12}

Y

e
_{11}

e
_{31}

d
_{11}

d
_{31}

CrSeBr 
3.656 
53.25 
12.37 
50.37 
70.87 
115.34 
1.733 
1.756 
CrTeI 
3.959 
42.74 
10.61 
40.10 
134.52 
91.59 
4.186 
1.716 
HMoS_{2} 
3.334 
134.3 
33.0 
126.5 
369 

3.64 

HMoS_{2}^{42} 

130 
32 
126.2 
364 

3.73 

MoSTe^{49} 

116 
28 
109 
450 
50 
5.1 
0.4 
Te_{2}Se^{27} 

38.82 
10.49 

461.4 
12.3 
16.285 
0.249 
3.2 Piezoelectricity
Electron localization function (ELF) theory is an effective tool for analyzing chemical bond and characterizing lone pairs.^{43,44} The ELF value ranges from 0.0 to 1.0. ELF values of 1.00 and 0.50 correspond to perfectly localized and electrongaslike electrons, respectively, whereas an ELF value of 0.00 refers to extremely low charge density. Taking CrSeBr as an example, we present the ELF in Fig. 1(c). The value around the Cr is approximately 0.25, and the minimum values between the Cr and Se, and Cr and Br are 0.078 and 0.034, respectively, indicating the presence of dominant ionic bonding in this material. An interesting feature is the presence of nonuniform distributions of ELF around the Se and Br sites. The maximum values are 0.88 and 0.78, thereby indicating that mean lone pairs may occur on the outside parts of Br and especially Se planes and suggesting an increased tendency for the emergence of the intrinsic polar electric fields along the normal direction to the monolayer plane due to broken inversion symmetry. This “stereochemical activity of the lone pair” is also a driving force for offcenter distortion in magnetoelectric materials.^{45}
Intrinsic polar electric field is a possible reason for the emergence of piezoelectricity. In the CrXY monolayer structure, the difference in atomic size and electronegativity of Te/Se (5.49/5.89, Pearson absolute electronegativity) and I/Br (6.76/7.59) atoms results in inequivalent Cr–Te/Se (2.754/2.521 Å) and Cr–I/Br (2.861/2.655 Å) bonding lengths and charge distributions. The Cr layer is positively charged, whereas I/Br and Te/Se layers are negatively charged. This charge imbalance between Te/Se and I/Br results in a net electric field pointing from the Te/Se layer to the I/Br layer. To identify the inherent electric field further, we plotted the planar charge densities and average of the electrostatic potential (ΔΦ) energy in Fig. 2. We observed changes in ΔΦ, which is associated with the work function change of the structure^{46,47} and is believed to be proportional to the magnitude of the dipole moment in material, according to the Helmholtz equation.^{48} The resultant net electric field points from the Te/Se atomic layer to the I/Br atomic layer. The magnitude of the net vertical electric fields are estimated to be 0.197 and 0.229 eV Å^{−1} for CrSeBr and CrTeI, respectively, following the equation = −∇Φ, that is, the slope of the planeaveraged ΔΦ between the two outer most atom minima (shown as the green dashed line in Fig. 2).

 Fig. 2 Planar average of the electrostatic potential for (a) CrSeBr and (b) CrTeI monolayers. The insets show the distribution of the charge density in the ground states of CrSeBr and CrTeI. The color indicates the relative amplitude of the local densities.  
The lack of inversion symmetry can result in piezoelectric property, which reflects the coupling between electric and mechanical fields in a martial that becomes electrically polarized when subjected to mechanical stress and conversely change shape under an applied electric field. The piezoelectric tensor (e_{ijk}) describes the coupling between electrical polarization (P_{i}) and strain (ε_{jk}) tensor. The thirdrank tensor is defined as e_{ijk} = ∂P_{i}/∂ε_{jk}, where i, j, and k correspond to the x, y, and z Cartesian directions, respectively. e_{ijk} consists of two parts, namely, ionic (e^{ion}_{ij}) and electronic (e^{elc}_{ij}) contributions. The piezoelectric strain tensors d_{ij} and stress tensor e_{ij} relate to each other through the relationship e_{ij} = d_{ik}C_{kj}, where C_{kj} is the elastic stiffness tensor. The number of independent components in e_{ij} and d_{ij} is reduced by the crystal symmetry. For 2D materials with C_{3v} symmetry, e_{ijk} has two independent piezoelectric coefficients, namely, e_{11}/d_{11} and e_{31}/d_{31}. In these equations, e_{31}/d_{31} describes the polarization induced along the z axis when the crystal is uniformly strained in the basal xy plane, and e_{11}/d_{11} measures the change in polarization along the x (or y) axis induced by strain in the same direction and accounts for the anisotropy of the electromechanical coupling in the basal xy plane. The d_{ij} coefficients of d_{11} and d_{31} can be expressed as follows: d_{11} = e_{11}/(C_{11} − C_{12}) and d_{31} = e_{31}/(C_{11} + C_{12}).^{23,26,49}
The calculated piezoelectric coefficients e_{ij} and d_{ij} are listed in Table 1. The 2D coefficient e^{2D}_{ij} must be renormalized by the z lattice parameter that corresponds to the spacing between 2D layers, that is, e^{2D}_{ij} = ze^{3D}_{ij}.^{26} As a benchmark test, e_{11} for singlelayer MoS_{2} is 369 pC m^{−1}, which is in excellent agreement with the perilous 364 pC m^{−1}.^{42} The calculated e_{11} value of 70.87/134.52 pC m^{−1} for CrSeBr/CrTeI is much smaller than those reported in 2D TMX_{2}^{26} and Janus TMXY.^{23,49} However, the obtained e_{31} values of 115.34 and 91.59 pC m^{−1} for CrSeBr and CrTeI, respectively, are much larger than those of most TMX_{2} and Janus TMXY. Given the broken inversion symmetry along the zdirection, the large piezoelectric coefficients of the CrXY monolayer, namely, e_{31} and d_{31}, are highly desired because they offer freedom to manipulate and design novel piezoelectric devices. The obtained d_{31} of 1.756/1.716 mp V^{−1} for CrSeBr/CrTeI is superior to those of most 2D buckled hexagonal III–V compounds,^{26,50} Janus TMXY,^{23} Janus group III chalcogenide monolayers, Te_{2}Se (0.249),^{27} Janus MoSTe (0.4),^{49} and GaInSe_{2} (0.46).^{51} Such a large outofplane piezoelectric effect in CrXY makes them potential materials for applications in diverse nanoelectromechanical devices.
3.3 Electronic structures and magnetic properties
3.3.1 Electronic structures.
To further explore potential applications in the field of spintronics, we illustrate the spinresolved electronic band structures calculated by the GGA + U in Fig. 3(a) and (b). The CrSeBr and CrTeI systems are semiconductors with the valence band maximum (VBM) at the Γ point and conduction band minimum (CBM) lying between the M and Γ points. The estimated indirect band gaps are 1.414 and 0.494 eV for CrSeBr and CrTeI, respectively. Considering the heavy atomic masses of Te, I, Se, and Br atoms, we calculated the band structure, with the inclusion of the SOC effects, as shown in Fig. 3a′ and b′. The inclusion of the SOC reduces the band gaps to 1.303 and 0.276 eV for CrSeBr and CrTeI, respectively. More interestingly, sizable Zeemantype valley spin splitting occurs at the Γ point, and the valley spin splitting is 117 and 207 meV for CrSeBr and CrTeI, respectively, at the VBM. The VBM arises mainly from the p_{xy} orbitals of anions, whereas the CBM is composed of the p_{z} state of anions and d_{xz+yz} state of Cr atom. To verify this physical picture, we visualized the realspace wave functions of the CBM and VBM of the monolayers, as shown in Fig. S2 (ESI†). The cooperative effect of the outofplane magnetic field and SOC of the system itself generates sizable band splitting for p_{xy} at the Γ point. Such large spin splitting provides a possible approach to design a new generation of spintronic devices at room temperature. In addition, we performed calculations within the HSE06 + U method with (and without SOC) to obtain more reliable band gaps. The predicted values are 2.119 (2.246) and 0.875 (1.115) eV for CrSeBr and CrTeI, respectively. The Zeemantype valley spin splitting changes the values to 127 and 197 meV for CrSeBr and CrTeI, respectively. All the corresponding band structures are available in Fig. S3 and S4 (ESI†). In addition to the band gap, we examined the carrier mobilities and optical absorption of the CrXY monolayers using firstprinciples calculations and deformation potential theory. Fig. S2, S6 and Table SI (ESI†) show the calculation results.

 Fig. 3 The calculated electronic band structures without and with SOC for (a/a′) CrSeBr monolayer and (b/b′) CrTeI monolayer.  
3.3.2 Magnetic moment.
The local magnetic moments on Cr sites are 3.510 and 3.729 μ_{B} for CrSeBr and CrTeI systems, respectively, as shown in Table 2. The elements in the VIA and VIIA groups are generally negatively bivalent and monovalent, respectively, which render the Cr ion to be in the 3+ state. For 3d ions the crystal field interaction is generally much stronger than the spin–orbit interaction. Therefore, Hund's third rule does not work well. In such distorted octahedral environment, crystal field interaction with ligands results in the quenching of the orbital moment (L = 0). Hence, the orbital moment can be effectively ignored. Thus, the effect magnetic moment for 3d ions can be estimated by rather than by . The trivalent Cr has a 3d^{3} shell with spin angular momentum (S) = 3/2. Hence, the effect magnetic moment is , which is consistent with the calculated values above. The local moments on anions are antiparallel to those in Cr sites, and most of them are distributed on the chalcogenide ions, with small contribution from halogen ions, as shown in Table 2 and Fig. 1(c). The antiparallel alignment of the spins between anion and cation sites results in a net magnetic moment of 3.00 μ_{B} per CrXY chemical formula and is responsible for FM coupling between Cr sites. The band structure projection of the CrSeBr monolayer further confirms that magnetism mainly originates from the imbalance of the Cr3d states between the downspin and upspin channels, as shown in Fig. 4.
Table 2 Calculated total magnetic moment (M_{total}), and local magnetic moments on Cr (M_{Cr}), X (M_{X}) and Y (M_{Y}) atoms; exchange parameters of nearest, and secondneighbor couplings (J_{1}, J_{2}); estimated T_{C} using MC simulations; MAE. The units of magnetic moment, exchange parameters and, MAE are μ_{B}, meV and μeV per Cr, respectively
Material 
M
_{total}

M
_{Cr}

M
_{X}

M
_{Y}

J
_{1}

J
_{2}

T
_{C} (K) 
MAE 
CrSeBr 
3.00 
3.510 
−0.399 
−0.075 
−6.700 
−1.110 
550 
−186 
CrTeI 
3.00 
3.729 
−0.504 
−0.117 
−8.264 
−2.970 
956 
−1192 
CrI_{3} 
6.00 
3.058 


−2.726 

47.7 
801 
CrI_{3}^{54} 
6.0 
3.0 


−2.667 


804 

 Fig. 4 Partial band structure projection of monolayer CrSeBr calculated by the PBE functional without including the SOC effect. The colored symbols show the bands from different atomic orbitals. The size of the symbol is proportional to the weight of the atomic orbitals. The Fermi level is shifted to zero. The arrows ↑ and ↓ corresponds to the up and downspin channels, respectively.  
3.3.3
T
_{C} and magnetic interaction.
The magnetic energy gain (ΔE_{sp}) values assessed by the total energy difference between the spinpolarized and nonspinpolarized states are 3.35 eV and 3.49 eV per Cr for CrSeBr and CrTeI, respectively, indicating that the magnetism in both systems are extremely robust. Hence, the exact magnetic ground state should be determined. Four possible collinear magnetic configurations in CrXY monolayers were considered using an orthogonal supercell, as illustrated in Fig. S5 (ESI†). The classical spin Heisenberg Hamiltonian is written as follows:
where H_{0} is the nonmagnetic Hamiltonian; M_{i} is the net magnetic moment at the i site; and J_{1} and J_{2} are the nearestneighbor (NN) and nextnearestneighbor (NNN) exchange coupling parameters, respectively. According to these magnetic configurations, the NN and NNN exchange coupling parameters J_{1} and J_{2} can be obtained, as shown in Table 2. The calculation details are provided in the ESI.† The derived J_{1} and J_{2} for CrSeBr/CrTeI are −6.750/−8.264 and −1.11/−2.97 meV, respectively. Therefore, the FM interaction between Cr atoms prevails over the antiferromagnetic (AFM) interaction, resulting in an energetically preferred FM order in both monolayers. The obtained exchange parameter allows us to further obtain T_{C} by means of meanfield theories or Monte Carlo (MC) simulations with the Metropolis algorithm.^{52,53} Meanfield theories ignore correlation and fluctuation, thereby overestimating the T_{C}. Hence, relatively reliable MC simulations are adopted. Here, the NN and NNN exchange interactions are only considered for simplification. Although MAE shows a negligible quantity relative to that of exchange coupling, it still is mandatory in physics. For Heisenberg spin Hamiltonian without easy axis, the T_{C} will be predicted to be zero according to the Mermin–Wagner theorem. In the MC simulations, we use 50 × 50 2D lattices that contain 2500 local magnetic moments. The simulations lasted for 5 × 10^{7} loops for each temperature. Fig. 5 presents the relationship of temperature with respect to the average magnetic moment and specific heat (C_{v}) per chemical formula for CrSeBr and CrTeI monolayers. Through the observation of the variation of the magnetic moment and C_{v} curves, the magnetic phase transition occurs at approximately 550 and 950 K for CrSeBr and CrTeI monolayers, respectively. This result suggests that the FM state can survive far above the room temperature. To verify the method's validity, we calculated the NN exchange parameter J_{1} for CrI_{3} monolayer on the basis of the parameter described in ref. 54. The exchange parameter J_{1} is 2.726 meV; then, according to triangular lattice, the estimated T_{C} is 47.4 K from MC simulation, which agrees well with the T_{C} value of ∼45 K in the experiment.^{12} When considered the MAE, the T_{C} was determined to be 35 K and 46 K at the PBE and HSE06 levels, respectivley.^{55}

 Fig. 5 Monte Carlo simulations the average magnetic moment and specific heat C_{v} with respect to the temperature calculated for the (a) CrSeBr and (b) CrTeI monolayers. Inset shows the wavefunctions at VBM of the downspin channel (at the upper left) and upspin channel (at the upper right). The schematic diagram of the exchange mechanism is illustrated at the lower right.  
To understand the origin of magnetism, we provide the partial charge density at the VBM for CrSeBr monolayer in Fig. 5. The VBM of the downspin channel is dominated by Se4p_{xy}, Br4p_{xy}, and Cr3d_{xz+yz} orbitals. The Cr3d_{xz+yz} orbitals hybridize with the Se4p_{xy} orbitals, which leads to partial spin pair. Therefore, the AFM coupling through superexchange interaction happens due to the ionicity of the Cr–Se bonds. As a result, the ⋯–Cr↑–Se↓–Cr↑–⋯ coupling chain induces robust FM order, as shown in the lower right panel of Fig. 5. For the upspin channel, the Cr3d_{xz+yz} orbitals hybridize with the Br4p_{xy} ones at the VBM. Thus, ⋯–Cr↑–Br↓–Cr↑–⋯ coupling chain works in the same manner.
3.3.4 MAE.
MAE inhibits magnetization reversal in materials; hence, it is important for stabilizing longrange magnetic order^{56} and closely related to information storage. To evaluate the magnetic anisotropy, we consider the SOC effect when the noncollinear magnetic calculations are carried out. To search the easy axis, which is an energetically preferred axis for the spontaneous magnetization, we first performed a series of selfconsistent calculations with magnetization direction parallel or perpendicular to the monolayers. MAE is generally defined as MAE = E_{‖} − E_{⊥}, where E_{‖} and E_{⊥} correspond to the total energies of the noncollinear nonselfconsistent calculations with magnetization directions parallel and perpendicular to the plane of monolayer, respectively. According to a rectangular cell of the 2D CrXY monolayer, as shown in Fig. 6, the angulardependent MAE can be expressed as follows:^{57} MAE = K_{1}sin^{2}θ + K_{2}sin^{4}θ, where K_{1} and K_{2} are the anisotropy constants and θ is the azimuthal angle of rotation. Anisotropy constants will be obtained by fitting the MAE–θ curve to this equation. As illustrated in Fig. 6, the energy is isotropic in the xy plane, whereas the MAE strongly is sensitive to the magnetization direction in either the yz or xz plane. Therefore, both systems are of magnetic anisotropy with inplane easy magnetization axis.

 Fig. 6 Dependence of the absolute value of MAE on the direction of magnetization for (a) CrTeI and (b) CrSeBr monolayers. The inset illustrates that the spin vector S on the xy, and xz plane is rotated with an angle θ about the x axis. The rectangular cell used for calculations contains two Cr atoms.  
As shown in Table 2, the MAE values are −186 and −1192 μeV per Cr atom for CrSeBr and CrTeI systems, respectively. Large MAE in CrTeI results from strong SOC because Te and I atoms are heavier than Se and Br atoms. Such MAE is larger than the reported values of other intrinsic 2D FM materials, such as VOCl_{2} (16.6 μeV per V),^{58} CrI_{3} (803.65 μeV per Cr),^{54} and CrGeTe_{3} (20 μeV per Cr).^{59} For a system containing a heavy element, the MAE stems from the cooperative effect of the anisotropy in the atom's orbital moment L and its interaction with the spin angular momentum S.^{60–62} For Crbased FM materials, Lado et al.^{63} suggested that large MAE in CrI_{3} stems from the strong SOC of the heavy iodine ions, whereas Zhang et al.^{64} attributed the MAE to the interaction of Ip_{x}/p_{y} and Crd_{xy}/d_{x2−y2} orbitals to Cr_{2}I_{3}X_{3} (X = Br, Cl) monolayers.
According to the perturbation theory, Wang et al.^{65} interpreted MAE as the competition between vertical and inplane SOC effects. Hence, MAE is defined as follows:
where ξ is the c strength of SOC; o and u denote the occupied and unoccupied states of the same spin, that is, up–up (↑↑) or down–down (↓↓), respectively; E_{o} and E_{u} are the energy levels of the occupied and unoccupied states, respectively; and L_{z} and L_{x/y} are the angular momentum operators along the z and x/y directions, respectively. To understand the origin of the large magnetic anisotropy, we demonstrate the projected density of states of the Cr3d and p orbitals of the anion in Fig. 7. For both systems, the interactions between the p_{z} and d_{z2} and between the d_{xz} and d_{yz} orbitals contribute to the positive MAE in the upspin channel. The positive contribution from the downspin channel is marginal due to the large value of E^{↓}_{u} − E^{↓}_{o} and small DOS near the Fermi level, especially for the 3d orbitals. Negative MAE is attributed to SOC interaction among the occupied p_{x/y}, d_{xz}, d_{yz}, and d_{x2+y2} orbitals in the downspin channel with those unoccupied orbitals in the upspin channel. E^{↑}_{u} − E^{↓}_{o} is much smaller than E^{↑}_{u} − E^{↑}_{o}, thereby indicating that negative MAE dominates over positive MAE.

 Fig. 7 The projected DOS of Cr3d orbitals and p orbitals of anions for (a) CrTeI and (b) CrSeBr monolayers.  
3.3.4 Controlling electronic structures and magnetism by carrier doping.
VBs in the downspin channel and CBs in the upspin channel simultaneously approach the Fermi level. Therefore, the CrXY monolayers are typical bipolar magnetic semiconductor materials.^{66,67} The unique electronic structures provide promising ways to realize halfmetallicity and manipulate fully spinpolarized currents in a specific spin channel by adjusting the position of the Fermi level. To reach such aim, we dope holes or electrons into these two systems with a realizable doping level at 0.1e per unit cell.^{11} As expected, under hole doping, both systems change form semiconductors into half metals with completely spin polarization in downspin channel (Fig. 8(a) and (b)). At the same time, the hole doping results in the same transition with full polarization in the upspin channel (Fig. 8(a′) and (b′)). In practice, electrical gating technique can be applied on the CrXY monolayers to tailor the position of the Fermi level.^{11,66,67}Fig. 8(c) shows the schematic of the bipolar fieldeffect spinfilter device on the basis of the CrXY systems. When a positive gate voltage (V_{G} > 0) is applied, the Fermi level shifts into the VB, and the fully polarized currents will appear in the downspin channel.^{67} Upon the application of a negative gate voltage (V_{G} < 0), the Fermi level moves up into the CB, and completely polarized currents will appear in the upspin channel. Therefore, completely spinpolarized currents with tunable spin polarization can be realized by applying a voltage gate upon the CrXY systems.

 Fig. 8 The total density of states for the (a) CrSeBr and (b) CrTeI under hole doping at a doping concentration of 0.1 carrier per unit cell. The right panels show the DOSs under electron doping at the same concentration as hole for (a′) CrSeBr and (b′) CrTeI. (c) Schematic of the bipolar fieldeffect spinfilter device and its I–V_{G} relationship. The Fermi levels are set to zero.  
Carrier doping can modify the electronic structures and induce magnetic phase transition for a magnetic system.^{11,66} Thus, whether the doped systems still retain room temperature FM ordering is an extremely interesting research topic. We provide the exchange parameters J_{1} and J_{2} in Table 3 with carrier doping concentrations varying from −0.2 to 0.2e per unit cell. Electron and hole doping result in equal numbers of increment and decrement of magnetic moments, respectively. Thus, magnetization is minimally affected because there is a total magnetic moment of 3.0 μ_{B} per unit cell. According to the variation of the exchange parameters J_{1} and J_{2}, FM interaction monotonically increases when the doping level changes from −0.2e to 0.2e per unit cell. Hole doping is favorable toward enhancing FM interaction, thereby promoting T_{C}. Meanwhile, electron doping plays the opposite role. Although the concentration of electron doping increases up to 0.1 electrons per unit cell, the estimated T_{C} (378 K) is well above room temperature for CrSeBr. The result reveals a feasible way to realize room temperature halfmetallicity by a suitable gate voltage.
Table 3 Calculated first and second nearest exchange couplings parameters J_{1} and J_{2} for CrSeBr and CrTeI monolayers doped with carriers at different doping level. The unit is meV for exchange couplings parameters
Structure 
Carriers doping concentration (e per unit cell) 
−0.2 
−0.1 
0.0 
0.1 
0.2 
J
_{1}

J
_{2}

J
_{1}

J
_{2}

J
_{1}

J
_{2}

J
_{1}

J
_{2}

J
_{1}

J
_{2}

CrSeBr 
−3.19 
1.29 
−4.77 
0.33 
−6.70 
−1.11 
−7.49 
−2.36 
−8.21 
−3.63 
CrTeI 
−4.60 
−0.27 
−6.16 
−1.04 
−8.26 
−2.97 
−8.49 
−3.18 
−8.22 
−3.34 
4. Conclusions
We have demonstrated stability, strong piezoelectric effects, high temperature ferromagnetism with robust magnetic anisotropy, and tunable electronic structures in Janus CrSeBr and CrTeI monolayers with triangular lattice through firstprinciples simulations. Given the lack of reflection symmetry, both Janus monolayers show vertical piezoelectric polarizations characterized by d_{31}, which are comparable to or even superior to most 2D piezoelectric materials, such as Janus TMXY. The two magnetic semiconducting materials also possess T_{C} that is much higher than room temperature on the basis of MC simulations and exhibit considerably large inplane MAE values of 186 and 1192 μeV per Cr atom. Carrier doping, regardless of the carrier type, induces transition from magnetic semiconductor to half metal. Particularly, hole doping improves FM interaction, thereby promoting the T_{C} of the systems. These results enrich the diversity of the Janus 2D materials, which have potential applications in electromechanical and spintronic devices in nanoscale.
Conflicts of interest
There are no conflicts to declare.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (No. 51701071), and the Scientific Research Fund of Hunan Provincial Education Department (No. 18A347 and 19C0487). We acknowledge the computational support provided by the computing platform of the Network Information Center of Hunan Institute of Engineering.
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Footnote 
† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0cp02293d 

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