Novel states of quantum anomalous Hall effect in XBiO₃ (X = Pd, Pt) magnetic monolayer

Abstract

The exploration of the regulation mechanism about Chern number (C) is crucial for acquiring high topological state in quantum anomalous Hall effect (QAHE). In this study, by symmetry analysis and first-principles calculations, monolayer XBiO₃ (X = Pd, Pt) are proven to be QAH insulators with tunable topological state. As the magnetization direction changes in the xy plane, monolayer XBiO₃ switch between QAH insulator with C=\left|1\right| and topological trivial semimetal with a period of 60°. It is caused by the breaking or protecting mirror symmetries for different polar angles. Comparatively, as the magnetization direction alters in the xz plane, monolayer XBiO₃ vary among QAH insulator with C=\left|3\right|, QAH insulator with C=\left|1\right| as well as mixed semimetal and QAH state with a period of 180°. The topological band gaps are as high as 114 and 132 meV for monolayers PdBiO₃ and PtBiO₃, respectively. The critical magnetic transition temperature of monolayers PdBiO₃ and PtBiO₃ reach up to 432 and 550 K, respectively. Notably, the QAH feature is robust for strains and U values. Our work provides an ideal platform to investigate the tunable high Chern number QAHE and design high performance QAH devices.

Graphical abstract

Keywords

quantum anomalous Hall effect / tunable Chern number / magnetization reorientation / topological semimetal

1 Introduction

Various novel states of matter have been induced in two-dimensional (2D) magnets because of their naturally breaking of time reversal symmetry under the intrinsic ferroic order [1, 2], which provides a new avenue for designing multifunctional devices [3, 4]. The on-site Coulomb interaction of d orbitals electric states results in the magneto-optical response in monolayer CrI₃ [5]. The Dirac states exist in 4H-SiC(0001) system are from the p_x and p_y orbitals [6]. A strong magnetoelectric coupling is theoretically confirmed in multiferroic 2D CuCrS₂ [7]. The multipiezo effect and noncollinear spin current are reported in the altermagnet monolayer V₂Se₂O [8]. In ferromagnetic (FM) VSe₂ monolayer [9], spontaneous valley polarization has been realized due to its intrinsic magnetic exchange interaction and spin-orbit coupling (SOC) effect, giving birth to the ferrovalley concept. Moreover, coexisting ferroelectric and ferrovalley polarizations in bilayer stacked VSi₂P₄, FeCl₂, and RuBr₂ with interlayer antiferromagnetic (AFM) coupling are found [10]. The high-temperature superconducting property of monolayer CoSb has been investigated in detail by first-principles calculations [11]. Interestingly, intrinsic quantum anomalous Hall effect (QAHE) is theoretically predicted and experimentally proved in topological insulators with magnetic doping [12], and intrinsic magnetic topological 2D MnBi₂Te₄ materials [13−15]. In addition, the odd-even-layer-dependent QAH and quantum spin Hall effects are demonstrated in 2D LiFeTe topological insulators [16]. Thus, 2D magnetic materials provide a suitable platform for investing novel states of matter and have become one of the hot spots for research.

For magnetic materials, the magnetization direction has a decisive impact on their physical properties because of their correlation with the symmetry and spin states of a system [17, 18]. The electric resistance value depends on the direction of magnetization in FM metals at saturation [19]. There is a cosine function dependence between valley polarization and magnetization direction in 2D valley materials [20−25]. Due to the coupling of magnetic and topological states, the topological gap of QAH insulator monolayer can be effectively adjusted by regulating the magnetization direction by using an external magnetic field [26]. In addition, under the influence of an external magnetic field in different directions, the spin states can be redirected [27]. Several studies have demonstrated the theoretical feasibility of achieving different topological states by regulating the magnetization direction in different Chern insulators, e.g., by tuning the magnetization direction, the systems can vary from a Weyl semimetal to a QAH insulator [28]. Furthermore, the Chern number C in topological insulators can be effectively regulated via manipulating the magnetization direction of system, which provides a novel way to acquire higher Chern number [29−33]. Specially, QAHE with adjustable Chern number based on metal-oxide lattice systems have been extensively studied [34−37]. Therefore, it will be interesting and meaningful to explore other novel physical states of matter in 2D magnetic materials by adjusting its magnetization direction through external magnetic field.

In this study, the magnetic, electronic, and topological properties of the monolayer XBiO₃ (X = Pd, Pt) are investigated. The monolayer XBiO₃ presents a FM coupling with in-plane magnetic anisotropy. The Berezinskii−Kosterlitz−Thouless transition temperatures of monolayers PdBiO₃ and PtBiO₃ are 432 and 550 K, respectively. Interestingly, monolayer XBiO₃ presents a topological phase transition behavior dependent on the magnetization direction. When the magnetization direction changes in the xy plane, monolayer XBiO₃ switch between QAH insulator with C=\left|1\right| and topological trivial semimetal with a period of 60°. It is caused by the breaking or protecting mirror symmetries for different polar angle Ψ. Comparatively, as the magnetization direction alters in the xz plane, monolayer XBiO₃ vary among QAH insulator with C=\left|3\right|, QAH insulator with C=\left|1\right| as well as mixed semimetal and QAH state with a period of 180°. Therefore, the Chern number of monolayer XBiO₃ can be regulated by changing the magnetization direction. The topological band gaps of monolayers PdBiO₃ and PtBiO₃ can reach to 114 and 132 meV, respectively. Notably, the QAH feature is robust for strain and different U values. This work presents two promising materials for practical implementation of spintronics devices.

2 Computational details

We employ the methodology of first-principles calculations, as implemented in the Vienna ab initio Simulation Package (VASP), based on density-functional theory (DFT) [38]. The projector-augmented-wave (PAW) potential [39] is utilized, with the exchange-correlation functional treated by generalized gradient approximation (GGA) using the Perdew−Burke−Ernzerhof (PBE) functional [40]. The plane-wave basis is employed with a kinetic energy cut-off of 500 eV. The convergence criteria for energy and force are set at 10⁻⁷ eV and 10⁻³ eV/Å, respectively. In order to obtain a more realistic electronic structure, SOC effect is also taken into account in the calculation. The Brillouin zone is sampled using a 15 × 15 × 1 Г-centered Monkhorst-Pack grid for accurate representation. The PBE+U method is used to treat the d orbitals, U is set to be 3.0 and 4.0 eV for Pd and Pt, respectively [41]. To confirm the results, the calculations are also checked using the Heyd−Scuseria−Ernzerhof (HSE06) method. To avoid the influence between adjacent layers, a vacuum layer in the z direction with a thickness of 18 Å is applied. The phonon spectrum is obtained by the PHONOPY code based on the density functional perturbation theory using a 4 × 4 × 1 supercell [42]. Ab initio molecular dynamics (AIMD) simulations are adopted using a Nosé-Hoover thermostat to control the system temperature with an NVT ensemble [43]. The VASPKIT package is used to process the calculation data [44]. A maximized localization function is created using the Wannier90 package [45], where the d orbitals of Pd/Pt atom and p orbitals of Bi and O atoms are considered as the projected orbitals for monolayer XBiO₃ (X = Pd, Pt). The AHC for monolayer XBiO₃ is evaluated by taking a very dense k-point mesh of 500 × 500 × 1 in the Brillouin zone. The edge states are calculated by using the WannierTools software package [46].

3 Results and discussion

3.1 Crystal structure and magnetic property

Fig.1(a) shows the crystal structure of monolayer (ML) XBiO₃ (X = Pd, Pt) with a space group of P\overline{3}1m, in which the middle layer Pd/Pt atoms are sandwiched by two layers of O atoms, leaving the Bi atoms on the outmost side. Fig.1(b) displays the first Brillouin zone with high symmetry points of ML XBiO₃. Notably, three orange vertical mirror planes are presented, corresponding to the C₃ rotation symmetry. The optimized lattice constants are 5.36 and 5.44 Å for ML PdBiO₃ and PtBiO₃, respectively, as presented in Table S1. The calculated energy difference between FM and three AFM configurations (ΔE = E_{\mathrm{N}é\mathrm{e}\mathrm{l}}_{/Stripe/Zigzag} ‒ E_FM) are 0.12, 0.43, and 0.18 eV for ML PdBiO₃ and 0.04, 0.44, and 0.26 eV for ML PtBiO₃, respectively, indicating the FM coupling of ML XBiO₃. The calculated magnetic moment is 2 μ_B/f.u.. The binding energy of ML XBiO₃ is computed as the energy difference between ML XBiO₃ and the Ni, Bi crystal, and O molecule, given by E_{\mathrm{b}}=E\left(X\mathrm{B}\mathrm{i}{\mathrm{O}}_3\right)-E\left(X\right) -E\left(\mathrm{B}\mathrm{i}\right)-3/2E\left({\mathrm{O}}_2\right). The obtained formation energies of −2.98 and −2.92 eV implies that the ML PdBiO₃ and PtBiO₃ are energetically stable and its experimental synthesis is achievable, as shown in Table S1. The dynamical and thermal stabilities of ML XBiO₃ is indicated by study their phonon dispersion spectrums and AIMD simulations, as depicted in Figs. S1(a)−(d). The absence of imaginary frequencies in the phonon spectra confirms the dynamical stability of the material. During the AIMD simulations, the total energy fluctuates within a small range, the structure of ML XBiO₃ remains intact, indicating its thermal stability. Moreover, the elastic constants C_{11}{,C}_{22},C_{12}\mathrm{a}\mathrm{n}\mathrm{d}{C}_{66} of ML XBiO₃ (see Table S2), which fulfill the Born criteria conditions C_{11}{,C}_{22}{,C}_{66}>0; C_{11}{C}_{22}-C_{12}^2>0, confirming the mechanical stability of ML XBiO₃ [47]. Figures S2(a) and (b) plot the electron localization function of ML XBiO₃. Notably, there are significant electron localization around the Bi and O atoms, suggesting an ionic bonding between X and Bi (O) atoms. During the forming of ML XBiO₃, the Pd/Pt and Bi atoms will contribute six electrons to the three O atoms, remaining seven valence electrons in the d orbitals of Pd/Pt atom. The Pd-Pd/Pt-Pt and Pd-O/Pt-O bond lengths are 3.09/3.14 and 2.09/2.10 Å, respectively.

Due to the localization of d orbitals of Pd/Pt, the direct-exchange should be weaker than the super-exchange [48]. The Pd/Pt atom in ML XBiO₃ is surrounded by six O atoms, forming a distorted octahedra [49, 50]. Under this crystal field, the d orbitals of Pd/Pt atom are split into a_g (d_{z^2}), e_g1 (d_{xy}, d_{x^2-y^2}) and e_g2 (d_{xz}, d_{yz}) orbitals. The seven electrons first fill the a_g (d_{z^2}), e_g1 (d_{xy}, d_{x^2-y^2}) orbitals, then the remaining one occupies one of the e_g2 (d_{xz}, d_{yz}) orbitals, as shown in Fig.1(d). This is consistent with the above analysis of the energy difference between FM and AFM configurations, and suggesting a strong and robust FM coupling in the present ML XBiO₃. The FM state is attributed to the Pd(Pt)−O−Pd(Pt) super-exchange interactions, as depicted in Fig.1(d). The d orbital of X (X = Pd, Pt) easily hybrids with the p orbital of O, which results in the super-exchange interactions through O atoms. Moreover, the Pd−O−Pd/Pt−O−Pt angle φ is relax to 92.9°/91.4°, closing to 90°, preferring FM coupling according to Goodenough−Kanamori−Anderson rule [51−53]. As a result, the same d orbitals of two Pd/Pt atoms can only be occupied by the electrons with the same spin state, thereby resulting in FM coupling. Figures S2(c) and (d) present the density of states (DOSs) for ML XBiO₃. The spin-up and spin-down TDOS values of the ML XBiO₃ are not symmetrical, indicating it is FM behavior. From the partial density of states (PDOS) of X atom, it is noted that the d orbital dominates the magnetism of ML XBiO₃, while the Bi and O atoms do not contribute to the magnetism.

Furthermore, the magnetic anisotropy energy (MAE) of ML XBiO₃ is calculated, which in defined as: MAE = E_θ – E_[100], where E_θ and E_[100] are the total energies of the system with the magnetization along θ and [100] directions, respectively, as shown in the inset of Fig.1(e) [54, 55]. In xy plane, the MAE is isotropic for different magnetization angel θ_xy. However, in xz and yz planes, the MAE varies with the magnetization angles of θ_xz and θ_yz. Thus, the easy axis of magnetization about ML PdBiO₃ and PtBiO₃ are located at in-plane with a MAE value of 5.09 and 22.8 meV/f.u., respectively, as displayed in Fig.1(e) and (f). According to the Mermin-Wagner theorem [56], the ML PdBiO₃ and PtBiO₃ should not be magnetically long-range ordered. Therefore, ML XBiO₃ belongs to XY magnets, which would present a Berezinskii−Kosterlitz−Thouless (BKT) transition at the critical temperature T_BKT [57, 58]. Based on this theoretical framework, an estimation can be made for T_BKT of ML XBiO₃ as T_BKT = 0.89 J/k_B [59, 60], where J represents the exchange integral and k_B denotes the Boltzmann constant. The T_BKT values of ML PdBiO₃ and PtBiO₃ are 432 and 550 K, respectively, which are higher than the observation temperature of ML MnBiTe (1.4 K) [61]. Thus, ML PdBiO₃ and PtBiO₃ are promising to be applied in the magnetic electronic devices at room temperature.

Notably, there exist some compounds made of Pd, Pt, Bi and O atoms, which are really existing. Such as the perovskite LaPdO₃ has been synthesized through the high-pressure techniques [62, 63]. The oxidized Pt_xBi_yO_z cluster is formed via an incipient wetness impregnation [64, 65]. Nanocomposites consisting of 5% Pt supported on activated carbon and promoted with 5% Bi or Sb are prepared by electroless deposition and microwave-assisted methods [66]. More importantly, ML XBiO₃ exhibits the same structure with SrRu₂O₆, which has prepared by using low-temperature hydrothermal synthesis and scalable technique of liquid exfoliation [67, 68]. Therefore, ML XBiO₃ can also be obtained from XBi₂O₆ in experiment using the aforementioned methods.

3.2 Electric structure and topological property

The energy band of ML XBiO₃ is used to investigate its electronical and spin polarization properties. Fig.2 shows the energy band structure of ML PdBiO₃. As displayed in Fig.2(a), without considering SOC effect, the spin-down state behaves as a semiconductor, while the spin-up state exhibits a semimetal state with a Dirac cone crossing the Fermi level. It induces a 100% spin polarization, showing a potential application in spintronics [69, 70]. Notably, due to C₃ rotation symmetry, there are a total of six Dirac cones all over the Brillouin zone, as indicated in Fig.2(b). As shown in Fig. S3, the Ψ and Φ represent the angle between the orientation of magnetization and x direction and the angle between the orientation of magnetization and the z direction, respectively. With considering SOC effect, the Dirac cone is opened as the magnetization direction angle Ψ = 0° in xy plane with a band gap 12 meV, as depicted in Fig.2(c). Moreover, both the d_xz and d_yz orbitals simultaneously appears in both conduction and valence bands, meaning energy inversion occurs. It should be noted that all of six Dirac cones are separated, as shown in Fig.2(d). These indicate that an in-plane QAHE is induced in present ML PdBiO₃ with a topological gap of 12 meV. Unexpected, ML PdBiO₃ also behaves as an out-of-plane QAHE as the magnetization direction angle Φ = 0° in xz plane, as indicated in Fig.2(e) and (f). Under this case, the topological gap describes to 114 meV, which exceeds thermal fluctuation energy ~ 25 meV at room temperature. As a result, the topological property of ML PdBiO₃ can be controlled through altering its magnetization direction. Notably, the topological feature of ML PtBiO₃ also varies with the change of magnetization direction, as indicated in Fig. S4. Furthermore, the band structure based on the HSE06 function is also carried out for comparison. It is found that the HSE06 scheme reveals the same electronic feature as that obtained from the above-mentioned PBE+U computation, as indicated in Fig. S5. Therefore, the PBE+U method is reliable and is selected for the present calculations.

The above content indicates that the topological characteristic is directly related with the magnetization direction in ML XBiO₃. Therefore, we furtherly investigate the QAH behavior of ML XBiO₃ under different M direction angles in the xy and xz planes, respectively. Fig.3 shows the topological property of ML PdBiO₃ under different magnetization direction in xy plane. As shown in Fig.3(a), all three Dirac cones along Γ-M1, Γ-M2 and Γ-M3 high symmetry lines are opened, and energy inversion occurs around the Dirac cones, indicating the QAH behavior in ML PdBiO₃ with Ψ = 0° in xy plane. To further determine the Chern number C, the corresponding anomalous Hall conductivity ({\sigma }_{xy}), Berry curvature ({\Omega }_z\left(\mathit{k}\right)) and chiral edge state are further studied, which can be calculated as [71, 72]

(1){\sigma }_{xy}=-{\displaystyle \frac{e^2}{\hslash }}{{\displaystyle \int }}_{\mathrm{B}\mathrm{Z}}{\displaystyle \frac{\mathrm{d}\mathit{k}}{{\left(2\pi \right)}^2}}{\Omega }_z\left(\mathit{k}\right)\hslash ,

(2)\begin{array}{rl}& {\Omega }_z\left(\mathit{k}\right)\\ & =-{{\displaystyle \sum }}_n{{\displaystyle \sum }}_{n\ne m}{f}_n\left(\mathit{k}\right){\displaystyle \frac{2\mathrm{I}\mathrm{m}⟨{\psi }_{n\mathit{k}}\left|{\hat{v}}_x\right|{\psi }_{m\mathit{k}}⟩⟨{\psi }_{m\mathit{k}}\left|{\hat{v}}_y\right|{\psi }_{n\mathit{k}}⟩}{{\left(E_{n\mathit{k}}-E_{m\mathit{k}}\right)}^2}},\end{array}

where e and ħ are the electronic charge and reduced Planck constant, respectively. The f_n\left(\mathit{k}\right) is the Fermi-Dirac distribution function and \mathit{k} being the electron wave vector, {\hat{v}}_x is the x component of velocity operator, and E_{n\mathit{k}} is the eigenvalue of Bloch wave function {\psi }_{n\mathit{k}} . As shown in Fig.3(b), the {\sigma }_{xy} is calculated as 1 {\mathrm{e}}^2/\mathrm{h} near the Fermi level, indicating a QAHE with C = 1. It is caused by one pair of negative extreme Berry curvatures and two pairs of positive Berry curvatures distributed throughout the Brillouin zone, which corresponds to Berry phases of −2π and 4π, respectively, resulting in a total Berry phase of 2π in ML PdBiO₃, being consistent with C = 1, as drawn in the insert of Fig.3(b). This quantum feature is also indicated by the appearing one chiral edge state connecting conduction and valence band along the high symmetry line, as plotted in Fig.3(c).

As the magnetization direction angle change to Ψ = 30°, the Dirac cone along Γ-M3 high symmetry line is closed, while the Dirac cones along Γ-M2 and Γ-M1 high symmetry lines are still open with energy inversion occurs, indicating an intrinsic semimetal state, as shown in Fig.3(d). It is induced by the protected M₊ and broken M and M₋ symmetries under magnetization direction angle Ψ = 30° [48, 73]. The obtained {\sigma }_{xy} is −0.02 {\mathrm{e}}^2/\mathrm{h}, losing the QAHE (C = 0). It is caused by one pair of negative extreme Berry curvatures and one pair of positive Berry curvatures distributed throughout the Brillouin zone, which corresponds to Berry phases of −2π and 2π, respectively, resulting in a total Berry phase of 0π in ML PdBiO ₃, being consistent with C = 0, as drawn in the insert of Fig.3(e). This intrinsic semimetal feature is also indicated by the zero disappearing chiral edge state, as plotted in Fig.3(f). For the case of Ψ = 60°, the Dirac cone along Γ-M1 high symmetry line is opened again with energy inversion occurs, returning to QAH insulator, as plotted in Fig.3(g). It is induced by breaking all the M, M₊ and M₋ symmetries with Ψ = 60° [48, 73]. At this moment, the {\sigma }_{xy} becomes −1 {\mathrm{e}}^2/\mathrm{h}, demonstrating the QAHE with C = −1. It is caused by two pairs of negative extreme Berry curvatures and one pair of positive Berry curvatures distributed throughout the Brillouin zone, which corresponds to Berry phases of −4π and 2π, respectively, resulting in a total Berry phase of −2π in ML PdBiO₃, being consistent with C = −1, as drawn in the insert of Fig.3(h). This intrinsic semimetal feature is also indicated by the appearing one chiral edge state along the high symmetry line, as plotted in Fig.3(i). As a result, ML PdBiO₃ will vary among QAHE with C = 1, topological trivial semimetal state and QAHE with C = −1 three topological phases with a period of 60° as the magnetization direction angle Ψ various in xy plane, as indicated by Fig. S6.

We also study the QAH behavior of ML PdBiO₃ under different magnetization direction angles Φ in the xz plane. As shown in Fig.4(a), all three Dirac cones along Γ-M1, Γ-M2 and Γ-M3 high symmetry lines are opened, and energy inversion occurs around the Dirac cones, indicating the topological feature of ML PdBiO₃ with Φ = 0°. As shown in Fig.4(b), the {\sigma }_{xy} is calculated as 3 {\mathrm{e}}^2/\mathrm{h} near the Fermi level, indicating a QAHE with C = 3. It is caused by three pairs of positive extreme Berry curvatures distributed throughout the Brillouin zone, which result in a total Berry phase of 6π, corresponding to C = 3, as drawn in the insert of Fig.4(b). This quantum characteristic is also indicated by the appearing three chiral edge state connecting conduction and valence band along the high symmetry lines, as plotted in Fig.4(c). As the magnetization direction angle change to Φ = 97.5°, the Dirac cones along Γ-M1 and Γ-M3 high symmetry lines are still opened, while the Dirac cone along Γ-M2 high symmetry line is closed with energy inversion occurs, as shown in Fig.4(d). Under this case, ML PdBiO₃ presents a mixed state of intrinsic semimetal and QAH insulator. The calculated {\sigma }_{xy} is −2 {\mathrm{e}}^2/\mathrm{h}, which is contributed by the anomalous Hall conductance of semimetal and QAH states. The QAH state is proven by the two pairs of negative extreme Berry curvature throughout the Brillouin zone, as shown in Fig.4(e). It is consistent with the appearing two chiral edge state along the high symmetry line, as plotted in Fig.4(f). Notably, this mixed state of intrinsic semimetal and QAH insulator is also indicated in ML PdBiO₃ with Φ = 86.5°, as shown in Fig. S7(d). For the case of Φ = 150°, all three Dirac cones along Γ-M1, Γ-M2 and Γ-M3 high symmetry lines are opened, ML PdBiO₃ behaves as a QAH insulator with a Chern number C = −3, as indicated in Fig.4(g)−(i). Under 0 ° − 86.5°, ML PdBiO₃ behaves as an QAH insulators with C = 3, as indicate in Fig. S7. Comparatively, under −86.5°−97.5°, presents an QAH states with C = 1. As a result, ML PdBiO₃ will vary among QAHE with C=\left|3\right|, intrinsic semimetal state and QAHE with C=\left|1\right| three topological phases as the magnetization direction angle Φ various in xz plane. Notably, this topological phase transition behavior depending on magnetization direction is also observed in PtBiO₃, as indicated in Fig. S8.

The magnetic and electric properties of 2D magnetic materials with transition metal elements is related to the electronic correlation strength of the system, which is affected by the strain and U value [74, 75]. Therefore, the topological characteristic of ML XBiO₃ under different strain and U values are studied. The strain can be defined as \epsilon =(\alpha -{\alpha }_0)/{\alpha }_0 [76, 77], where {\alpha }_0 and \alpha are lattice constants of ML XBiO₃ without and with strain, respectively. As shown in Fig.5(a), the energy difference (ΔE) between AFM and FM states increases gradually from −3% to 3% strains. Moreover, ΔE is always positive during the entire range of strains, indicating a FM coupling of ML XBiO₃. As described in Fig.5(b), the MAE also decreases from −3% to 3% strains, and keep a positive value, keeping an in-plane magnetic anisotropy. Furthermore, during the strain progress, the topological band gap of ML XBiO₃ gradually decreases, as plotted in Fig.5(c) and Fig. S9(a). Under 0−5 U values, the ΔE enlarge gradually with positive values and MAE reduce gradually with positive values, demonstrating a FM coupling and an in-plane magnetic anisotropy of ML XBiO₃, as described in Fig.5(d) and (e). Moreover, throughout the U values, ML XBiO₃ also keeps a QAHE, and the topological gap decreases gradually with the increases of U value, as indicated in Fig.5(f) and Fig. S9(b). The various U values would change the electronic correlation strength for ML XBiO₃ (X = Pd, Pt) [61, 78, 79]. In a word, the QAHE of ML XBiO₃ is robust for strain and U value. It has been established that the growth of materials on specific substrates inherently gives rise to strain [80]. For illustration, we construct a XBiO₃-MoS₂ heterostructure (1 × 1 XBiO₃ and \sqrt{3} \times \sqrt{3} MoS₂), in which the ML XBiO₃ is above the ML MoS₂ (see Fig. S10). After the structural relaxation, the in-plane lattice constants of ML PdBiO₃ and PtBiO₃ are enlarged by 1.3% and 0.3%, respectively.

Fig.6(a) shows the schematic of the QAHE measurement by varying the orientation of magnetization. Under the action of in-plane electric field, the transverse voltage (conductivity) can be detected by the regulating the magnetization of Pd/Pt atoms in ML XBiO₃ in xy and xz planes using an external magnetic field. The detected platformized conductivity indicates the QAH state of ML XBiO₃. Fig.6(b) displays the topological phases of ML XBiO₃. It can be found that ML XBiO₃ changes among QAHE with C=\left|3\right|, QAHE with C=\left|1\right| and topologically trivial semimetal state for different magnetization directions. Based on these analysis and above research results, the variation of Chern number and topological band gap of ML XBiO₃ with the magnetization direction can be obtained. As shown in Fig.6(c), when magnetization direction angle Ψ varying in the xy plane, the Chern number alternatively changes between C=1 and C=-1 with a period of 60°. Moreover, as Ψ = 30°, 90°, 150°, 210°, 270° and 330°, ML XBiO₃ presents a semimetal state. As shown in Fig.6(d), ML XBiO₃ presents a Chern number of 3 and −3 as Φ located in the range of 277.5°−86.5° and 97.5°−266.5°, respectively. Furthermore, ML PdBiO₃ presents a Chern number of 1 and −1 as Φ located in the range of 86.5°−97.5° and 266.5°−277.5°, respectively. Under Φ = 86.5°, 97.5°, 266.5° and 277.5°, ML XBiO₃ behave as a mixed state of semimetal and QAH state. As a result, the Chern number of ML PdBiO₃ can be regulated by controlling the magnetization direction, which provides an example for tuning the Chern number of QAH insulators.

4 Conclusion

In conclusion, the magnetic properties, electronic structures, and topological characteristics of monolayer XBiO₃ (X = Pd, Pt) are studied through the first-principles calculations. The monolayer XBiO₃ present ferromagnetic ground states with in-plane magnetic anisotropy. The critical magnetic transition temperature of monolayers PdBiO₃ and PtBiO₃ reach up to 432 and 550 K, respectively. The magnetic coupling feature of monolayer XBiO₃ are induced by the super-exchange interaction along X−O−X path. Interestingly, monolayer XBiO₃ can exhibit QAHE with tunable Chern number. When the magnetization direction changes in the xy plane, monolayer XBiO₃ switch between QAH insulator with C=\left|1\right| and topological trivial semimetal with a period of 60°. It is caused by the breaking or protecting mirror symmetries for different polar angle Ψ. Comparatively, as the magnetization direction alters in the xz plane, monolayer XBiO₃ vary among QAH insulator with C=\left|3\right|, QAH insulator with C=\left|1\right| as well as mixed semimetal and QAH state with a period of 180°. Therefore, the Chern number of monolayer XBiO₃ can be regulated by adjusting the magnetization direction. The topological band gap is as high as 114 and 132 meV for monolayers PdBiO₃ and PtBiO₃, respectively, matching with C=\pm 3. Notably, under −3% to 3% strains or 0−5 eV U values, QAHE keeps stable for the present systems. Our work provides an ideal platform to investigate the tunable high Chern number QAHE.

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