Introduction

Advancements in condensed matter physics, materials science, and electronics have paved the way for the discovery and exploration of various innovative Hall effects, including quantum spin Hall effect1,2,3, quantum anomalous Hall effect4,5,6,7, valley Hall effect

$${H}_{{mono}}=\sum _{ < {ij} > ,n,{n}^{{\prime} }}{c}_{i,n}^{+}{t}_{i,j}^{n{n}^{{\prime} }}{c}_{j,{n}^{{\prime} }}+{\triangle }_{m}\sum _{i,n}{c}_{i,n}^{+}{\sigma }_{z}{c}_{i,n}+{\rm{\lambda }}\sum _{i,n}{c}_{i,n}^{+}\left({\boldsymbol{l}}\,\cdot \,{\boldsymbol{\sigma }}\right){c}_{i,n}.$$
(1)
Fig. 1: Single-layer ferromagnetic orbital-active honeycomb lattice.
figure 1

a Honeycomb lattice structure with A and B representing the two sublattices. b Tight-binding model band structure with and without spin-orbit coupling. c Band structure and anomalous Hall conductance for opposite ferromagnetic orientations. In both b and c, the color bar indicates the magnitude of the associated Berry curvature for the bands, with red signifying positive values and blue indicating negative values.

Full size image

In this expression, i and j index the atomic sites, while n and n’ specify the atomic orbitals. \({c}_{i,n}^{+}({c}_{i,n}\)) denotes the creation (annihilation) operator at ith-site and n-orbital of the monolayer. \({t}_{i,j}^{n{n}^{{\prime} }}\) represents the NN hop** energy between distinct orbitals on different sites. Δm signifies the Zeeman splitting associated with the ferromagnetic order, and σz refers to the Pauli matrix in the z direction for the spin subspace. The final term represents the on-site SOC effect with λ denoting its strength.

Within the C3v point group symmetry of the honeycomb lattice, three distinct orbital doublets manifest: (px, py), (dxz, dyz), and (dxy, \({d}_{{{\rm{x}}^{2}}-}{\,}_{{\rm{y}}^{2}}\)), which correspond to the same 2D irreducible representation30. Consequently, they display identical electronic band structures and wavefunction forms according to Eq.(1), establishing them as symmetrical equivalents. Given the pronounced correlation interactions inherent to the d orbitals, our study primarily focuses on the (dxz, dyz) doublet, i.e., \(n,{n}^{{\prime} }\in \{{{d}}_{{\rm{xz}}},{d}_{{\rm{yz}}}\}\). Considering the strong π-bond interaction between the dxz/dyz orbitals of different lattice points, we set Vddπ = t and Vddδ = 0.1 t. Taking into account the ferromagnetic ground state and the strong on-site spin-orbit coupling, we set Δm = 4 t and λ = 0.3 t. The band structure, depicted in Fig. 1b, reveals Dirac points positioned at the corners of Brillouin zone (K1 and K2) in the absence of SOC. The incorporation of SOC induces a topologically nontrivial band gap at these Dirac points, accompanied by the emergence of Berry curvature monopoles11 and chiral edge states, which shown in Supplementary Fig. 1. By evaluating the integral of the Berry curvature of the occupied states over the entire Brillouin zone, we can determine the Chern number (C), which corresponds to the anomalous Hall conductance of the system. As illustrated in Fig. 1c, varying the Fermi level within the band gap results in a Chern number of 1, thus characterizing the system as a 2D QAHI. Notably, reversing the magnetization of the system by altering the Zeeman splitting (Δm) leads to the reversal of both the Berry curvature and anomalous Hall conductance, implying the potential of achieving layer-locked Berry curvature in this 2D QAHIs.

We then consider a bilayer model comprising two 2D QAHI monolayers described by Eq. (1). The two monolayers have identical lattice structures but opposite magnetizations, as depicted in Fig. 2a. The A site in the upper layer perfectly aligns with the B site in the lower layer, showcasing the characteristic of Bernal stacking pattern. The TB Hamiltonian of the bilayer 2D QAHI model is expressed as,

$$\begin{array}{ll}{H}_{{bi}}={H}_{{mono}}^{{top}}+{H}_{{mono}}^{{bottom}}+\mathop{\sum}\limits_{ < {ij} > ,n,{n}^{{\prime} }}\left({\widetilde{c}}_{i,n}^{+}{t}_{\perp }{c}_{j,{n}^{{\prime} }}+{c}_{j,{n}^{{\prime} }}^{+}{t}_{\perp }{\widetilde{c}}_{i,n}\right)\\\qquad\qquad\qquad+\,\frac{\delta }{2}\sum \limits_{i,n}\left({\widetilde{c}}_{i,n}^{+}{\widetilde{c}}_{i,n}-{c}_{i,n}^{+}{c}_{i,n}\right).\end{array}$$
(2)
Fig. 2: Bernal-stacked orbital-active bilayer honeycomb lattice.
figure 2

a Lattice structure. Dashed lines and solid lines indicate the top and bottom layers, respectively. b Tight-binding model band structure and anomalous Hall conductance with opposite interlayer potentials of antiferromagnetic bilayer Bernal-stacked orbital-active honeycomb lattice. The color bar represents the magnitude of the associated Berry curvature for the bands, with red indicating positive values and blue denoting negative values. In the anomalous hall conductance plot, the red and blue dashed lines, respectively, represent the projected values for the top and bottom layers. c Schematic diagram of electrically manipulating layer-polarized net Berry curvature and anomalous Hall conductance.

Full size image

Here, \({t}_{\perp }\) denotes the NN interlayer hop**, \({\widetilde{c}}_{i,n}^{+}{(\widetilde{c}}_{i,n})\) and \({c}_{i,n}^{+}({c}_{i,n}\)) represents the creation (annihilation) operators at ith-site and n-orbital of the top and bottom monolayers, respectively. Considering the weak interlayer vdW interaction, we set \({t}_{\perp }\,\)= 0.1 t. The last term accounts for the effects of electric polarization along the out-of-plane direction arising from the intrinsic electric polarization or external electric field, with δ being the electrostatic potential difference between the two layers.

In the absence of electric polarization (δ = 0), the Berry curvatures and Hall quantum conductance from the two antiferromagnetically stacked monolayers cancel out each other, due to the constrains of PT symmetry, in analogous to the cases of quantum spin Hall insulators (QSHIs)44. Further layer-resolved anomalous Hall conductance calculations indicate that the interlayer hop** has a minimal impact on the quantum anomalous Hall conductance of each 2D QAHI monolayer.

Interestingly, upon introducing an interlayer potential difference |δ| = 0.3 t, as depicted in Fig. 2b, the bands of the two layers undergo a relative shift, lifting the P symmetry of the system. When the Fermi level lies between the band edges of the two layers, as shown in the shaded region of Fig. 2b, the Berry curvature of the two monolayers cannot cancel out each other, leading to net layer-polarized Berry curvature and consequently nonzero Hall conductance within this energy region. Notably, the net Berry curvature exhibits dependence on the orientation of electric polarization. Specifically, reversing the electric polarization orientation of the bilayer leads to the reversal of both the net Berry curvature and anomalous Hall conductance, accompanied by the switch of layer degree between the two layers, as shown in Fig. 2c. Considering that the out-of-plane electric polarization in a bilayer can be induced by interlayer interaction and regulated by interlayer sliding41, this intriguing scenario offers a promising approach for achieving QALHE.

It should be mentioned that the magnitude of the net Berry curvature in the bilayer depends on the distribution of the Berry curvature in the two monolayers and cannot guarantee an integral Chern number. The maximal Hall conductance given by the Hamiltonian of Eq. (2) is nearly the half of integral quantum anomalous Hall conductance corresponding to C = 1. Nevertheless, this value significantly exceeds that of classical Hall effects. In view of its quantum origination, we will continue to refer to it as QALHE.

Atomic and electronic structure of FeS monolayer

To validate the feasibility of the model described above, we explored the properties of a 2D vdW single-layer ferromagnetic material known as FeS, as designed in the computational 2D Materials Database (C2DB)45,46. This material has been predicted to exhibit a correlation-enhanced substantial topological band gap31. The lattice structure of FeS monolayer comprises two buckled honeycomb FeS sublayers linked by interlayer FeS bonds, as depicted in Fig. 3a. This configuration aligns with the previously synthesized 2D vdW MnSe material47. Furthermore, the phonon spectrum and molecular dynamics calculations for the monolayer FeS, presented in Supplementary Fig. 3, demonstrate its structural and thermodynamic stability, supporting its feasibility for experimental synthesis. The transition metal atom Fe is exposed to a triangular prism crystal field with a C3v point group. This field divides the 3d orbitals into a singlet state \({d_{z^{2}}}\) and two doublet states (dxz, dyz) and (dxy, \({d}_{{{\rm{x}}^{2}}-}{\,}_{{\rm{y}}^{2}}\)), as shown in Fig. 3d. Given the d6 electronic configuration of Fe2+ and the on-site Coulomb interaction, the doublet state (dxz, dyz) is partially occupied, forming spin-polarized Dirac points at the Fermi level in the K1 and K2 valleys, as illustrated in Fig. 3b. The inclusion of SOC induces a substantial band gap of ~0.63 eV at these Dirac points, as depicted in Fig. 3c, aligning with previously reported findings31. The 2D QAHI aspects of the ferromagnetic FeS monolayer can be verified from the quantized anomalous Hall conductance within the band gap corresponding to C = 1, as illustrated in Fig. 3c, the Berry curvature distribution at the K1 and K2 valleys, as shown in Fig. 3e, and the presence of chiral edge states that connect the valence and conduction bands of FeS monolayer, as depicted in Fig. 3f. Additionally, we employed the HSE06 hybrid functional to compute the electronic structure of the monolayer FeS, as depicted in Supplementary Fig. 4, which validate the existence of large topological nontrivial band gap.

Fig. 3: Lattice and electronic structures of FeS monolayer.
figure 3

a Lattice structure of FeS monolayer with the red dashed line indicating the unit cell. b Spin-polarized band structure without spin-orbit coupling. c Band structure and anomalous Hall conductance upon introducing spin-orbit coupling. d Schematic illustration of the occupation of the d orbitals of the Fe atom. e Distribution of Berry curvature in reciprocal space, with the red dashed line denoting the first Brillouin zone. f Edge states of the monolayer nanoribbon.

Full size image

Ferroelectricity and tunable QALHE in FeS bilayer

We then considered a FeS bilayer, consisting of two FeS monolayers arranged in different stacking patterns. Our calculations reveal a preference for an antiferromagnetic interlayer alignment over a ferromagnetic one, with an energy difference of ~130 meV per Fe atom. Exploring different stacking configurations for the FeS bilayer, we identified two energetically most favorable options, denoted as AB and BA, which are energetically degenerate, as depicted in Fig. 4a, similar to the scenario observed in MnSe bilayer48. In the AB stacking configuration, the Fe atoms in the upper monolayer align directly above the S atoms in the lower monolayer, whereas the Fe atoms in the lower monolayer are positioned directly beneath the hexagonal centers of the upper monolayer. In both stacking patterns, the inversion symmetry (P) is lifted, resulting in charge transfer between the monolayers, as illustrated in the inset of Fig. 4b. Consequently, an out-of-plane electric polarization of 4.1 pC m−1 emerges, comparable to that of the MnSe48 and BN41 bilayers. The inherent electric polarization in the FeS bilayer is manifested as an electrostatic potential difference between the monolayers, as shown in Fig. 4b.

Fig. 4: Antiferromagnetic FeS bilayer.
figure 4

a Energy profile of stacking as a function of interlayer sliding, referenced to the AA stacking in an antiferromagnetic FeS bilayer. b Plane-averaged electric potential for both AB and BA stacking patterns. Insets showcase the associated charge density difference; red and green isosurfaces, respectively, represent electron accumulation and depletion. c Pathway of ferroelectric switching and the variation in electric polarization during interlayer sliding between AB and BA configurations. d Spin-polarized band structure with SOC for both AB and BA stacking patterns.

Full size image

The BA stacking configuration presents a similar scenario, albeit with a reversal in both charge transfer direction and electrostatic potential difference. Utilizing the nudge-elastic-band (NEB) method, depicted in Fig. 4c, we ascertained that the FeS bilayer can transition between the AB and BA configurations through interlayer sliding, encountering an energy barrier of ~6.4 meV per formula. The observed variation in electric polarization during this sliding process demonstrates the feasibility of electrical switching between the two Bernal configurations. Furthermore, as depicted in Fig. 4d, the interlayer potential difference of ~0.13 V induces a comparable level of splitting in the monolayer bands with distinct spin polarizations. A significant band splitting is also evident at the K valley edges. Given the pronounced Berry curvature at the K valley in the monolayer FeS, which aligns with layer-locked Berry curvature, this suggests the potential for realizing an intrinsic Layer Hall effect in a ferroelectrically stacked FeS bilayer. Moreover, as depicted in Supplementary Fig. 6, the results obtained with the PBE + D3 functional demonstrate that the impact of different van der Waals corrections on the electronic structure of the ferroelectric polarization structure can be deemed negligible. This observation underscores the robustness of the layer Hall effect induced by the intrinsic electric field.

It’s worth noting that both the AB and BA stacked FeS bilayers exhibit an indirect band gap, as shown in Fig. 4d. The conduction band minimum (CBM) at the K (K1 and K2) valleys arises from the doublet state (dxy, \({d}_{{{\rm{x}}^{2}}-}{\,}_{{\rm{y}}^{2}}\)) of the Fe atoms, whereas the valence band maximum (VBM) originates from the p orbitals of the S atoms and is located at the Γ point, as illustrated in Supplementary Fig. 7. This electronic band structure differs significantly from the electronic band structure given by the Hamiltonian of Eq. (2). Fortunately, the electronic band structure of the FeS bilayer can be effectively regulated by applying external strain. A subtle biaxial tensile strain of ~2% can shift the edge of the valence band at the K1 and K2 valleys, originating from the Fe atom’s doublet states (dxz, dyz), to a position above the valence band edge at the Γ point, as depicted in Supplementary Fig. 8a. Additionally, applying compressive strain along the out-of-plane direction can enhance the interlayer electric potential difference, as depicted in Supplementary Fig. 8b.

We computed the phonon spectrum of the FeS bilayer under the combined action of in-plane and out-of-plane strains, as shown in Supplementary Fig. 10a. The absence of any imaginary frequencies indicates the structural stability of the bilayer FeS under these conditions. Additionally, we have calculated the energy variation along the ferroelectric flip** pathway under strain in Supplementary Fig. 10b. The flip** barrier is found to be 13 meV per formula, slightly larger than that without applied strain, yet still within an experimentally achievable range.

We therefore examined a strained Bernal-stacked FeS bilayer subjected to a 2% in-plane biaxial tensile strain and a 2% compressive strain along the out-of-plane direction, as showcased in Fig. 5a. The electronic band structures of both AB and BA stacked FeS bilayers near the K1 and K2 valleys are consistent with the results of our bilayer model. Specifically, both the VBM and CBM, stemming from the Fe atom’s doublet states (dxz, dyz), are situated at the K1 and K2 valleys, accompanied by the presence of topologically nontrivial band gap due to SOC, as highlighted in Supplementary Fig. 9. Further computations unveiled that when the Fermi level is regulated to the region between the band edges of the two FeS monolayers, anomalous Hall conductance approaching quantized values emerges, as depicted in the shaded region of Fig. 5a. The regulation of the Fermi level position can be achieved by electron/hole do** to the FeS bilayer via a dual-gated device15. When the Fermi level falls within the band gap of the FeS bilayer, the Berry curvature of the two FeS monolayers cancels each other out, resulting in zero anomalous Hall conductance. Changing the orientation of the intrinsic electric polarization by interlayer sliding, which transforms the stacking pattern from AB to BA, reverses both the net Berry curvature and Hall conductance signs, as shown in Fig. 5b. Notably, the layer contributing to the layer-polarized net Berry curvature also switches, indicating that the intrinsic QALHE in the FeS bilayer system can be modulated by ferroelectric polarization. These findings align well with the predictions of our TB model.

Fig. 5: FeS bilayers under strains.
figure 5

a Band structure, associated Berry curvature, and anomalous hall conductance of AB and BA stacked FeS bilayer subjected to a 2% in-plane tensile strain and a 2% out-of-plane compression. The red and blue dashed lines, respectively, represent the projected anomalous hall conductance for the bottom and top layers of FeS. b Schematic diagram of intrinsic ferroelectric polarization induced layer-polarized net Berry curvature and anomalous Hall conductance in Bernal-stacked FeS bilayer.

Full size image

To access the robustness of the topological properties against the electron/hole do** effects, we calculated the Berry curvature-projected band structures and anomalous Hall conductivity of ferroelectric FeS bilayers, as shown in Supplementary Fig. 11. The selected electron/hole do** concentrations were set at 2.3 × 1014 cm−2, regulating the Fermi level to the positions suitable for achieving anomalous Hall effect. The results demonstrate that the topological properties remain well-preserved in the doped FeS bilayer.

Finally, it is important to highlight that the quantum anomalous Hall conductance exhibits two distinct peaks with opposite signs in the valence and conduction band regions, as depicted in Figs. 2 and 5. These dual peaks correspond to the opposite spin polarization and distinct electron layer degrees, presenting an alternative method for regulating the QALHE.