Tailoring the quantum anomalous layer Hall effect in multiferroic bilayers through sliding

doi:10.21203/rs.3.rs-3497689/v1

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Tailoring the quantum anomalous layer Hall effect in multiferroic bilayers through sliding

https://doi.org/10.21203/rs.3.rs-3497689/v1

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published 03 Jun, 2024

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Layer Hall effect (LHE) initially discovered in the magnetic topological insulator MnBi₂Te₄ film expands the Hall effect family and opens a promising avenue for layertronics applications. In this study, we present an innovative ferroelectric bilayer model to attain a tunable quantum anomalous layer Hall effect (QALHE). This model comprises two ferromagnetic orbital-active Dirac monolayers stacked antiferromagnetically, accompanied by out-of-plane electric polarization. The interplay between the layer-locked Berry curvature and the intrinsic out-of-plane electric polarization leads to layer-polarized near-quantized anomalous Hall conductance. Using first-principles calculations, we have identified a promising material for this model, namely FeS bilayer. Our calculations demonstrate that the intrinsic out-of-plane electric polarization in the Bernal-stacked FeS bilayer can induce QALHE by regulating the layer-locked Berry curvature of FeS monolayers. Importantly, the intrinsic electric filed can be reversed through interlayer sliding. The discovery of ferroelectrically modulated QALHE paves the way for the integrability and non-volatility of layertronics, offering exciting prospects for future applications.

Physical sciences/Physics/Condensed-matter physics/Ferroelectrics and multiferroics

Physical sciences/Physics/Condensed-matter physics/Topological matter/Topological insulators

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 effect^1–3, quantum anomalous Hall effect^4–7, valley Hall effect^{8, 9} and so on. These investigations not only expand our understanding of the Hall effect family but also hole the potential for practical applications in fields like quantum computing, spintronics, and more. The concept of Berry curvature, which describes the geometric phase of the electrons in momentum space^{10, 11}, plays a pivotal role in these phenomena. Specifically, the intricate interplay between Berry curvature and diverse electron degrees of freedom, such as charge, spin, and valley underlies the extraordinary electron transport behaviors observed in these exotic Hall effects, offering avenues for information encoding, processing, and storage^12–14. Consequently, the precise control of Berry curvature becomes imperative for achieving tunable Hall effects.

In some layered materials, electrons gain an additional layer degree of freedom corresponding to their confinement in the out-of-plane direction. The interplay between the Berry curvature and this layer degree gives rise to layer-locked Berry curvature, resulting in the layer Hall effect (LHE), as demonstrated in the topological antiferromagnetic MnBi₂Te₄ (MBT) thin film¹⁵. The layer-locked Berry curvature stems from their topologically non-trivial phases confined to the bulk or multilayer-stacked thin films^16–18, Notably, the induction of a layer-polarized net anomalous Hall conductance in these materials requires an external electric filed applied along the out-of-plane direction, effectively breaking the spatial inversion (P) and time reversal (T) symmetry. The quantum anomalous layer Hall effect (QALHE) proposed in MnBi₂Te₄ and other materials^19–21, holds great potential for the development of next-generation high-performance information devices.

Beside the MBT materials, there have been immense works on two-dimensional (2D) quantum anomalous Hall insulators (QAHIs) featured by the presence of Berry curvature monopoles and chiral edge states within a monolayer ^22–28. One notable subgroup within this category is the ferromagnetic orbital-active Dirac materials, in which the spin-orbit coupling (SOC) effect induces a topologically nontrivial band gap ^{29, 30}. The low-energy effective model of these materials can be described by the Haldane model⁴. Moreover, recent research has highlighted that electronic correlations can significantly enhance the effective SOC effects, resulting in an expansion of the topological bandgap and enhancing robustness against disorder and thermal perturbations³¹. The 2D QAHIs demonstrate the intriguing possibility of layer-locked Berry curvature, opening up a promising avenue for realizing QALHE. However, this phenomenon is rarely reported³².

Additionally, in the realm of nonvolatile layertronics devices, intrinsic electric polarization in QALHE materials is required to break the PT symmetry, in order to render a detectable anomalous Hall conductance. It has been demonstrated that the interlayer interaction in Van der Waals (vdW) materials can induce intrinsic out-of-plane electric polarization whose orientation can be dynamically reversed by interlay sliding (known as sliding ferroelectricity).^33–40 This inherent capability effectively enables the switch of layer-polarized net anomalous Hall conductance between different layers.^41–43 Therefore, investigating the intrinsic QALHE in ferroelectrically stacked 2D QAHI bilayers would offer significant prospects for non-volatile layertronics applications.

In this study, we propose a ferroelectric bilayer model composed of two ferromagnetic orbital-active Dirac monolayers for achieving intrinsic QALHE. The two monolayers in this mode are stacked antiferromagnetically, giving rise to a distinctive out-of-plane electric polarization. We demonstrate that the interplay between the layer-locked Berry curvature monopoles and the intrinsic out-of-plane electric polarization leads to layer-polarized near-quantized anomalous Hall conductance. Using first-principles calculations, we have identified a promising material, FeS monolayer ³¹, for this model. Our calculations verified that the intrinsic out-of-plane electric polarization in the Bernal-stacked FeS bilayer can induce QALHE by regulating the layer-locked Berry curvature of FeS monolayers. Furthermore, we demonstrated that the interlayer sliding can reverse the orientation of electric polarization and subsequently modulate the QALHE. These results not only offer a promising principle for design of QALHE materials but also pave the way for the integrability and non-volatility of layertronics.

TB Hamiltonian of the bilayer 2D QAHI

We begin by employing a Slater-Koster type tight-binding (TB) model to describe a single-layer ferromagnetic orbital-active honeycomb lattice, as depicted in Fig. 1a. This lattice is composed of two identical sublattices, A and B, with each site accommodating two distinct orbitals. Taking into account the nearest-neighbor (NN) hopping interactions between adjacent orbitals, we can express the Hamiltonian for this model as,

In this expression, i and j index the atomic sites, while a and b specify the atomic orbitals. \(t_{{ij}}^{{ab}}\) represents the NN hopping energy between distinct orbitals on different sites. \({\Delta _m}\) signifies the Zeeman splitting associated with the ferromagnetic order, and \({\sigma _z}\) refers to the Pauli matrix in the z direction for the spin subspace. The final term represents the on-site SOC effect with\(\lambda\)denoting its strength.

Within the C_3v point group symmetry of the honeycomb lattice, three distinct orbital doublets manifest: (p_x, p_y), (d_xz, d_yz), and (d_xy, d_x²_–y²), 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 (d_xz, d_yz) doublet. Considering the strong π-bond interaction between the d_xz/d_yz orbitals of different lattice points, we set \({V_{dd\pi }}\)= t and \({V_{dd\delta }}\)= 0.1t. Taking into account the ferromagnetic ground state and the strong on-site spin-orbit coupling, we set \({\Delta _m}\)= 4t and \(\lambda\)= 0.3t. 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. S1. 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 \(({\Delta _m})\) 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,

Here, \({t_ \bot }\) denotes the NN interlayer hopping. Considering the weak interlayer vdW interaction, we set \({t_ \bot }\)= 0.1t. 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 filed, with \(\delta\) being the electrostatic potential difference between the two layers.

In the absence of electric polarization (\(\delta =0\)), 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 hopping has a minimal impact on the quantum anomalous Hall conductance of each 2D QAHI monolayer.

Interestingly, upon introducing an interlayer potential difference \(\left| \delta \right|\) = 0.3t, 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, the 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. This material has been predicted to exhibit a correlation-enhanced substantial topological bandgap.³¹ The lattice structure of FeS monolayer comprises of two buckled honeycomb FeS sublayers linked by interlayer Fe-S bonds, as depicted in Fig. 3a. This configuration aligns with the previously synthesized 2D vdW MnSe material.⁴⁵. The transition metal atom Fe is exposed to a triangular prism crystal field with a C_3v point group. This field divides the 3d orbitals into a singlet state d_z² and two doublet states (d_xz, d_yz) and (d_xy, d_x²_–y²), as shown in Fig. 3d. Given the d⁶ electronic configuration of Fe²⁺ and the on-site Coulomb interaction, the doublet state (d_xz, d_yz) 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 substential band gap of about 0.63 eV at these Dirac points, as depicted in Fig. 3c, aligning with previously reported findings.³¹ 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.

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 of an antiferromagnetic interlayer alignment over a ferromagnetic one, with an energy difference of approximately 130 meV/Fe. 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 bilayer⁴⁶. 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 2.9 pC/m emerges, comparable to that of the MnSe⁴⁶ 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.

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 approximately 3.2 meV/f.u. 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 about 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 ferroelectrically stacked FeS bilayer.

It’s worth noting that both the AB and BA stacked FeS bilayers exhibit an indirect band gap, as shown in Fig. 4d. The the conduction band minimum (CBM) at the K (K₁ and K₂) valleys arises from the doublet state (d_xy, d_x²_–y²) 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. 3. 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 approximately 2% can shift the edge of the valence band at the K₁ and K₂ valleys, originating from the Fe atom’s doublet states (d_xz, d_yz), to a position above the valence band edge at the Γ point, as depicted in Supplementary Fig. 4a. Additionally, applying compressive strain along the out-of-plane direction can enhance the interlayer electric potential difference, as depicted in Supplementary Fig. 4b.

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 (d_xz, d_yz), 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. 5. 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. 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 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.

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

In summary, we proposed a ferroelectric bilayer model, which comprises two ferromagnetic orbital-active Dirac monolayers, to achieve a tunable QALHE. The two Dirac monolayers are stacked antiferromagnetically, accompanied by out-of-plane electric polarization. We demonstrated that the interplay between the layer-locked Berry curvature monopoles and the intrinsic out-of-plane electric polarization leads to layer-polarized quantized anomalous Hall conductance. Based on first-principles calculations, we also identified a multiferroic FeS bilayer for this model. Our calculations demonstrate that the intrinsic out-of-plane electric polarization in the Bernal-stacked FeS bilayer can induce QALHE by regulating the layer-locked Berry curvature of FeS monolayers. Importantly, the intrinsic electric filed can be reversed through interlayer sliding. The discovery of ferroelectrically modulated QALHE paves the way for the integrability and non-volatility of layertronics, offering exciting prospects for future applications.

We conducted first-principles calculations using the Vienna Ab Initio Simulation Package (VASP) within the framework of density functional theory (DFT)^{47, 48}. The exchange-correlation functional employed the generalized gradient approximation of Perdew-Burke-Ernzerhof (GGA-PBE)⁴⁹. Interactions between ions and valence electrons were described using the projector-augmented wave (PAW) method⁵⁰. The electron wavefunctions utilized a plane-wave basis with a kinetic energy cutoff set at 500 eV. To eliminate unwanted interactions between images, we introduced a 20 Å vacuum space in the perpendicular direction. For geometry optimization and electronic structure calculations, we sampled the Brillouin zone using Г-centered 11×11×1 and 13×13×1 Monkhorst-Pack k meshes⁵¹, respectively. We ensured the geometry was fully optimized when forces on each atom were below 0.01 eV/Å, and set the electronic self-consistency convergence threshold at 10^− 6 eV. To address on-site Coulomb interactions, we employed the GGA + U method⁵² with a U_eff value of 3.0 eV for Fe's 3d electron, aligning with prior studies. The climbing image nudged elastic band (CINEB)⁵³ technique estimated the ferroelectric switching energy barrier, while the Berry phase method^{54, 55} assessed electric polarization. We constructed the Wannier representation by projecting first-principles Bloch states using Wannier90^56–58. The tight-binding models, built from the Wannier representation, facilitated calculations of edge states, Berry curvature, and anomalous hall using the WannierTools package⁵⁹.

DATA AVAILABILITY

The data supporting the findings of this study are available within this article and its Supplementary Information. Additional data that support the findings of this study are available from the corresponding author on reasonable requests.

CODE AVAILABILITY

The related codes are available from the corresponding author on reasonable requests.

ACKNOWLEDGEMENTS

This work is supported by the National Natural Science Foundation of China (No. 12074218) and the Taishan Scholar Program of Shandong Province.

AUTHOR CONTRIBUTIONS

M.Z. conceived the project. K. L. and X.M. performed the first-principles calculations and conducted theoretical analysis. Y.L. conducted the theoretical analysis. K. L. and M.Z. wrote the manuscript.

COMPETING INTERESTS

The authors declare no competing interests.

ADDITIONAL INFORMATION

Supplementary information The online version contains supplementary material available at

Correspondence and requests for materials should be addressed to Mingwen Zhao.

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Tailoring the quantum anomalous layer Hall effect in multiferroic bilayers through sliding

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INTRODUCTION

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TB Hamiltonian of the bilayer 2D QAHI

Atomic and electronic structure of FeS monolayer

Ferroelectricity and tunable QALHE in FeS bilayer

DISCUSSION

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