Electrically tunable anomalous Hall conductivity in ferrovalley–ferroelectric heterostructure VSe₂/Sc₂CO₂

Abstract

Electrical control of anomalous Hall conductivity (AHC) and valley polarisation are necessary to realize energy-efficient devices for information storage that exploit the valley degrees of freedom in a material. Robustness in electrical control is easily realizable in ferroelectric materials having bistable electric polarisations, while manipulation of valley properties is possible in ferrovalley materials. Combining two such materials would make a valleytronics device that is electrically controllable. In this work, we investigate the possibility of the electric field-induced switching of the AHC and spin–orbit coupling-induced valley polarization in the two-dimensional (2D) ferrovalley/ferroelectric VSe₂/Sc₂CO₂ heterostructure. We find that the valley degeneracy in this system is lifted by intrinsic spin–orbit coupling, leading to a substantial valley polarization and AHC. Upon application of an electric field, valley polarisation is significantly modified along with a sign reversal of AHC when the direction of the electric field is reversed. Modifications in the band structure leading to changes in the valley occupancy under the influence of an electric field are responsible for these. Our finding suggests that electric field-induced AHC switching is possible in this 2D heterostructure without the introduction of dopants or defects.

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1. Introduction

The presence of multiple local energy extrema or valleys in the electronic band structures of hexagonal two-dimensional (2D) materials and their manipulation by external stimuli enable faster information processing, increased storage capacity, and reduced energy consumption. In these materials, the control and manipulation of valleys often require lifting of valley degeneracy at high symmetry K and K′ points in the Brillouin zone.¹ The lifting of valley degeneracy or valley splitting leads to novel valley-selective transport phenomena and anomalous Hall effect (AHE).^2,3 The AHE requires the breaking of time-reversal symmetry (TRS) as it makes one valley dominant in transport over the other. Introduction of external dopants,^4,5 substrates,⁶ magnetic fields⁷ are the conventional methods for breaking TRS. However, the chemical way of influencing the valley polarisation introduces impurities in the structure. On the other hand, even a high magnetic field, in spite of facilitating high power consumption, leads to valley splitting of only a few meV, reducing the possibility of realizing devices based on valley transport. For example, efforts to effect valley-dependent transport have been made in 2D transition metal dichalcogenides (TMDCs) where TRS is broken due to intrinsic spin–orbit coupling (SOC) in them.^8–11 The valley polarisation obtained is of a few meV order. Thus, precise control of valley-dependent phenomena is difficult this way.

Coexistence of valley polarisation and ferromagnetism, controllable with external perturbative effects leading to sizable valley-dependent properties like contrasting Berry curvature and Anomalous Hall conductivity (AHC) are desirable. To this end, TMDC based van der Waals(vdW) heterostructures such as WS₂/CrI₃,^12,13 WSe₂/CrInSe₃,¹⁴ MoTe₂/CrBr₃,¹⁵ WSe₂/1T-VSe₂¹⁶ are investigated. In order to obtain significant valley-dependent effects, (i) the components of the heterostructures should have a small lattice mismatch, (ii) the ground state of the heterostructure should be semiconducting, and (iii) sizable valley polarisation should be observed. The TMDC-based vdW heterostructures of ref. 12–16, however, either have metallic ground states or the components have significant lattice mismatch; the valley polarization in them is of the order of a few meV. A solution to this can be obtained in ferrovalley (FV) materials that exhibit intrinsic ferromagnetism and valley polarisation spontaneously.^17,18 Among 2D TMDCs, the H-phase of VSe₂ is the first to exhibit ferrovalley characteristics.^7,19 This implies that it can be used to design magneto-valley (MV) logic devices without the application of an external magnetic field. The ferrovalley materials, thus, open up new avenues to realize valleytronics devices with desired qualities.

On the other hand, precise control of switching AHC, valley population, inter-valley transitions, and magneto-valley coupling (MVC) can be done with the application of an external electric field.^20–24 The electrical tuning becomes more accessible when a 2D ferrovalley material combines with a ferroelectric (FE) to form vdW heterostructure.^25,26 This vdW heterostructure is expected to break both inversion and time-reversal symmetry. The external electric field, applied on the heterostructure, can easily switch the direction of valley-dependent anomalous properties due to the presence of an FE component.

In this work, we have adopted this approach by constructing a heterostructure comprising of monolayers of H-VSe₂, the FV constituent, and Sc₂CO₂, a member of the MXene family, the FE constituent. The motivations behind such choice are as follows: (a) H-VSe₂, exhibiting room temperature ferromagnetism, is synthesized experimentally,²⁷ (b) Sc₂CO₂ MXene turns out to be one with a large electric polarisation of 1.6 µC cm⁻² (ref. 28–31) obtained by first-principles density functional theory (DFT)³² calculations. Though this MXene is not yet synthesized but has a distinct possibility of being exfoliated from Sc-based MAX compounds, (c) the lattice mismatch between the constituents is only 2.4% and (d) the ground state is semiconducting for a fixed FE polarization. By switching the polarisation, a semiconductor to half-metal transition is obtained.³³ This adds to the required flexibilities for tuning the valley-related properties in this heterostructure.

In what follows, with the help of DFT-based calculations, we examine the degree of valley polarisation, Berry curvature, and AHC and their tunability in the presence of an external electric field. We find intrinsic switching of directions and spontaneous valley-dependent phenomena in the absence of any external stimulus, indicating strong MVC in this system. We exploit it to control the ground state electronic structure and subsequent valley transport properties under the influence of an external electric field.

2. Computational details

All calculations are carried out using the framework of density functional theory (DFT).³² The electron–ion interactions are modeled by the projector augmented wave (PAW) method³⁴ as implemented in the Vienna Ab Initio Simulation Package (VASP).³⁵ The exchange–correlation part of the Hamiltonian is approximated using the Perdew–Burke–Ernzerhof (PBE) parameterization³⁶ of the generalized gradient approximation (GGA). The spin–orbit coupling (SOC) is included in the calculations. Plane waves up to 550 eV are used throughout. For structural optimizations, the energy and force convergence criteria are set to 10⁻⁶ eV and 0.01 eV Å⁻¹, respectively. The DFT-D3 method³⁷ is employed to account for van der Waals interactions. The Brillouin zone is sampled with a 14 × 14 × 1 Monkhorst–Pack k-point grid.³⁸ Dipole corrections are incorporated into all calculations. A 20 Å vacuum is introduced along the z-direction to prevent interactions between adjacent layers. The dynamical stability of the systems considered is verified by computing their phonon dispersion relations by the supercell method as implemented in the Phonopy package.³⁹ A 3 × 3 × 1 supercell is used to compute the phonons. The convergence criteria for energy is kept at a high value of 10⁻⁸ eV to achieve good convergence.

The Hall conductivity and Berry curvature are investigated using Wannier tight binding Hamiltonians with projections from the d orbitals of V and Sc, p orbitals of Se, C, and O atoms, computed using Wannier90.⁴⁰ The adaptively refined k-mesh is set to 5 × 5 × 1 to transform the Bloch functions into maximally localized Wannier functions (MLWF).⁴¹ The AHC and Berry curvature are subsequently calculated with a dense k-mesh of 600 × 600 × 1 using WannierTools.⁴² The ferroelectric switching pathway and energy barrier of monolayer Sc₂CO₂ and the vdW heterostructure are calculated using climbing-image nudged elastic band (CI-NEB) method.⁴³

3. Results and discussions

3.1. Structure and stability

Monolayer 2H-VSe₂ is a quasi-2D material with a layered structure. The structure, from two perspectives, is shown in Fig. 1(a). In this structure, each V(Se) atom is surrounded by six(three) Se(V) atoms. The optimized lattice parameter from our calculations is 3.33 Å, close to the available result.⁴⁴ The calculated magnetic moment of VSe₂ is 1µ_B. The crystal structure of Sc₂CO₂ (both top and side view) is shown in Fig. 1(b). As can be observed, the top(bottom) surface passivating O atom occupies the hollow site associated with the C(Sc) atom; the top (bottom) refers to the surface along the z(−z) direction. Due to such asymmetry with regard to the occupation of O atoms on the surfaces, inversion symmetry is broken, giving rise to a large out-of-plane electric polarisation.³⁰ After structural optimization, the lattice parameter of Sc₂CO₂ is found to be 3.41 Å, in good agreement with the existing results.⁴⁵

Fig. 1 Top and side view of (a) VSe₂ (b) Sc₂CO₂ crystal structure. (c)–(e) show heterostructure stacking in model AA, BB and CC respectively. (f) The phonon spectra of the heterostructure in model AA.

The lattice mismatch of only 2.4% between these two makes the construction of a vdW heterostructure by stacking these two possible. Ferroelectric Sc₂CO₂ has two states of polarisation: P↑ when C atoms are closer to the top surface, and P↓ when they are closer to the bottom surface. From our previous work,³³ we find that when monolayer VSe₂ is interfaced with P↓ (P↑) state of monolayer Sc₂CO₂, the electronic ground state of the heterostructure is a half-metal(semiconductor). In this work, we consider the heterostructure formed with P↑ state of Sc₂CO₂, as a semiconductor is required for Valleytronics device application.

Three different stacking models are considered to find out the optimized configuration of the heterostructure: (a) model AA, where the O atom closest to VSe₂ is directly above the V atom (Fig. 1(c)); (b) model BB, where the O atom closest to VSe₂ is directly above the Se atom (Fig. 1(d)); and (c) model CC, where the O atom closest to VSe₂ is above the V–Se bond (Fig. 1(e)). The stability of the heterostructures is assessed by calculating their binding energies given by:

E_b = (E_hetero − E_VSe2 − E_Sc2CO2)/A

The total energy of the heterostructure is denoted as E_hetero. E_VSe2 and E_Sc2CO2 represent the energies of the individual monolayers of VSe₂ and Sc₂CO₂, respectively. A refers to the surface area of the heterostructure. The binding energies are presented in Table 1. From the results, it is evident that model AA is the lowest energy configuration. The dynamical stability of this heterostructure is assessed by computing the phonon dispersion. The absence of any imaginary phonon mode (Fig. 1(f)) in the phonon spectra indicates the dynamical stability of the heterostructure in AA stacking configuration. The magnetic moment of V-atoms in the heterostructure stays almost equal to 1µ_B since the charge transfer across the interface is only 0.004e.³³

Table 1 Binding energies of VSe₂/Sc₂CO₂ vdw heterostructure in different stacking configurations

Stacking	E b (meV Å^−2)
AA	−19.84
BB	−13.27
CC	−17.8

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3.2. Electronic structure and valley polarisation

In Fig. 2(a), we show the total and atom-resolved densities of states of the heterostructure. The gaps at both spin channels cutting across the Fermi level indicate the semiconducting nature of the heterostructure. The states close to the gap, on either side, are mostly contributed by the V-atoms. In Fig. 2(b), we present the spin-resolved band structure of the heterostructure. The valence band maxima(VBM), at the Γ point, is in the spin-up band, while the conduction band minima (CBM), at the M point, is in the spin-down band. This implies that the system is an indirect band gap semiconductor; the band gap is only 127.5 meV. Prominent valleys, degenerate at K and K′, are observed in both occupied and unoccupied parts of the spectrum. The valley degeneracy is lifted upon the inclusion of SOC (Fig. 2(c)). The band structure suggests that though valleys associated with both valence and conduction bands split, the splitting is more prominent in the valence band. This implies that the valley polarization will be more significant in the valence band. The valley splitting in the valence band, , is −79.2 meV. ΔE_v < 0 implies greater stability of K′ valley in the valence band. The valley splitting in the conduction band, is only 14.0 meV. Consequently, the valley polarization Δ_KK′ = ΔE_c − ΔE_v = 93.2 meV, significantly large in comparison with several ferrovalley heterostructures (Δ_KK′ for WS₂/CrI₃, WSe₂/CrSnSe₃, WS₂/CrBr₃, WSe₂/CrBr₃, MoSe₂/CrBr₃,MoS₂/CrI₃,MoSe₂/CrI₃,MoTe₂/CrI₃, MoTe₂/CrBr₃,MoS₂/CrBr₃ and WSe₂/VSe₂ are 1.6 meV,^12,13 9 meV,¹⁴ 1.4 meV,¹⁵ 15.2 meV,¹⁵ 5.6 meV,¹⁵ 2.1 meV,¹⁵ 3.5 meV,¹⁵ 5.9 meV,¹⁵ 28.7 meV,¹⁵ 1.3 meV¹⁵ and 9 meV,¹⁶ respectively.) The large valley polarisation driven by the SOC shows the presence of substantial MVC in our system.

Fig. 2 (a) Total and atom projected densities of states of VSe₂/Sc₂CO₂ heterostructure. (b) and (c) are the electronic band structure of the system without and with SOC, respectively.(d) The variations in electronic band gap and charge transfer as a function of applied electric field.

3.3. Electric field tunable valley polarisation

In this section, we investigate the effects of an external electric field on the valley polarization. We consider an out-of-plane external electric field that is applied along VSe₂ → Sc₂CO₂ (along +z -direction). We compute the relevant quantities by varying the external electric field in the range of −0.5 V Å⁻¹ to +0.5 V Å⁻¹ (The '+' sign means the field is directed along +z direction while the '−' sign means the field is reversed). Within this range, the state of ferroelectric polarization of the heterostructure does not change from P↑ to P↓ as it requires overcoming a potential barrier of 0.96 eV as shown in the Fig. S5. In Table 2, we show the valley splittings in valence and conduction bands and the resulting valley polarisations as the applied electric field changes its magnitude and direction. We find that ΔE_c, the splitting in the conduction band valley, irrespective of the direction of the electric field, is almost unchanged. ΔE_v, on the other hand, changes substantially. When the electric field is along +z direction, ΔE_v undergoes little change. Upon reversal of the field, the splitting reduces significantly in the range −0.2 V Å⁻¹ to −0.5 V Å⁻¹; the minimum ΔE_v being −24.07 meV for an electric field of −0.5 V Å⁻¹. Consequently, Δ_KK′, the valley polarisation reduces considerably; while it is 92.87 meV for an electric field of 0.5 V Å⁻¹, a low value of 38.08 meV is obtained when the field is −0.5 V Å⁻¹. The band structures demonstrating the splittings in the bands for electric fields of 0.5 V Å⁻¹ and −0.5 V Å⁻¹ are shown in Fig. S1, SI.

Table 2 Valley splitting of valence (ΔE_v) and conduction band (ΔE_c), and the valley polarisation (Δ_KK′) at different values and directions of applied electric field. The values are in meV unit

Electric field (V Å^−1)	ΔEv	ΔEc	ΔKK′	Electric field (V Å^−1)	ΔEv	ΔEc	ΔKK′
−0.5	−24.07	14.01	38.08	0.5	−78.77	14.10	92.87
−0.4	−33.61	14.18	47.79	0.4	−72.70	12.86	85.56
−0.3	−44.46	14.10	58.56	0.3	−76.68	13.91	90.59
−0.2	−65.13	14.44	79.57	0.2	−77.46	14.44	91.90
−0.1	−76.92	13.91	90.83	0.1	−76.68	14.18	90.86

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A clue to understanding the suppression of valley splitting in the valence band and subsequent reduction in valley polarisation can be discovered by examining Fig. S1, SI. While the valleys associated with the VBM are distinct when the system is subjected to an electric field of 0.5 V Å⁻¹, they are broader and less distinct due to the presence of bands with nearly the same energies when the field is reversed to −0.5 V Å⁻¹. Moreover, the CBM at the M point crosses the Fermi level and protrudes into the occupied part of the spectrum. The VBM at Γ is pulled towards higher energy, indicating the band gap may be closed. We investigate it further by computing the variations in the electronic band gap and charge transfer across the interface with the variation in the applied electric field (Fig. 2(d)). We find extreme non-monotonic variations in the electronic band gap. The sharp reduction in the band gap occurs when the electric field is reversed. The gap vanishes when the electric field (in the reverse direction) is greater than 0.2 V Å⁻¹. We find a concurrent increase in the charge transfer across the interface, which is quantitatively, an order of magnitude larger upon maximum electric field reversal. To understand this, we look at the dependence of total and atom-projected densities of states (DOS) (Fig S2, SI). We find that the electronic ground state of the heterostructure undergoes a transformation from semiconducting to half-metallic when the electric field is reversed, and its magnitude is greater than 0.2 V Å⁻¹. This is brought about by the modifications in the spin-down band as the gap in this spin-band closes when the field is greater than 0.2 V Å⁻¹ in the reverse direction. This is consistent with the variations in the CBM, and the VBM presented in Fig. S1(a), SI discussed above. In a half-metal, spin polarisation is significant. In these materials, the spin channels dominate the physics, not the valley indices. Therefore, the valley splitting and, consequently, the valley polarisation under the applied electric field is less. Since the valleys associated with the valence band at the symmetry points are affected substantially, as demonstrated in Fig. S1(a), SI, the effects on ΔE_v are substantial.

The variations in the electronic band gap can be explained by the variations in the densities of states. Since the CBM and VBM are due to two different spin channels, both coming from VSe₂, the band gap vanishes when the semiconducting ground state transforms into a half-metallic one. The reason behind semiconductor to half-metal transformation at a critical value of external electric field along −z can be traced back to the relative directions of the electric fields at play. The electrostatic potential associated with the two surfaces of VSe₂ is identical, while the surface of Sc₂CO₂ along positive z direction is at a higher electrostatic potential than the other one.³³ As a result, an internal electric field acts in the heterostructure along the −z direction. When the external electric field is along −z, the two fields reinforce each other and try to change the electric polarization of the heterostructure from P↑ to P↓. When the polarization is P↓ in this heterostructure, a transition from semiconducting to the half-metallic ground state has been reported.³³ The variations in the electronic ground state with the external electric field along −z direction only, therefore, must be because of the effect of deviation from P↑ polarisation of the heterostructure. The rise in the charge transfer with an increase in the magnitude of the electric field applied along −z direction is also due to this as significant charge transfer has been observed³³ when this heterostructure is in P↓ state of electric polarisation as opposed to almost no charge transfer when the polarization is P↑. The external electric field is dependent on valley splitting and valley polarisation due to changes in the electronic ground state, which suggests that the MVC in this system can be tuned by the application of an external electric field.

3.4. Berry Curvature and anomalous hall conductivity (AHC)

Substantial and persistent valley polarisation in this heterostructure obtained under a positive external electric field implies that this system can generate a non-vanishing AHC. The presence of AHC is essential for using this heterostructure as a valleytronics device as AHC can convert the valley polarisation into a charge current or voltage, enabling the reading of information in a non-volatile memory device. The valley polarisation denotes the unequal population of K and K′ valleys. This, combined with a non-zero Berry curvature that arises due to broken inversion symmetry and/or broken TRS, leads to a non-zero AHC. We, therefore, first calculate the Berry curvature and AHC in this heterostructure without subjecting it to any external stimulant.

The Berry curvature can be obtained using the Kubo formula⁴⁶

where
f_z, v_x,y and Ψ_mk,nk denotes the Fermi–Dirac distribution function, velocity operator, and the Bloch electron wavefunction corresponding to energy eigenvalues E_mk and E_nk, respectively. The AHC can be obtained from the Berry curvature of Bloch electrons as⁴⁷

In Fig. 3, we show the calculated Berry curvature of the heterostructure along the symmetry directions in the Brillouin zone. The results are obtained when no external field is applied. We find valley contrasting,substantially unequal Ω_z(k) at K and K′ valley. The density plot of Ω_z(k) (Fig. 3(b)) corroborates this. These results confirm the presence of significant valley polarisation. In the absence of any external field, this is an artifact of the broken TRS. The calculated values of σ_xy, shown in Fig. 3(c), turn out to be substantial. At the Fermi level, the AHC is 153.33 S-cm⁻¹, more than double the value obtained in WSe₂/VSe₂ heterostructure,¹⁶ for example.

Fig. 3 Calculated (a) Berry curvature along high symmetry direction, (b) density plot of Berry curvature in the first Brillouin zone and (c) AHC of VSe₂/Sc₂CO₂ heterostructure. The calculations are done with zero external electric field.

3.5. Electric field tunable Berry curvature and Anomalous Hall conductivity

Manipulation of the AHC by an external electric field, particularly reversal of AHC, is significant as it brings in many possibilities for next-generation spintronics and valleytronics device applications. The AHC is associated with the efficiency of the material in converting valley polarisation into transverse charge currents or voltages. This is crucial for the detection of electrical signals and thus storing the information into bits. Reversal of the AHC implies that the valley polarisation is reversed. If this happens under the application of an electric field, it means that the valley states can be electrically switched. This trait can be exploited to make transistors, logic, or memory devices based on valley transport. The electric field-induced reversal of AHC has been observed in several bulk and thin film structures.^48–50 This aspect is much less explored in 2D materials. We, therefore, investigate the effect of the electric field on the Berry curvature and AHC of our ferrovalley–ferroelectric system.

In Fig. 4, we present the results on Ω_z and σ_xy of VSe₂/Sc₂CO₂ heterostructure for external electric fields of the same magnitude but in opposite directions. Under an external field of 0.1 V Å⁻¹, the Berry curvature (Fig. 4(a)) changes considerably in comparison to the case when no external field is applied. Noticeably large value of Ω_z is observed at the Γ point when the field is applied. Unlike the zero-external-field case, the Berry curvature now consists of negative and positive contributions of near equal magnitude along Γ–K and Γ–K′ directions. The density plot of Ω_z (Fig. 4(b)) shows more contributions from positive Ω_z in the first Brillouin zone. As a result, a negative AHC of −118.4 S-cm⁻¹ is obtained at the Fermi level. When the field is reversed to −0.1 V Å⁻¹, there are substantial changes in Ω_z (Fig. 4(d)); the highlights being larger contributions around K and K′ valleys and large negative values near Γ point. In contrast to the case when the electric field is 0.1 V Å⁻¹, the density plot (Fig. 4(e)) in this case shows more distribution of dark green spots across the Brillouin zone, signifying more negative contributions. As a result, the AHC is positive at the Fermi level with a magnitude of 89.53 S-cm⁻¹ (Fig. 4(f)). The results clearly show that the sign reversal of AHC can be effected by reversing the electric field. This result carries immense significance as the tuning of AHC by an external electric field is obtained without any carrier doping, in stark contrast to the case of WSe₂/VSe₂ heterostructure.¹⁶ A bigger implication of this is that it opens up the possibility of inverting spin⁷ and ferroelectric polarization⁵¹ in such vdW heterostructures.

Fig. 4 Calculated (a) Berry curvature along high symmetry direction, (b) density plot of Berry curvature in the first Brillouin zone and (c) AHC of VSe₂/Sc₂CO₂ heterostructure for an external electric field 0.1 V Å⁻¹ along +z -direction. (d)–(f) show the corresponding results for the electric field of same magnitude but along −z direction.

We further examine the robustness of AHC inversion by computing the Ω_z and σ_xy for the entire range of external electric fields considered in this work. The result for σ_xy at different electric fields for no-carrier-doping is shown in Fig. 5. The variations of Ω_z and σ_xy with electric field are shown in Fig. S3 and S4, SI, respectively. We find that σ_xy does not change monotonically as the electric field changes its direction. However, the reversal of σ_xy is obtained at all magnitudes of the electric field (upon reversing its direction) except for their magnitudes being 0.2 V Å⁻¹ and 0.3 V Å⁻¹. However, the magnitudes of σ_xy undergo large changes when the directions of the field with magnitudes 0.2 V Å⁻¹ and 0.3 V Å⁻¹ reverse. In both cases, σ_xy have smaller magnitudes when electric fields are in the −z direction, in comparison to all other cases. This originates from the differences in Ω_z when compared for a fixed magnitude but opposite signs of the electric field. For the case of field with magnitude 0.2 V Å⁻¹, Ω_z is dominated by a large (small) positive (negative) contribution near the Γ point along Γ–K (Γ–K′) leading to a negative σ_xy. Upon reversal of the field, such features vanish; Ω_z has a net positive value due to a large positive contribution along Γ–K. Since this stand-out contribution is not as large as the one for the field of equal magnitude, along +z direction, the value of σ_xy is smaller. Similar is the case for the field of magnitude 0.3 V Å⁻¹. We find that the magnitude of σ_xy grows continuously when the value of the electric field along −z direction increases beyond 0.2 V Å⁻¹, producing the largest σ_xy when the field is −0.5 V Å⁻¹. That such behavior of σ_xy can be correlated with the changes in occupations of the valence bands as is evident from the changes in electronic ground state beyond a critical value of the electric field along −z direction, can be understood upon inspection of Ω_z (Fig. S3, SI) and the band structures (Fig. S1, SI) for electric fields of ±0.5 V Å⁻¹. In this case, when the direction of the electric field is reversed to −z direction, large positive contributions to Ω_z are observed near Γ point along both Γ–K and Γ–K′. In the case of field along +z, larger contributions to Ω_z are found at and near M point, K′ point, and along M–K′. The band structures (Fig. S1(a) and (b), SI) corresponding to these cases show that when the field is +0.5 V Å⁻¹, more states are there at around −0.75 eV (with respect to Fermi level) along K–M and M–K′. Upon reversal of the field, there are more states near Γ. Since Ω_z is obtained from contributions of the occupied states, these features in the band structure can be considered responsible for the features in Ω_z. Moreover, the heterostructure system has intrinsic asymmetric response and with the application of electric field it controls the intrinsic asymmetry nonlinearly. This asymmetric response can possibly result in AHC change, where only a limited range between 0.1 V Å⁻¹ is critical for sign reversal and is subtly balanced at small fields. But beyond 0.1 V Å⁻¹, the system saturates deeper into the asymmetric regime leading to no sign reversal. Thus, the tuning of the AHC can be obtained by tuning the band structure of this system through electrical control.

Fig. 5 Variations in the calculated AHC with the external electric field in the range −0.5 V Å⁻¹ to 0.5 V Å⁻¹.

4. Conclusions

Control of information processing and storage through electrical means is an energy-efficient process that has been exploited to design memory and storage devices that work on new technologies like spintronics. In comparison, exploration of the same in the context of new physics and technology that manipulate the valley degrees of freedom in a material is significantly less in number. In this work, we perform a detailed investigation of the tunability of valley polarisation and anomalous Hall conductance, quantities that can be directly used for memory and storage devices, by electric control in 2D heterostructure VSe₂/Sc₂CO₂, combination of a ferrovalley and a ferroelectric material. We find that the system has a large intrinsic valley polarisation brought about by the intrinsic spin–orbit coupling in the VSe₂ layer. The SOC significantly alters the valence band and the states in the symmetry-related K and K′ valleys. The symmetry breaking and subsequent lifting of degeneracy in the valleys generate a non-zero Berry curvature and, consequently, a significant anomalous Hall conductance (AHC). An external electric field, applied perpendicular to the heterostructure, modifies the valley polarization, the Berry curvature, and the AHC substantially. The most important finding is that sign contrasting AHC in this system can be obtained by reversal of the applied electric field. This feature can be used for efficient storage and reading of data in a non-volatile memory device. The origin of this wonderful feature is the changes in the band structure of the system near symmetry points and the valleys. The results carry immense significance due to two reasons: (a) there has been hardly any investigation on a 2D ferroelectric-ferroalloy heterostructure that exhibits switching of AHC under an external electric field obtained by exploiting the valley characteristics, and (b) although TMDCs have been explored towards valleytronics, compounds from MXene family, are almost not investigated upon. MXenes are compounds with immense flexibility in structure and composition and can offer a lot of tunability in valley properties under external effects. This is the first work that unleashes the potential of an MXene-TMDC vdW heterostructure in obtaining electrical control of valley-related properties that can be easily implemented in robust memory and storage devices. This, thus, can pave the way to explore more such heterostructures and their science and technological prospects.

Conflicts of interest

The author declare that there is no conflicts to declare.

Data availability

The code VASP for calculation of electronic structures can be found at https://www.vasp.at/ ref. 35 in the main text shows the ones citing the code. The version of the code employed for this study is version 5.4.1. The code PhonoPy for calculation of phonons can be found at https://phonopy.github.io/phonopy/. Ref. 39 in the main text shows the ones citing the code. The version of the code employed for this study is version 2.21.2. The code Wannier90 can be found at https://wannier.org/download/ ref. 40 in the main text shows the ones citing the code. The version of the code employed in this study is 3.1.0. The code WannierTools can be found at https://www.wanniertools.com/ ref. 42 in the main text shows the ones citing the code. The version of the code employed in this study is 2.7.0.

The data supporting this article have been included as part of the supplementary information (SI). Supplementary information is available. See DOI: https://doi.org/10.1039/d5cp02847g.

Acknowledgements

MB acknowledges SERB, Government of India for providing financial support through National Postdoctoral Fellowship (File No. PDF/2023/000717). The authors gratefully acknowledge the Department of Science and Technology, India, for the computational facilities under Grant No. SR/FST/P-II/020/2009 and IIT Guwahati for the PARAM supercomputing facility.

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