Electric-field-induced topological phase transition in CrS₂/CrSe₂ heterostructures

Abstract

Electric fields can induce topological phase transitions from normal insulators to topological insulators, offering a promising pathway for designing low-power electronic devices based on two-dimensional topological insulators with tunable and practical functionalities. In this work, we demonstrate a dynamically tunable topological insulator realized in van der Waals (vdW) heterostructures composed of two-dimensional transition metal dichalcogenides — specifically, a CrS₂/CrSe₂ junction. Remarkably, this system exhibits significant electric-field tunability of its topological properties. First-principles calculations reveal that an external electric field induces a band inversion at a critical field strength of −2.0 eV/nm, signaling the emergence of nontrivial topological phases. These topological characteristics remain robust, as confirmed by van der Waals-corrected DFT-D2 calculations. Furthermore, we systematically explore strain engineering as an additional control parameter. Under 1.45% biaxial strain, the band inversion threshold shifts significantly to −0.3 eV/nm while maintaining a finite bandgap of 9.6 meV. This dual tunability, achieved through both electric fields and mechanical strain, establishes the CrS₂/CrSe₂ heterostructure as a versatile platform for developing topological field-effect transistors and other quantum devices requiring dynamic control of edge conduction states.

Graphical abstract

Keywords

band inversion / transition metal dichalcogenides / topological states / valley quantum spin Hall

1 Introduction

Since the groundbreaking isolation of graphene in 2004 [1], the field of two-dimensional (2D) materials has rapidly expanded with discoveries including h-BN [2], transition metal dichalcogenides (TMDs) [3], phosphorene [4], and MXenes [5]. These 2D crystals exhibit extraordinary tunability of properties, spanning ferroelectricity [6−8], topological domain walls [9], superconductivity [10], quantum spin-Hall (QSH) effects [11, 12], quantum anomalous Hall effects [13−15], and various other topologically nontrivial phases [16−19]. Topological insulators (TIs) have attracted extensive attention in condensed matter physics and material science due to their time-reversal symmetry-protected conducting edge/surface states. Theoretically, these topological edge states provide dissipationless transport channels and are immune to scattering by disorder, impurities, or vacancies [20]. Numerous 2D materials have been predicted to exhibit characteristics of 2D TIs, including MnBi₂Te₄ [21−23], Pt₂HgSe₃ [24], the MX compound family [25], stanene [26], Bi(111) bilayers [27], and WTe₂ [28, 29].

The ability to controllably tune topological invariants remains a highly sought-after objective, as it unlocks new avenues for both fundamental research and practical applications. To date, numerous methods have been employed to modulate topological states, including magnetization rotation [30], application of external magnetic fields [31], mechanical strain engineering [32], and electric fields [33]. Notably, external electric fields represent a promising approach owing to their inherent reversibility, ultrafast temporal response, and energy-efficient operation-attributes that are particularly advantageous for low-power device applications. The electric-field-induced topological phase transition phenomenon is of fundamental importance, as it enables the dynamic reconfiguration of quantum states, thereby unlocking transformative functionalities for beyond-Moore electronics. The realization of such transitions requires achieving a critical electric field strength capable of mediating the trivial-to-nontrivial band inversion. Remarkably, state-of-the-art experiments have successfully demonstrated sustained electric fields exceeding 0.3 V·Å⁻¹ [34], providing compelling experimental validation of this paradigm’s technological feasibility. Theoretical predictions indicate that a perpendicular electric field can induce a transition from normal insulators to TIs in specific 2D materials. For instance, the MnBi₂Se₄/2Bi₂Se₃/MnBi₂Se₄ sandwich vdW heterostructure is a good candidate for the electric field control of topological phase transition [35]. Similarly, six-layer germanane undergoes a topological phase transition at a critical field of 0.2 V·Å⁻¹, reopening a nontrivial bandgap of 6 meV [36]. Furthermore, electric fields can modulate topological phase transitions in AlSb/InSe heterobilayers [37] and tune bilayer van der Waals stanane from a normal insulator to a TI [38]. Tunable topological phases can also be induced in graphene nanoribbons by transverse electric fields [39]. Significantly, in halide perovskite superlattices, an electric field triggers a normal insulator to a TI transition accompanied by a pronounced Rashba effect [40].

Recent breakthroughs in synthetic chemistry have enabled the controlled growth of monolayer 1T, 1T′, and 2H phases of CrS₂ via chemical vapor deposition (CVD) [41]. Theoretical studies further reveal that 2D chromium dichalcogenides (CrX₂, X = S, Se) possess unique functionalities, including valley polarization, piezoelectric coupling, and phase-dependent magnetism. For instance, the CrSe₂ monolayer is a direct-bandgap semiconductor (~1.51 eV) with high dynamic stability [42]. CrS₂ exhibits remarkable phase-driven property diversity: the 1T phase is computationally predicted as an antiferromagnetic (AFM) metal, the 2H phase as a nonmagnetic semiconductor, and the 1T′ phase as an FM semiconductor with an ultrahigh predicted Curie temperature (~1000 K) [43]. Notably, strain engineering can dynamically modulate the 1T′-CrS₂ phase, inducing a transition to a spin-polarized half-metallic state under tensile or compressive strain. This tunability originates from strain-induced lattice deformation, which selectively enhances spin-up or spin-down orbital interactions near the valence band maximum [43]. Experimental characterizations, such as X-ray magnetic circular dichroism, combined with density functional theory (DFT) calculations, provide critical insights: molecular beam epitaxy (MBE)-grown monolayer 1T-CrSe₂ exhibits antiferromagnetism, whereas monolayer Cr₂Se₃ is ferromagnetic with a Curie temperature of ~200 K [44]. Despite these advances, the bandgap characteristics and potential for electric-field-induced topological phase transitions in CrS₂/CrSe₂ heterostructures remain unexplored.

In this study, we employ density functional theory systematically to investigate the electronically tunable topological states in CrS₂/CrSe₂ heterostructures. Our calculations reveal interfacial charge transfer that critically modifies the band topology through layer hybridization. External-electric-field modulation induces a topological phase transition at −2 eV/nm. Remarkably, applying 1.45% biaxial strain reduces the electric field to −0.3 eV/nm while maintaining a topological nontrivial bandgap of 9.6 meV. This dual electric-field and strain control establishes CrS₂/CrSe₂ as a promising platform for voltage-gated topological transistors and adaptive quantum devices.

2 Computational methods

All structural relaxation calculations were performed within the framework of DFT as implemented in the Vienna ab initio simulation package (VASP) [45, 46]. The electron-core interaction was described by the projected augmented wave pseudopotential [47, 48] with the general gradient approximation (GGA) parameterized by Perdew, Burke, and Ernzerhof [49]. The electron wave function was expanded in plane waves up to a cutoff energy of 450 eV, and the 18 \times 18 \times 1 Monkhorst–Pack k-grid [50] was used to sample the Brillouin zone of the supercell. The convergence criterion for the electronic energy was set to be 10⁻⁵ eV. A vacuum buffer space of 25 Å was used in the calculations to avoid interaction between adjacent slabs. The GGA + U [51] method was used to describe the exchange and correlation effects of localized transition metal d electrons. In this approach, an effective onsite interaction parameter was defined as U_eff = U − J, where J = 1 eV and U = 3.8 eV described the exchange interaction and the Hubbard onsite Coulomb repulsion, respectively. Moreover, the van der Waals force was included by adopting the DFT-D3 method in all structural optimizations [52]. The Wannier90 software package [53] was employed to construct a tight-binding (TB) model based on maximal localized Wannier functions, and the Wannier Tools [54] software was used to calculate the Wannier charge center (WCC), Z₂ invariant, and boundary states to characterize the topological features of the system.

3 Results and discussion

As illustrated in Fig. 1(a), the monolayer 2H-CrS₂ is stacked by the S−Cr−S sandwich layers along the Z-direction. Its primitive cell belongs to the P63/mmc group. Each Cr cation is surrounded by six S anions. Similarly, the monolayer CrSe₂ is enveloped by two layers of selenium (Se) atoms. The lattice constants for 2H-CrS₂ are a = b = 3.05 Å, and the interlayer distance is h = 2.886 Å. In comparison, the monolayer of 2H-CrSe₂ has lattice constants of a = b = 3.21 Å, with an interlayer distance of h = 3.186 Å, which are in agreement with the previous study [55]. In the ground state of the CrS₂ and CrSe₂, the 2H phase is a nonmagnetic semiconductor. When designing the heterostructures, the average lattice constant of CrS₂ and CrSe₂, specifically a = b = 3.13 Å, was adopted as the lattice constant for the heterostructures, resulting in a lattice mismatch of approximately 2.6%. There are six stacking configurations, encompassing both H-type and R-type arrangements, denoted as {\mathrm{H}\left(\mathrm{R}\right)}_{\mathrm{X}}^{\mathrm{M}}. As depicted in Fig. 1(b), the H-type and R-type configurations are organized in antiparallel and parallel orientations with respect to the upper and lower layers, respectively. X represents S or Se atoms, M represents Cr atoms. Each arrangement allows for three alignment methods: (i) alignment of X with X ({\mathrm{H}\left(\mathrm{R}\right)}_{\mathrm{X}}^{\mathrm{X}}), (ii) alignment between M and X ({\mathrm{H}\left(\mathrm{R}\right)}_{\mathrm{X}}^{\mathrm{M}}), (iii) alignment of M with M ({\mathrm{H}\left(\mathrm{R}\right)}_{\mathrm{M}}^{\mathrm{M}}). The different configurations and the stacking orders are shown in Fig. 1(b). The total energies of different stacking orders indicate that the {\mathrm{H}}_{\mathrm{X}}^{\mathrm{M}} configuration is the most stable. The antiparallel structure preserves the original spatial symmetry of the system, and its interlayer spacing is the closest among the six configurations. This arrangement significantly enhances the interlayer coupling effect in heterostructures. Figure 1(c) shows the phonon dispersion of the CrS₂/CrSe₂ heterostructures with the stacking configuration {\mathrm{H}}_{\mathrm{X}}^{\mathrm{M}}, which indicates that it is dynamically stable.

To investigate the electronic and topological characteristics of the CrS₂/CrSe₂ heterostructure with {\mathrm{H}}_{\mathrm{X}}^{\mathrm{M}} configuration, we analyze its band structure in Fig. 2, with the corresponding first Brillouin zone illustrated in Fig. 1(a). Figure 2(a) presents the band structures without spin-orbit coupling (SOC), revealing spin-degenerate states for both spin-up and spin-down channels. The energy valleys at the K and K' points exhibit robust valley degeneracy, confirming the heterostructure as a direct bandgap semiconductor with a gap of 107.9 meV (valence and conduction band extrema located at Γ). Incorporating SOC [Figs. 2(b) and (c)], significant spin-splitting emerges due to strong interlayer hybridization. The valence band near the Fermi level undergoes pronounced splitting (163.7 meV), while the conduction band exhibits a markedly smaller splitting (9 meV). This asymmetry in SOC-induced splitting reduces the global bandgap to 47.2 meV, a 56.25% reduction compared to the non-SOC case. The substantial bandgap narrowing highlights the critical role of interfacial charge redistribution and layer-coupled spin-orbit interactions, which amplify the system’s tunability under external stimuli. The preserved valley degeneracy at K/K' under SOC further suggests robustness against intervalley scattering, a prerequisite for topological edge state coherence.

When two dissimilar materials form a heterostructure, interfacial charge transfer typically occurs, leading to significant modifications of the electronic structure. In the study of CrS₂/CrSe₂ heterostructures, the charge redistribution at the interface was analyzed by calculating the planar-averaged charge density difference along the out-of-plane (Z) direction. As shown in Fig. 2(d), the charge density difference reveals distinct interfacial polarization: the CrS₂ monolayer exhibits electron accumulation (positive density regions), while the adjacent CrSe₂ layer displays electron depletion (negative density regions). This spatial charge redistribution indicates a net electron transfer from the CrSe₂ layer to the CrS₂ layer upon heterostructure formation, which fundamentally modifies the interfacial electronic properties. In van der Waals heterostructures, interfacial charge transfer indicates significant electronic coupling, which is crucial for understanding the physical and chemical properties at the interface. This charge transfer generates a built-in electric field. We applied external electric fields along the direction of this field to induce band structure modulation.

Figures 3(a)–(g) systematically track the electrically driven band structure evolution in the CrS₂/CrSe₂ heterostructure. The electronic band structures were calculated under vertical electric fields (E) with intensities of (a) −0.5 eV/nm, (b) −1.5 eV/nm, (c) −1.8 eV/nm, and (e) −2 eV/nm, respectively. At an electric field of −0.5 eV/nm [Fig. 3(a)], the global bandgap exhibits a significant narrowing to 35.3 meV, corresponding to a 25.2% reduction compared to the zero-field reference configuration [Fig. 2(b)]. When the field strength increases to −1.5 eV/nm [Fig. 3(b)], the bandgap undergoes further suppression down to 10.4 meV. This systematic modulation stems fundamentally from the electric-field-induced interfacial charge redistribution along the vertical direction. When the field reaches −1.8 eV/nm [Figs. 3(c) and (d)], degeneracy occurs with the conduction band minimum at the K-point positioned at 12.1 meV and the valence band maximum at the K′-point at 12.3 meV, resulting in a vanishing bandgap. Remarkably, at E = −2.0 eV/nm [Fig. 3(e)], a bandgap of 5.8 meV reopens alongside momentum-resolved band inversion. Figure 3(f) quantitatively demonstrates the nonmonotonic evolution of bandgap as a function of applied vertical electric field, revealing a critical transition threshold near −1.8 eV/nm. Figure 3(g) presents the layer-projected band structure decomposition of the CrS₂/CrSe₂ heterostructure under −2 eV/nm vertical electric field. The magenta-coded spectral weights represent CrS₂ layer contributions, while light blue denotes CrSe₂ orbital projections, revealing distinct interlayer charge redistribution effects. To validate this transition, we constructed a maximally localized Wannier function tight-binding model, achieving excellent agreement between Wannier90-interpolated bands [red dashed, Fig. 3(h)] and first-principles VASP calculations (black solid). The energy spectra [Fig. 3(i)] of a semi-infinite ribbon of CrS₂/CrSe₂ heterostructure illustrate that there is a pair of Kramers-degenerate nontrivial edge channels [56], indicating the formation of a valley quantum spin Hall state. The computed Z₂ invariant [ν = 1, Fig. 3(j)] confirms topological nontriviality, confirming this heterostructure as an electrically tunable topological property with a 5.8 meV bulk gap.

The preceding computational results were obtained using the DFT-D3 dispersion correction scheme. To evaluate the sensitivity of our conclusions to vdW treatment, we further perform calculations employing the DFT-D2 functional to investigate the robustness of the topological property against different dispersion corrections. As illustrated in Fig. S1, under the electric field strength of −0.5 eV/nm, the bandgap decreases to 34.3 meV [Fig. S1(a)]. At −1.0 eV/nm, the conduction band minimum at the K point lies at 12.6 meV, while the valence band maximum at the K' point lies at 12.8 meV [Fig. S1(b)]. Increasing the field further to −1.5 eV/nm shifts the K-point conduction band minimum to 15.8 meV and the K'-point valence band maximum to 15.9 meV [Fig. S1(c)]. Crucially, band inversion consistently emerges at an electric field strength of −2.0 eV/nm [Fig. S1(e)]. The bulk bandgap calculated with DFT-D2 functional reaches 6.6 meV, compared to 5.8 meV obtained with DFT-D3 functional, representing a 13.79% enhancement attributable to modified interlayer charge transfer. The persistence of edge states [Fig. S1(h)] confirms that the system’s tunable topological properties are preserved regardless of the change of dispersion correction scheme.

The pervasive influence of biaxial strain in two-dimensional material systems — stemming from substrate mismatch or mechanical processing — presents a vital avenue for customizing electronic quantum states. Next, we investigated the band structure for the strain without electric fields. Through first-principles calculations, we meticulously evaluate the strain response of CrS₂/CrSe₂ heterostructures subjected to biaxial strains ranging from −2% (compressive) to 2% (tensile), as illustrated in Figs. S2−S5. The global bandgap shows a consistent decline from 120.6 meV at −2% strain to 14 meV at 0.8% strain, followed by oscillatory behavior within the range of 0.8% to 2%. Notably, band inversion initiates at 1.45%, marking the onset of a strain-induced topological phase transition. By applying vertical electric fields (E) to the −1.45% configuration (see Fig. 4), there is a reversed band ordering, where inverted bands emerge at E = −0.3 eV/nm, accompanied by an enhanced bulk gap of 9.6 meV. To further substantiate our findings, tight-binding models constructed via Wannier90 accurately reproduce the DFT band structures with a deviation of less than 1 meV near the Fermi level [Fig. 4(g)]. As shown in Fig. 4(h), valley-resolved transport occurs, with K' and K valleys forming linear-dispersive edge channels that exhibit contrasting group velocities — a characteristic feature of quantum spin Hall transport. The definitive nontrivial Z₂ invariant [ν = 1, Fig. 4(i)] corroborates the presence of robust nontrivial edge states under the combined influence of strain and electric field. This dual-field control (strain combined with electric field) positions CrS₂/CrSe₂ as a voltage-tunable platform for topological states, where biaxial strain reduces the critical electric field necessary for the topological phase transition by 85% when compared to unstrained systems (from −2 eV/nm to −0.3 eV/nm).

4 In conclusions

In this work, we developed heterostructures composed of monolayer CrS₂ and CrSe₂, revealing significant charge transfer between the layers. First, we assess the binding energies of six different configurations and found that the {\mathrm{H}}_{\mathrm{X}}^{\mathrm{M}} configuration exhibited the highest stability. We then investigate the topological state of the CrS₂/CrSe₂ heterostructure, observing a band inversion occurring at an electric field of −2 eV/nm. Further exploration of the heterostructure’s topological characteristics indicates that the application of an electric field led to the emergence of two edge states, each characterized by distinct slopes. Moreover, we examine the DFT-D2 van der Waals corrections and discovered that these topological properties remained unchanged. Finally, we evaluate the impact of stress on the band structure. Our calculations suggest that applying a biaxial strain of 1.45% induces a band inversion at an electric field of −0.3 eV/nm, accompanied by a bandgap of 9.6 meV. These findings highlight the promising potential of these heterostructures for advanced electronic applications.

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