Strain and U driven phase transitions in monolayer intrinsic ferrovalley NbIn₂As₂Se₂

Abstract

The manipulation of the valley degree of freedom presents opportunities for both research and practical application. In this work, we theoretically demonstrate that the intrinsic valley anomalous Hall effect can exist in monolayer NbIn₂As₂Se₂. Due to time-reversal symmetry breaking, monolayer NbIn₂As₂Se₂ is an out-of-plane magnetization semiconductor with a Curie temperature of 232 K. The ability to induce phase transitions in the material through strain and the U value leads to different electronic states like the valley quantum anomalous Hall effect and the half-valley-metal state. The chiral-spin-valley locking of edge states and the band inversion of the d orbital of Nb at the K/K′ valley offer insights into the mechanisms behind these transitions. These findings not only contribute to the fundamental understanding of topology, spintronics, and valleytronics, but also pave the way for potential practical applications and experimental investigations in this exciting and rapidly evolving field.

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Valleys, as regulated information storage similar to charge and spin, have garnered significant interest in recent years due to their outstanding properties and promising prospects for applications in valleytronic devices.^1–3 Valleytronic materials typically feature at least two valleys in momentum space, with these valleys representing either local minima or local maxima on the band structure.^4,5 Typically, these two valleys possess degenerate energy and inequivalent Berry curvatures, located at the K and K′ points.^6,7 Experimental observations have shown that optical pumping can induce valley polarization in transition metal dichalcogenides (TMDCs), which can be explained by valley-contrasting optical selection rules.^8–10 Additional methods, such as manipulating magnetic fields, utilizing magnetic doping, and capitalizing on the magnetic proximity effect, can provide alternative pathways to achieve and sustain valley polarization.^11–13 Nevertheless, the magnetic field exhibits low efficiency (0.1–0.2 meV T⁻¹), magnetic doping tends to elevate impurity scatterings across distinct valleys, and a magnetic substrate obscures the valley physics within the host materials.^14–16 Hence, the quest for intrinsic valley polarization is crucial.

Furthermore, the progress in valleytronic materials has led to a new insight into 2D ferromagnetic (FM) systems, known as ferrovalley materials.^17–19 FM semiconductors inherently break the time-reversal symmetry, and when coupled with spin–orbit coupling (SOC), they exhibit spontaneous valley polarization phenomena.²⁰ The quantum anomalous valley Hall effect (QAVHE) is easily observed experimentally in ferrovalley materials, making them have great application potential in information storage.²¹ Achieving spontaneous valley polarization requires out-of-plane magnetism, whereas in-plane magnetic materials are more commonly found in nature.^22–24 Therefore, a difficult-to-modulate magnetic axis is necessary for realizing spontaneous valley polarization.^25,26 To date, several ferrovalley materials have been theoretically predicted, including VSi₂P₄, VSiGeN₄, NbX₂, Fe₂SSe, Nb₃I₈, 2H-VSe₂, and GdI₂.^6,27–30 On the other hand, valleys combined with other physical properties can give rise to new physical phenomena, such as the valley quantum anomalous Hall effect (VQAHE), which emerges from the coupling between valley polarization and topology insulators (TI).^31,32 Despite their high value, materials that combine the quantum anomalous Hall effect (QAHE) and the VQAHE are exceedingly rare. Consequently, the design of multi-functional multi-valley materials holds great significance in materials science and condensed matter physics.

Recently, a two-dimensional van der Waals material, MoSi₂N₄, with seven atomic layers, was experimentally synthesized using the chemical vapor deposition (CVD) method.³³ The monolayer MoSi₂N₄ can be viewed as composed of monolayer 2H-MoS₂-like structure MoN₂ and bilayer silicene-like structure SiN, exhibiting K valley polarization phenomena similar to 2H-MoS₂.³⁴ Spontaneous valley polarization can be achieved by introducing magnetism to break the time-reversal symmetry. Here, we theoretically predict a 2D FM material, NbIn₂As₂Se₂, with a structure similar to MoSi₂N₄. Based on first-principles, the monolayer NbIn₂As₂Se₂ is a semiconductor with out-of-plane FM at U = 3 eV and has a Curie temperature (T^M_C) of 232 K. We also confirm that the VQAHE can be observed in this material. The material can achieve phase transition by applying in-plane biaxial strain engineering or adjusting the value of U. These findings indicate that monolayer intrinsic ferrovalley NbIn₂As₂Se₂ holds promise for applications in spintronics and valleytronics.

This work was performed as implemented in the Vienna ab initio simulation package, which was based on density functional theory.^35,36 The exchange–correlation interaction between electrons was described by the Perdew–Burke–Ernzerhof (PBE) type generalized gradient approximation (GGA).³⁷ The energy cutoff of 500 eV and 13 × 13 × 1 k-meshes for Brillouin-zone (BZ) sampling were chosen. The calculation results of the convergence criterion for force and energy were selected as 10⁻³ eV Å⁻¹ for and 10⁻⁶ eV. Due to the strong correlation effect, we use the GGA+U method in the 3d orbital of Nb element.³⁸

To verify the stability of the material, calculations of the phonon dispersions of monolayer NbIn₂As₂Se₂ were performed based on density functional perturbation theory (DFPT) by using the PHONOPY code. The molecular dynamics simulation based on the NVT ensemble was conducted to demonstrate the thermal stability by using the Nosé–Hoover method at 300 K. A 5 × 5 × 1 supercell and a 3 × 3 × 1 Monkhorst–Pack grid were employed in phonon dispersions and MD simulations.

The FM Curie temperature was determined through Monte Carlo (MC) simulations using a 30 × 30 × 1 supercell and 10⁸ loops with the Mcsolve code.³⁹ The Fukui method was employed in the Berry curvature calculation using the VASPBERRY code.^40,41 We calculated the valley Hall effect through the use of the WANNIER90 and WANNIERTOOLS packages.^42,43

Fig. 1(a) shows the side and the top view of monolayer NbIn₂As₂Se₂, which consists of Se–In–As–Nb–As–In–Se seven atom layers in a unit cell. The monolayer NbIn₂As₂Se₂ structure can be viewed as consisting of three parts, in which the upper/lower part is a β-P-like structure InSe layer, the middle part NbAs₂ shares the same crystal structure with 2H-MoS₂, and the upper/lower middle layers are connected by In–As binding. It possesses a 2D hexagonal lattice which has P6m2 symmetry with space group no. 187. The monolayer NbIn₂As₂Se₂ structure maintains the M_z symmetry about the Nb atom, which can lead to many novel properties, such as hidden physics. We consider the influence of the U value on lattice constant a, optimizing the lattice constants a for different U values, shown in Fig. S1(a) (ESI†). It can be found that the lattice constant a increases with the increase of the U value, from 3.918 Å to 3.950 Å.

Fig. 1 Structure and stability analyses. (a) Top and side views of monolayer NbIn₂As₂Se₂, where the dashed line marks the unit cell. Phonon dispersion (b) and molecular dynamics simulation (c) for monolayer NbIn₂As₂Se₂.

To estimate the stability of monolayer NbIn₂As₂Se₂, the formation energy E_form and the cohesive energy E_coh were calculated through the equations: E_form = (E_NbIAS − μ_Nb − 2μ_In − 2μ_As − 2μ_Se)/7 and E_coh = (E_NbIAS − E_Nb − 2E_In − 2E_As − 2E_Se)/7, where E_NbIAS, μ_Nb/In/As/Se and E_Nb/In/As/Se are the total energy of monolayer NbIn₂As₂Se₂ and the energy of Nb, In, As and Se atoms in a stable bulk phase structure or isolated state. The calculated values of E_form and E_coh are −0.23 eV and −4.51 eV per atom for monolayer NbIn₂As₂Se₂. The negative value of E_form indicates that monolayer NbIn₂As₂Se₂ can be synthesized from the pure phase Nb, In, As, and Se elements. Our E_coh is lower than the E_coh values of some theoretically predicted two-dimensional materials, such as Cu₂Ge (−3.17 eV per atom)⁴⁴ and CeI₂ (−3.14 eV per atom),⁶ indicating that the monolayer NbIn₂As₂Se₂ is stable.

To evaluate the thermodynamic stability of monolayer NbIn₂As₂Se₂, the phonon dispersion calculations and molecular dynamics simulations have been performed. Phonon dispersion can describe the dynamic stability of a material, which needs all phonons across the whole Brillouin zone. Fig. 1(b) clearly displays that there is no virtual frequency in the Brillouin zone at U = 3 eV, confirming the dynamic stability of monolayer NbIn₂As₂Se₂. The MD simulation can evaluate the thermal stability of monolayer NbIn₂As₂Se₂. Fig. 1(c) demonstrates that the structure remains stable for at least 7.5 ps at a temperature of 300 K. The Born–Huang criteria can be used to evaluate the mechanical stability,⁴⁵ which requires C₁₁C₂₂ > C₁₂² and C₆₆ > 0 for a 2D hexagonal lattice. The value of the elastic constant is about C₁₁ = 109.5 N m⁻¹, C₁₂ = 44.5 N m⁻¹ and C₆₆ = 38.7 N m⁻¹ at U = 3 eV. We checked these values for the Born–Huang criterion and confirmed monolayer NbIn₂As₂Se₂ is mechanically stable.

In order to define the magnetic ground state of monolayer NbIn₂As₂Se₂, we constructed a magnetic configuration by using a 2 × 2 rectangular supercell in Fig. 2(a). The ground state of the system can be represented by the total energy difference (ΔE) between FM/antiferromagnetic (AFM) and nonmagnetic (NM) states and a graph of ΔE–U is plotted in Fig. 2(b), demonstrating that the system is in the FM state at different U values. Spontaneous valley polarization requires an out-of-plane magnetic direction, so it is crucial to understand the magnetic orientation of the system. Here, we plotted the MAE at different U values in Fig. 2(c), which can be calculated as E_MAE = E₁₀₀ − E₀₀₁ within GGA+SOC+U. The MAE can determine magnetic anisotropy, where a positive value indicates out-of-plane magnetism as the ground state, while a negative value indicates in-plane magnetism as the ground state. In the range of U = 0–3 eV, in-plane magnetism is the ground state. As the U value increases (U ≥ 3), the magnetic direction changes from in-plane to out-of-plane. Therefore, in the following discussion, our calculation results are primarily obtained under the condition of U = 3 eV. The magnetic moment is a crucial parameter in a ferromagnetic system, with the magnetism primarily originating from the Nb atom in our materials. Hence, we employed two methods, namely the electron localization function (ELF) and Bader charge, to analyze the environment surrounding the Nb atom. Fig. S1(b) (ESI†) illustrates the ELF in the (110) plane of monolayer NbIn₂As₂Se₂. The Nb atom has a lower electron density, in contrast to the Se atom which has a higher electron density, indicative of an ionic bond between Nb and Se atoms, implying a transfer of charge from Nb to Se. Through Bader charge analysis, it was verified that Nb atoms transferred 0.6 electrons to each Se atom. Consequently, the calculation results indicate a magnetic moment per Nb of 1μ_B at U = 3 eV.

Fig. 2 Analysis of magnetism. (a) The FM and AFM configurations of monolayer NbIn₂As₂Se₂. (b) The total energy difference (ΔE) between FM and AFM/NM as a function of the U value. (c) The MAE at different U values; the MAE can be expressed as E_MAE = E₁₀₀ − E₀₀₁ within GGA+SOC+U, where E₁₀₀ and E₀₀₁ are the total energies of in-plane and out-of-plane magnetization. (d) The Curie temperature of a ferromagnetic at a U value of 3 eV.

T ^M_C is an important parameter in magnetic materials, which can be estimated using the Heisenberg model:

where J, A, S_i/j and S^z_i represent the nearest-neighbor exchange parameter, MAE, spin vectors and spin component along the c direction, respectively. J with normalized spin vector (|S| = 0.5) can be obtained by comparing the energies of the E_AFM and E_FM configurations with a rectangular supercell, and the corresponding J can be written as

E_FM = E₀ − 6JS² (2)

E_AFM = E₀ + 2JS² (3)

The calculated normalized J is 74.9 meV at U = 3 eV, and the normalized magnetic moment and auto-correlation vs. temperature are plotted in Fig. 2(d). The predicted T^M_C is 232 K. On the other hand, the transition temperature can be expressed as by using the mean-field approximation (MFA), where J and K_B are the nearest-neighboring exchange parameter and Boltzmann constant, respectively. The value of T^MF_C is 579 K, which is higher than T^M_C (232 K). Fig. S2 (ESI†) shows the T^M_C of NbIn₂As₂Se₂ at different values of U and the result that the T^M_C decreases as the U value increases.

Fig. 3(a) shows the spin polarized band structure of monolayer NbIn₂As₂Se₂ without the SOC effect. It can be seen that both the conduction band (CB) and the valence band (VB) near the Fermi level come from the spin-up channel for M↑. When the magnetic direction changes to M↓, as shown in Fig. S4 (ESI†), the CB and VB come from the spin-down channel, which is in contrast to the result when the magnetic direction is M↑. Theoretical calculations reveal that the conduction band minimum (CBM) and valence band maximum (VBM) both reside at the K or K′ point, with a band gap value of 104 meV, indicating that monolayer NbIn₂As₂Se₂ is a FM semiconductor. When the inversion symmetry is broken, the two K valleys will no longer be equal. Under the effect of the SOC, bands at the K and K′ points experience a shift, resulting in spontaneous valley polarization. The calculated band gap value is 29 meV at the K point, which is lower than the band gap of 182 meV at the K′ point, as depicted in Fig. 3(b). As shown in Fig. S3 (ESI†), external magnetic fields can be used to control band gap. When the magnetic direction becomes M(↓), both the VB and CB near the Fermi surface come from the spin-down channel, and the band gap values at K and K′ points are exchanged.

Fig. 3 The electronic structure of monolayer NbIn₂As₂Se₂. (a) and (b) The spin-polarized band structure of monolayer NbIn₂As₂Se₂ without/with the SOC effect. (c) Orbital projected band structure considering SOC effects. (d) The Berry curvature along the whole Brillouin zone at U = 3 eV.

The above results confirm that the monolayer NbIn₂As₂Se₂ is a FM semiconductor with an out-of-plane magnetic orientation. In the presence of SOC, it can spontaneously generate valley polarization. As is well-known, the SOC effects can be written as

Ĥ_soc = αL̂_z (5)

where L̂_z is the orbital angular momentum along the c direction. The orbital projection of Nb atoms near the Fermi level is shown in Fig. 3(c). At the K point, the VBM is mainly contributed by the d_x²−y² + d_xy orbital, while the CBM is almost entirely the d_z² orbital. Unlike the orbital projection at the K valley, the orbital contributions of the Nb atoms at the K′ point are opposite to those at the K point. Therefore, it can be considered that the different valley polarization at the K and K′ points originates from their different orbital contributions near the Fermi level. Since the magnetic quantum number for d_z² is 0, the basis functions at K and K′ points can be represented by d_x²−y² and d_xy orbitals. Considering the presence of C_3h symmetry operations in the system, the basis functions can be written asHere, τ = ±1 represents the valley at the K or K′ point.

E^τ = 〈ϕ^τ|Ĥ_soc|ϕ^τ〉 (7)

Therefore, due to the SOC effect, the d_x²−y² and d_xy orbitals near the Fermi level shift, resulting in different energy eigenvalues and leading to spontaneous valley polarization, which can be expressed as

The above equation helps us understand that valley polarization arises from the SOC effect causing the degenerate K/K′ valleys to shift, resulting in the two valleys no longer being equivalent.

In 2D materials, the Chern number C can be used to characterize the quantum anomalous Hall effect, which can be expressed as^46,47

where Ω_n is the Berry curvature in momentum space and it can be expressed as^48,49where Φ_nk is the Bloch wave function and υ_a/υ_b is the velocity operator along the a/b direction. The calculated Berry curvature of monolayer NbIn₂As₂Se₂ is shown in Fig. 3(d), which shows that the values of the Berry curvature at the K and K′ points are unequal in magnitude and opposite in sign. Here, C is obtained by integrating the Berry curvature of the filled electron bands in momentum space of monolayer NbIn₂As₂Se₂. We obtain C = 1/−1 for M(↑)/M(↓) in monolayer NbIn₂As₂Se₂. The magnitude of C is directly related to the number of chiral edge states along any edge of the monolayer NbIn₂As₂Se₂ system. The chiral edge states with no dissipation in various magnetization orientations correspond to one-way electron conduction pathways connecting the VB and CB within nontrivial band gaps, plotted in Fig. 4(a) and (b). Electrons demonstrate conductivity within the bandgap's edge states, with their spins and momentum being locked. It is evident from Fig. 4(a) and (b) that the edge state will conduct spin-up electrons in the case of M(↑) and spin-down electrons in the case of M(↓). These unique edge states that are paired together are referred to as chiral spin valley locking edge states. In addition, the anomalous Hall conductivity (AHC) can be calculated using the formula: σ_xy = Ce²/h, where C, e and h are the Chern number, electronic charge and Planck's constant. It can be found that a non-zero quantization platform within the nontrivial band gap serves as evidence that monolayer NbIn₂As₂Se₂ is a nontrivial TI exhibiting the VQAHE, as shown in Fig. 4(c). The σ_xy value is a result of dissipation-free transport occurring within the distinctly topologically protected chiral spin valley locking edge states. Due to the valley polarization effect, non-zero anomalous Hall conductivity σ_xy can still exist when the Fermi level shifts between the VB and CB. This information demonstrates the presence of the VQAHE in monolayer NbIn₂As₂Se₂. In real space, the system exhibits insulating behavior within its core but demonstrates metallic properties at its edges due to the nontrivial band gap induced by SOC, resulting in an edge Hall current. Fig. 4(d) illustrates the VQAHE featuring dissipationless chiral spin valley locking edge states. The presence of SOC causes the spin signal and electron motion to be intertwined. Spin-down electrons move clockwise, generating an edge Hall current at the edge of the monolayer NbIn₂As₂Se₂. Consequently, when flipping the magnetic direction, the movement direction and spin signal of the electrons also reverse, leading to a corresponding reversal of edge Hall current. These results indicate that the monolayer NbIn₂As₂Se₂ is an intrinsic valley polarization material with the QAHE, which can be applied in quantum computing and information storage.

Fig. 4 Valley quantum anomalous Hall effect analysis. (a) and (b) Edge states of monolayer NbIn₂As₂Se₂ at M(↑) and M(↓). (c) The energy as a function of the corresponding calculated AHC σ_xy. (d) Schematic diagram of the VQAHE.

In order to understand the relationship between strain and electronic structure, we apply biaxial in-plane strain in the system monolayer NbIn₂As₂Se₂ and plot the band gap at the K or K′ valley as a function of strain ε in Fig. 5(a). Under the influence of strain, it can be observed that the system undergoes a transition from a half-metal to a TI to a half-metal. Notice that the systems transform into a half-valley-material (HVM) state at ε = 0.5%. This state means that one valley is a semiconductor and the other is a metal. When strain ε > 1.4%, the value of band gap at the K or K′ valley increases and few electrons occupy the CB at the K valley, and the systems become half-metal. Akin to tensile strain, as compressive strain increases, the system transitions to a half-metal state at ε = −2%. Until the strain is greater than −2%, few electrons occupy the VB at the K′ valley, and the systems become half-metal. As shown in Fig. S4 (ESI†), the d_z² and d_x²−y² at K and K′ have a competitive relationship via strain engineering. The inversion of the d_z² and d_x²−y² orbitals occurs at the valleys at the K or K′ point after a process of closing and reopening. This process can be explained by the deformation of Nb atoms under strain. Tensile strain reduces the overlap between the d_z² orbital of Nb and the p_z orbital of Se, while compressive strain increases the overlap between the d_z² orbital of Nb and the p_z orbital of Se, resulting in a decrease in energy during tension and an increase in energy during compression.

Fig. 5 Strain and the U value regulate valley polarization. The band gap of monolayer NbIn₂As₂Se₂ at K or K′ as a function of the in-plane strain (a) and the U value (b). The pink, blue and yellow areas represent the half-metal, Tl and ferrovalley semiconductor.

We have also considered the influence of the U value on the band gap at the K or K′ valley, plotted in Fig. 5(b). It clearly demonstrates that monolayer NbIn₂As₂Se₂ is in the ferrovalley phase for U = 0.0–2.2 or 3.4–4.0 eV and in the TI phase for U = 2.2–3.4 eV, and the corresponding topological edge states are shown in Fig. S6 (ESI†). On the other hand, the U value can also regulate the band gap value at the K or K′ valley. As the U value increases, the band gap at the K or K′ valley will undergo a process of opening–closing–opening state. Similar to strain engineering, when U = 2.2 or 3.4 eV, the band gap at the K or K′ point closes, and monolayer NbIn₂As₂Se₂ exhibits HVM behavior. We have also observed that the system transitions from the TI state to a ferrovalley state after passing through the HVM state, which is akin to strain effects. The results above demonstrate two methods to manipulate the performance of monolayer NbIn₂As₂Se₂, thereby enhancing its potential applications in spintronic devices.

In summary, monolayer NbIn₂As₂Se₂ is confirmed to be a ferrovalley material with TI via first-principles calculations. We discussed the structural stability in terms of energy, phonon dispersion, molecular dynamics, and mechanical properties. The FM state is the ground state for monolayer NbIn₂As₂Se₂ with a T^M_C of 232 K at U = 3 eV, and the MAE confirms the out-of-plane magnetization direction, thus inducing spontaneous valley polarization. As the system exhibits the VQAHE, reversing the magnetic direction allows for the observation of chiral spin-valley edge states. Strain engineering and the U value can control the state of the system, such as the half-metal, TI and ferrovalley states. Our work provides theoretical guidance for exploring interactions among the FM, valleys, and TI, and offers candidate materials for valleytronic devices.

Data availability

The data that support the findings of this study are available with the corresponding author and can be obtained upon reasonable request.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the Natural Science Foundation of China (Grant No. 22175150 and U2002217). We thank Dr Botao Fu for valuable discussion.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4tc03637a

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