Quantum anomalous Hall effect in monolayers Ti₂X₂ (X = P, As, Sb, Bi) with tunable Chern numbers by adjusting magnetization orientation

Abstract

Despite extensive research, the achievement of tunable Chern numbers in quantum anomalous Hall (QAH) systems remains a challenge in the field of condensed matter physics. Here, we theoretically proposed that Ti₂X₂ (X = P, As, Sb, Bi) can realize tunable Chern numbers QAH effect by adjusting their magnetization orientations. In the case of Ti₂P₂ and Ti₂As₂, if the magnetization lies in the x−y plane, and all C₂ symmetries are broken, a low-Chern-number phase with C = 1 will manifest. Conversely, if the magnetization is aligned to the z-axis, the systems enter a high-Chern number phase with C = 3. As for Ti₂Sb₂ and Ti₂Bi₂, by manipulating the in-plane magnetization orientation, these systems can periodically enter topological phases (C = ±1) over a 60° interval. Adjusting the magnetization orientation from +z to −z will result in the systems’ Chern number alternating between ±1. The non-trivial gap in monolayer Ti₂X₂ (X = P, As, Sb, Bi) can reach values of 23.4, 54.4, 60.8, and 88.2 meV, respectively. All of these values are close to the room-temperature energy scale. Furthermore, our research has revealed that the application of biaxial strain can effectively modify the magnetocrystalline anisotropic energy, which is advantageous in the manipulation of magnetization orientation. This work provides a family of large-gap QAH insulators with tunable Chern numbers, demonstrating promising prospects for future electronic applications.

Graphical abstract

Keywords

tunable Chern number / ferromagnetic / quantum anomalous Hall effect / Ti₂X₂ (X = P, As, Sb, Bi) / first-principles

1 Introduction

In recent years, thanks to the development of topological physics theory and two-dimensional (2D) material preparation, two-dimensional topological materials have received keen attention, enriching condensed matter theory and facilitating the discovery of next-generation quantum functional materials [1–3]. Among them, the quantum anomalous Hall (QAH) effect is a special 2D topological phase of great research significance, which is characterized by the presence of a non-zero Chern number and spin-polarized helical edge electron states within the energy gap of the bulk band structure [4]. The quantum anomalous Hall insulator, also known as a Chern insulator, is distinguished from a conventional insulator due to its unique properties. In the absence of an external magnetic field, internal magnetic exchange interactions can disrupt time-reversal symmetry and open a gap caused by spin−orbit coupling, resulting in quantized anomalous Hall conductivity [5]. Due to the absence of dissipation in its chiral edge states, the QAH effect holds promise for the development of advanced electronic devices with low power consumption. Additionally, it offers an advantageous platform for the exploration of emerging quantum phenomena, such as Majorana fermions and topological magnetoelectric effects [6–10].

Hence, achieving the QAH insulators has become an exceedingly significant focal point within the realm of condensed matter physics. To date, many methodologies and approaches have been proposed to achieve QAH insulators [11–24]. The first observation of the QAH effect occurred in thin films of (Bi/Sb)₂Te₃ doped with Cr at extremely low temperatures (below 30 mK) due to the weak magnetic coupling in the doped Cr atoms and the narrow band gap of the material [25]. However, the impracticality of these extreme experimental conditions limits the potential application of this effect. Since the QAH insulator is a particular type of two-dimensional ferromagnetic semiconductor characterized by a non-trivial bandgap opened by spin−orbit coupling (SOC), both strong ferromagnetic (FM) coupling and SOC strength are crucial to reaching the high-temperature QAH state. Hence, the objective of increasing the observation temperature is to discover an inherent magnetic topological insulator. Fortunately, the QAH effect has been observed in a thin film of MnBi₂Te₄ at a temperature of 1.4 K. By applying an external magnetic field to induce a transition from antiferromagnetic to ferromagnetic interlayer magnetic coupling, this temperature can be raised to 6.5 K [26]. This finding strongly suggests that the intrinsic magnetic insulator has the potential to support the QAH effect at high temperatures, making it suitable for practical applications.

Furthermore, apart from the temperature challenge, the effective control and manipulation of the topological state of Chern insulators is another crucial concern, particularly regarding increasing the Chern number. Enhancing the Chern number in high-Chern-number QAH systems can significantly enhance functionality by inducing the additional chiral edge channels, which exhibit dissipationless properties [27–30]. So far, several scientific studies have shown that the QAH effect can be achieved theoretically with adjustable Chern numbers in different magnetic topological materials, such as MnBi₂Te₄ and multilayer structures of magnetically doped topological insulators [31, 32]. In addition, by altering the direction of magnetization, the Chern number of the materials can also be tuned [33–36]. Topological states are typically protected by different symmetries in the system, which means that different topological states require different symmetries. However, the low symmetry of 2D systems poses a significant challenge to the emergence of multiple topological states. Therefore, it is imperative to undertake in-depth investigations into novel 2D materials that exhibit the QAH effect and possess tunable Chern numbers.

In recent years, a series of honeycomb-type lattice structures, such as MX [37], MX₂ [17, 38–47], and M₂X₃ [48–51] families, have been theoretically predicted to host the QAH effect. Inspired by the above research, we propose a family of 2D monolayer materials called Ti₂X₂ (X = P, As, Sb, Bi) in buckled honeycomb lattice structures. Based on first-principles calculations, we predict they are all QAH insulators. It is crucial to note that all four proposed materials with thermal and dynamical stability have robust FM ground states, beneficial to practical applications and experimental observation. We show that Ti₂P₂ and Ti₂As₂ (Ti₂Sb₂ and Ti₂Bi₂) are x–y easy-plane (z easy-axis) ferromagnetic half-metals. For Ti₂P₂ and Ti₂As₂, due to C₃ and inversion symmetries, six Dirac points appear in the first Brillouin zone. After considering SOC, the nontrivial band gaps open at the Dirac points, allowing for the hosting of the QAH effect. Remarkably, we find that the Chern number depends on the orientation of magnetization. When the magnetization lies in the x–y plane and all C₂ symmetries are broken, C=\pm 1. When the magnetization aligns with the z-axis, a high-Chern-number C=\pm 3 phase arises. For Ti₂Sb₂ and Ti₂Bi₂, a spin-polarized Dirac point exists at each K and {\mathrm{K}}^{\prime } point in the first Brillouin zone. When SOC is introduced, different orientations of magnetization lead to distinct topological states. By tuning the orientation of in-plane magnetization, the system can periodically enter topological phases (C=\pm 1) over a {60}^{\circ } interval. By tuning magnetization orientations from +z to –z, the Chern number of the system alternatively changes between ±1. Finally, we have also discovered that biaxial strain can effectively adjust the magnetocrystalline anisotropic energy (MAE), enabling the manipulation of magnetization orientation and the achievement of tunable Chern numbers for the QAH insulators.

2 Results and discussion

The Ti₂X₂ (X = P, As, Sb, Bi) crystal structure exhibits a double-layered hexagonal lattice of buckled honeycomb with the space group of P-3m1 (No. 164), the same as that of MnSe monolayer [52]. As depicted in Fig.1(a), each unit cell contains two Ti atoms and two X atoms, where the lower layer X is situated beneath the upper layer Ti, while the lower layer Ti is located directly beneath the upper layer X. Tab.1 lists the values of the lattice constants (a), Ti1−Ti2 bond length (d), and Ti1−X−Ti3 angle (\alpha). There are C₃ rotation axes through z(C_{3z}), space inversion (P), and C₂ rotation axes through x(C_{2x}) in the corresponding first Brillouin zone. Especially, due to the existence of C_{3z}, there are three C₂ rotation axes, i.e., C_{2x}, C_{2+}, and C_{2-}, as shown in Fig.1(b).

To confirm the stability of the Ti₂X₂, we first calculated the cohesive energy E_c as E_c=(2E_{\mathrm{T}\mathrm{i}}+2E_{\mathrm{X}}-E_{{\mathrm{T}\mathrm{i}}_2{\mathrm{X}}_2})/4, in which E_{{\mathrm{T}\mathrm{i}}_2{\mathrm{X}}_2} is the energy of each unit cell of Ti₂X₂, and E_{\mathrm{T}\mathrm{i}}, E_{\mathrm{X}} are energies of free Ti and X atoms, respectively. The cohesive energies for Ti₂P₂, Ti₂As₂, Ti₂Sb₂, and Ti₂Bi₂ are 3.51, 3.16, 2.64, and 2.35 eV, respectively. Besides, the dynamic stability is confirmed by the phonon spectra. All phonon modes of Ti₂X₂ display positive frequencies across the momentum space [Figs. S1(a)−(d)], indicating the stability of each structure. Moreover, the thermodynamic stability could be checked by ab initio molecular dynamics simulations. After annealing for three ps at 300 K, the structures remain intact with the fluctuating temperature and total energy [Figs. S1(e)−(h)], further confirming the thermodynamic stability of Ti₂X₂.

Then, we investigate the magnetic ground state of the monolayer Ti₂X₂ (X = P, As, Sb, Bi). The calculations were performed in a 2×1 supercell, with four magnetic phases considered, including one FM and three AFM configurations, i.e., AFM-Néel, AFM-Zigzag, and AFM-Stripy [Fig. S2]. The total energies listed in Table S1 indicate that the ferromagnetic state is preferred in all Ti₂X₂. In the monolayer Ti₂X₂, each Ti atom provides three electrons to the corresponding X atom, resulting in partially-occupied valence-electron shells of 3d^14s^0 for Ti atoms and fully-occupied stable shells for X atoms. According to the crystal field theory, the 3d orbitals of Ti atoms are split into three groups: (d_{xz}, d_{yz}), d_{z^2}, and (d_{xy}, d_{x^2-y^2}). Additionally, after the exchange interaction is introduced, the spin-up and spin-down channels are split, and only one 3d electron occupies one of the (d_{xz}, d_{yz}) orbitals, resulting in a net magnetic moment of 1 {\mu }_B per Ti atom [Fig.1(c)].

To uncover the underlying mechanism of magnetic coupling, we calculated the magnetic exchange parameters using the total-energy mapping method (see details in Section 2 in the supplementary material). The results are presented in Tab.1. J_1 and J_2 represent the parameters of the exchange interactions between the nearest-neighbor (NN) and the next-nearest-neighbor (NNN) Ti atoms, respectively. They are determined by comparing the total energies of the materials with the FM, AFM-s, and AFM-z magnetic structures. Clearly, J_1 is positive and significantly larger than J_2, giving rise to ultrastable FM ground states in all the Ti₂X₂ (X = P, As, Sb, Bi) monolayers. Because of the short distance between NN Ti atoms (~3.0 Å), the Ti−Ti hybridization is strong even for 3d orbitals, which is also demonstrated through the dispersive 3d states near the Fermi level [Fig.2(e)−(h)]. Combined with the specific orbital occupation of Ti 3d orbitals, it leads to a ferromagnetic (FM) state in Ti₂X₂. For NNN Ti atoms, the magnetic exchange parameter J_2 is determined by the superexchange (SE) mechanism. According to the Goodenough−Kanamori−Anderson (GKA) SE rule [53–55], the SE interaction between the NNN Ti atoms via the X p orbitals is FM for a bond angle \alpha of {90}^{\circ } (Ti1−X−Ti3) and AFM for a bond angle of {180}^{\circ }. As presented in Tab.1, the bond angles of Ti−X−Ti in Ti₂X₂ (X = P, As, Sb, Bi) are {114.0}^{\circ }, {111.4}^{\circ }, {105.4}^{\circ }, and {102.7}^{\circ }, respectively, gradually approaching {90}^{\circ }. So, the NNN exchange parameter (J_2) changes from antiferromagnetic coupling for Ti₂P₂ to ferromagnetic coupling for Ti₂X₂ (X = As, Sb, Bi). As listed in Tab.1, D denotes the MAE per Ti ion, defined as \mathrm{\Delta }E=E_{\uparrow }-E_{\to }, where E_{\uparrow }/(E_{\to }) is the total energy when the magnetic moment is oriented along the out-of-plane (in-plane) direction. The MAE illustrates that the magnetization lies in the x−y plane for Ti₂P₂ and Ti₂As₂, and is aligned with the z-axis for Ti₂Sb₂ and Ti₂Bi₂.

We now investigate the electronic band structures of the Ti₂X₂ (X = P, As, Sb, Bi) monolayers. In Fig.2(a)−(d), one can observe that all monolayers are half-metals with wide spin windows (1.55, 1.41, 0.98, and 0.76 eV for Ti₂P₂, Ti₂As₂, Ti₂Sb₂, and Ti₂Bi₂, respectively). The spin-down bands exhibit a large energy gap around the Fermi level, while the spin-up bands display a metallic electronic structure, promising them candidates in spintronics, such as magnetic tunnel junctions and spin valves [56, 57]. For Ti₂P₂ and Ti₂As₂, a spin-polarized Dirac point appears along the \mathrm{\Gamma }−\mathrm{K} high symmetry line at the Fermi level. The presence of C_{3z} rotation and inversion symmetries results in six spin-polarized Dirac points in the first Brillouin zone. But in terms of Ti₂Sb₂ and Ti₂Bi₂, at the Fermi level, there exists a spin-polarized Dirac point located at each \mathrm{K} and {\mathrm{K}}^{\prime } points, respectively. Consequently, Ti₂P₂ (Ti₂As₂) and Ti₂Sb₂ (Ti₂Bi₂) exhibit distinct electronic structural properties. Subsequently, we will categorize them into two groups and analyze each individually.

Firstly, we will investigate the topological properties of Ti₂P₂ and Ti₂As₂. Owing to their similar electronic structure, we take monolayer Ti₂As₂ as an example in the following. After the spin-orbit coupling (SOC) is turned on, various topological phases can be formed depending on the orientation of magnetization. As illustrated in Fig.1(b), there exist three C₂ rotation axes along the high symmetry lines of \mathrm{\Gamma }−\mathrm{K}({\mathrm{K}}^{\prime }). When the magnetization aligns with the C₂ rotation axis (e.g., x-direction), the spin-polarized Dirac points are still preserved along the \mathrm{\Gamma }−{\mathrm{K}}^{\prime } line [see Fig. S3(a)]. In order to open a gap in the in-plane magnetization, it is crucial to break all C₂ rotation symmetries. When the magnetization direction is oriented at a {30}^{\circ } angle with the x-axis, \phi ={30}^{\circ } [Fig. S4], as shown in Fig.3(a), a global gap of about 54.4 meV opens in the monolayer with a Chern number of C = 1. As plotted in Fig.3(b), a single chiral edge state connecting the valence and conductance bands demonstrates the nontrivial band topology with an exotic QAH effect. Additionally, from Tab.1, the tiny MAE (~0.7 meV) suggests that by applying a magnetic field of a few Tesla, it is possible to experimentally orient the magnetization from the x−y plane to the z direction [33, 45]. As illustrated in Fig.3(e), the magnetization aligned with the z direction leads to the monolayer Ti₂As₂ entering a high-Chern-number phase (C = 3) with a band gap of 8.4 meV. As shown in Fig.3(f), there are three chiral gapless edge states connecting the valence and conduction bands in direct agreement with the high Chern number. Further, we continuously adjust the orientations of magnetization to investigate the topological phase transition. Fig.3(c) illustrates that when the magnetization lies in the plane and aligns with the C₂ rotation axis, i.e., \phi =0^{\circ } ({180}^{\circ }), {60}^{\circ } ({210}^{\circ }), {120}^{\circ } ({300}^{\circ }), the band gaps close at the \mathrm{\Gamma }−\mathrm{K}({\mathrm{K}}^{\prime }, \mathrm{K}{}^{″}) [Fig. S5] high symmetry line [Fig. S6]. Consequently, the Chern number of the system alternates between ±1 with a periodicity of {60}^{\circ }. In the case of the out-of-plane magnetization, starting from z direction (\theta ={90}^{\circ }), along the path of \phi = {30}^{\circ }, the global band gap gradually decreases with the decrease of polar angle \theta. Through the first-principles calculations, it has been determined that the gap will completely close at \theta ={85.6}^{\circ }, {98.6}^{\circ } ({265.6}^{\circ }, {278.6}^{\circ }) [Fig. S7], and subsequently reopen. The critical polar angle determines the phase boundaries between low and high Chern numbers. Fig.3(g) represents the phase diagram with distinct Chern numbers. Additionally, it is observed that the maximum gap value does not occur at \theta = 0^{\circ } but rather slightly deviates, manifesting at \theta ={10}^{\circ }.

For other paths (\phi =0^{\circ } and \phi =5^{\circ }), the phase diagrams are presented in Fig. S8. In the case of Ti₂P₂, phase diagrams of the Chern number as a function of azimuthal angle \phi and polar angle \theta are shown in Fig. S9.

In order to explain the results mentioned above, a two-band effective Hamiltonian can be employed using the \mathit{k}\cdot \mathit{p} model. In the absence of the SOC, the spin and orbital degrees of freedom become independent, resulting in the spin being a well-defined quantum number within its own subspace. As such, the \mathrm{\Gamma }−K path has the little group C₂, whose symmetry operators are E and C_{2x}. The effective Hamiltonian H_{\mathrm{e}\mathrm{f}\mathrm{f}} in an effective \mathit{k}\cdot \mathit{p} model can be constrained by C_{2x}{H}_{\mathrm{e}\mathrm{f}\mathrm{f}}(k_x, k_y)C_{2x}^{-1}=H_{\mathrm{e}\mathrm{f}\mathrm{f}}(k_x, -k_y). The H_{\mathrm{e}\mathrm{f}\mathrm{f}} can be written as H_{\mathrm{e}\mathrm{f}\mathrm{f}}= h_0(\mathit{q})+h(\mathit{q})\cdot \sigma, where \mathit{q} is referenced to the Dirac Point, and the Pauli matrices \sigma act on the orbital space (see detail in Section 3.1 in the supplementary material), there, h_0(\mathit{q})=C{q}_x, h(\mathit{q})=(A{q}_x,B{q}_y,0).

When considering the SOC, out-of-plane ferromagnetism breaks the C₂ symmetry as the magnetization direction is perpendicular to the C₂ rotation axes. The SOC introduces a Dirac mass term H^{\prime } =M{\sigma }_z into the effective Hamiltonian, leading to the formation of a Dirac gap. This phenomenon remains valid even when the magnetization direction lies in the plane but does not align with the C₂ rotational axes. When the magnetization direction is oriented along the C₂ rotational axes, the rotational symmetry of C₂ remains preserved, preventing the introduction of a Dirac mass term by the SOC. To understand the Chern number variation, we computed the distribution of Berry curvature in the Brillouin zone. Fig.3(d) and (h) present the distribution of Berry curvature for in-plane and out-of-plane magnetizations, respectively. In the case of in-plane magnetization, two-thirds of the Brillouin zones exhibit Berry curvatures with consistent signs, while the remaining one-third of the zone displays Berry curvatures with opposite signs. Conversely, for out-of-plane magnetization, the Berry curvatures give the same contribution in all Brillouin zones. Since each gapped Dirac point in a system contributes to a Chern number of ±1/2, consequently, the in-plane magnetization results in a small net value of low Chern number (C = 1), while the out-of-plane magnetization leads to a high Chern number (C = 3). Therefore, multiple Dirac cones within the first Brillouin zone are necessary to achieve tunable Chern numbers in the Chern insulator by adjusting magnetization orientation.

Next, we will analyze the topological characteristics of Ti₂Sb₂ and Ti₂Bi₂. Given their similar properties, Ti₂Sb₂ will be utilized as an illustrative example. As previously mentioned, in the absence of the SOC, there exists a spin-polarized Dirac point at each \mathrm{K} and {\mathrm{K}}^{\prime } point. However, when SOC is introduced, different orientations of magnetization lead to distinct topological states. When the magnetization aligns with the C₂ rotation axis, the spin-polarized Dirac points still exist but slightly deviate from the {\mathrm{K}}^{\prime }(\mathrm{K}) point on the {\mathrm{K}}^{\prime }(\mathrm{K})−\mathrm{\Gamma } path [Fig. S3(b)]. On the other hand, when the magnetization direction is at an angle of \phi ={30}^{\circ }, as shown in Fig.4(a), a global gap of approximately 10.2 meV emerges in the system, and as plotted in Fig.4(b), one gapless chiral edge state within the SOC gap is clearly visible, accompanied by a Chern number of C = 1. By tuning the orientation of in-plane magnetization, similar to Ti₂As₂, the system can periodically enter topological phases with alternative Chern numbers of C=\pm 1 in the interval of {60}^{\circ } [Fig.4(c)]. When the magnetization direction is oriented towards the z-axis, as shown in Fig.4(d), a global gap of 60.8 meV emerges in the system, and as plotted in Fig.4(e), one chiral gapless edge states connect the valence and conduction bands, concomitant with a Chern number of C = 1. We continuously tune magnetization orientation from +z to −z (along \phi ={30}^{\circ }). As shown in Fig.4(f), the Chern number of the system also alternatively changes between ±1. However, it is worth noting that the phase transition point does not precisely occur at \theta = 0^{\circ }, but rather at \theta = -{11.6}^{\circ }. In the case of Ti₂Bi₂, phase diagrams of the Chern number as a function of azimuthal angle \phi and polar angle \theta are shown in Fig. S10. The difference in Chen number in Ti₂P₂ (Ti₂As₂) and Ti₂Sb₂ (Ti₂Bi₂) can be analyzed by their electronic structure properties. As shown in Fig.2(e)−(h), as the atomic number of elements in the group VA increases, the relative positions of two different irreducible energy levels near the Fermi level at K point change, until Ti₂Sb₂, where the two energy levels undergo energy level inversion at \mathrm{K} point, which results in Ti₂P₂ (Ti₂As₂) and Ti₂Sb₂ (Ti₂Bi₂) having different Chern number when magnetization direction along +/−z.

The above results can also be described using a two-band effective Hamiltonian through the \mathit{k}\cdot \mathit{p} method. The d_{xz} and d_{yz} orbitals of Ti ions form the basis functions of the 2D irreducible representation at the \mathrm{K}({\mathrm{K}}^{\prime }) valley located around the Fermi level. Owing to the constraint of the corresponding little-group, the effective Hamiltonian H_{\mathrm{e}\mathrm{f}\mathrm{f}} can be written as H_{\mathrm{e}\mathrm{f}\mathrm{f}}=A_1{k}_x{\sigma }_x+A_1{k}_y{\sigma }_y (see detail in Section 3.2 in the supplementary material). Consequently, a Dirac cone with linear dispersions emerges at the \mathrm{K}({\mathrm{K}}^{\prime }) valley, resembling the characteristic Dirac cones observed in graphene. When including the SOC and magnetization direction aligns with the C₂ rotational axes (e.g., x-direction), the C₂ rotational symmetry is preserved and introduces a A_0{\sigma }_x term into the effective Hamiltonian, results in the Dirac point deviate slightly from {\mathrm{K}}^{\prime } point on the {\mathrm{K}}^{\prime }−\mathrm{\Gamma } path. However, in the other cases, the SOC will introduce a Dirac mass term and open a Dirac gap.

Because of the inherent flexibility of two-dimensional materials, the manipulation of strain has proven to be an effective method for modifying various material properties [58–62]. In this study, we investigate the impact of strain on monolayer Ti₂X₂ (X = P, As, Sb, Bi) by subjecting it to biaxial strain ranging from −6% to 6%. As the results depicted in Fig.5(a)−(d), the MAE decreases monotonically from tensile strain to compressive strain, suggesting that strain plays a beneficial role in facilitating the transition of the QAH effect between different topological phases.

3 Conclusion

In conclusion, we systematically investigate these four materials’ structural stability, magnetic properties, and topologically electronic properties based on density functional theory, \mathit{k}\cdot \mathit{p} model, and symmetry analysis. The phonon calculations and molecular dynamics simulations confirmed the dynamical and thermal stabilities of the system. Their FM ground states are determined by both the NN Ti−Ti direct exchange coupling and NNN Ti−Ti superexchange coupling. We find that Ti₂X₂ (X = P, As, Sb, Bi) can realize the QAH effect with tunable Chern numbers by adjusting the orientation of magnetization. For Ti₂P₂ and Ti₂As₂, the QAH effect with low Chern numbers is formed for the in-plane magnetization, while the QAH effect with high Chern numbers arises when the magnetization is aligned along the z direction. Similarly, for Ti₂Sb₂ and Ti₂Bi₂, by tuning the orientation of in-plane magnetization, the system can periodically enter topological phases with Chern numbers of C=\pm 1 within a {60}^{\circ } interval and tuning magnetization orientations from +z to −z, the Chern number of the system alternatively changes between ±1. By subjecting the material to a stretching force, the magnitude of the MAE can be reduced, which is beneficial for controlling the orientation of magnetization using an external magnetic field. Our work provides an ideal platform to realize tunable Chern numbers for the QAH effect via adjusting magnetization orientation, which is expected to be applied in fields such as spintronics, topological superconductivity, and quantum computing.

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