Intrinsic valley-polarized quantum anomalous Hall effect in a two-dimensional germanene/MnI₂ van der Waals heterostructure

Valley, another degree of freedom of carriers analogous to spin and charge, has attracted widespread interest since it uniquely combines low energy consumption and high storage density,^[1–3] which is characterized by the energy extrema on the valence/conduction band.^[4–8] In recent years, new phases of matter have emerged from the coupling of two-dimensional (2D) hexagonal lattices to them due to experimental discoveries of valleytronic materials such as 2D graphene and transition metal dichalcogenides (TMDs). A defining characteristic of this coupling is the valley Hall effect (VHE), which is determined by its intrinsic inversion (P) symmetry breaking.^[9,10] Graphene loses its VHE and valley-dependent optical selection rule since it crystallizes in an P-symmetric honeycomb lattice.^[11] In this regard, germanene is an ideal valley material to achieve valley-dependent property owing to its intrinsic P-symmetry breaking.^[12] VHE links Berry curvature with P-symmetry broken related valley degree, which produces a transverse Hall current due to carriers moving in opposite directions under an in-plane (IP) electric field. Physically, time reversal (T) symmetry is also an important factor for various Hall effects. Quantum spin Hall effect (QSHE), for instance, is topologically protected due to its T-symmetry.^[13] The quantum anomalous Hall effect (QAHE) occurs when the T-symmetry is broken with magnetic doping.^[14] Valley-polarized quantum anomalous Hall effect (VQAHE) combines valleytronics and spintronics with nontrivial band topology.^[15–17] Removing the T-symmetry to generate VQAHE is a pressing issue. Several alternative methods have been proposed to achieve VQAHE, such as decorating magnetic transition metal atoms,^[18] statically magnetic field,^[19] optical pumping,^[20] and magnetic proximity effects.^[21,22] Evidently, to overcome this difficulty, one must go beyond the existing paradigm to generate the VQAHE.

Recently, the approach of magnetic proximity effects in ferromagnetic (FM) van der Waals (vdW) heterostructures (HTSs) is the most attractive and effective way to lift the valley degeneracy.^[23–25] In particular, the vdW HTS composed of 2D layered materials offers a promising approach to minimize lattice mismatch, thereby providing multiple opportunities for manipulating the valley splitting through the utilization of diverse magnetic substrates.^[22] With the development of growth technology for 2D intrinsic FM materials, for example, Gr₂Ge₂Te₆,^[26] VS₂,^[27] CrI₃,^[28] MoTe₂/EuO(111),^[29,30] an attractive opportunity for seeking the combination of valleytronics and band topology into FM vdW HTSs likely triggers emerging physics and novel applications.

Here, with the aid of magnetic exchange effect, we propose that germanene/MnI₂ HTS can realize valley-dependent topological phase. We can see that the contrasting valley features responses to K/K′ valleys in spin splitting with the P-symmetry broken. Furthermore, with introduction of the magnetic proximity effect of an FM substrate, the valley-splitting in K valley still preserves the trivial feature, while the bands in K′ valley undergo a band inversion phenomenon and then become a nontrivial phase. With first-principles supports, germanene/MnI₂ HTS favors out-of-plane (OOP) magnetization and possesses both QAHE and anomalous valley Hall effect (AVHE) in nature. Herein, VQAHE with a valley-dependent chiral edge state could enrich valley-related physics.

Our first-principles calculations are performed using Vienna ab initio simulation package (VASP)^[31,32] based on the density functional theory (DFT). The electronic structure is described by the projector augmented wave (PAW) potentials.^[33] The generalized gradient approximation (GGA) in the form of Perdew–Burke–Ernzerhof (PBE)^[32] is adopted for the exchange–correlation. The force and energy convergence criteria are 10⁻³ eV/Å and 10⁻⁶ eV, respectively. The energy cutoff of the plane wave basis set is 500 eV. A Monkhorst–Pack grid of 15 × 15 × 1 is adopted to sample the Brillouin zone (BZ) and a grid of 25 × 25 × 1 is used for magnetic anisotropy energy (MAE) calculation. A vacuum space of 25 Å is adopted to avoid interactions between adjacent layers. The vdW corrected functional Grimme (DFT-D3) method is employed.^[34] PBE + U method is adopted and U = 4.0 eV due to the correlation effects of Mn-3d electrons, as previously reported.^[35–37] We construct a maximally localized Wannier (MLWF) function by the WANNIER90 package^[38] based on the band structures in the presence of SOC calculated by using DFT. The chiral edge states and anomalous Hall conductivity (AHC) are calculated by the iterative Greens method as implemented in the WannierTools.^[39]

Here, we construct an FM vdW HTS by depositing the monolayer (ML) germanene onto a MnI₂ substrate. The electronic and topological properties resulting from magnetic exchange interactions in germane/MnI₂ FM vdW HTS are elucidated through first principles calculations. Figure S1 presents the crystal structures of ML-germanene and FM ML-MnI₂ and their electronic band structures. As demonstrated in experiments, some bulk metal dihalides possess a naturally layered structure.^[40] Thus, akin to MoS₂ and WSe₂, individual layers of dihalides can potentially be extracted from the bulk through exfoliation techniques. We optimize all atomic positions and lattice constants to ensure the minimum-energy structure for each ML. The structure of ML-dihalides bears conceptual resemblance to that of well-established TMDs. Each metal dihalide consists of three atomic layers, specifically the Mn atom layer sandwiched between two I atom layers. Namely, six I atoms encompass the Mn atom to form an octahedral [MnI₆]⁴⁻ unit with each Mn ion possessing 5 μ_B magnetic moment.^[35] Three possible configurations are considered, as shown in Fig. S2 and the lattice mismatch is 3.63% by the formula η=|a1−a2|(a1+a2)/2×100%.

Subsequently, the stability of germanene/MnI₂ HTS is assessed by binding energy, and its charge transfer mechanism is investigated. To evaluate the stability of germanene/MnI₂ HTS, we define E_b = E_HTS – E_germanene – E_MnI2, where E_HTS, E_germanene, and E_MnI2 are the total energies of germanene/MnI₂ HTS, ML-germanene, ML-MnI₂, respectively. The findings of E_b (refer to Table S1) indicate that all configurations of germanene/MnI₂ HTS under consideration are experimentally viable. Grmanene/MnI₂ HTS features an interlayer distance of approximately 3.6 Å, which exceeds the sum of the covalent radii of I atom (1.33 Å) and Ge atom (1.22 Å). This discrepancy suggests a lack of chemical bonding between the layers, exhibiting typical vdW interaction in germanene/MnI₂ HTS.^[41] Hence, we will utilize the most stable structure C–I, as shown in Figs. 1(a) and 1(b), as a representative example to further investigate the physical properties of germanene/MnI₂ FM vdW HTS in subsequent calculations.

Subsequently, our focus shifts towards the magnetic properties of germanene/MnI₂ HTS based on the C–I structure. In order to determine the magnetic ground state, we consider two collinear magnetic configurations within a 2 × 2 × 1 supercell, encompassing both an FM state and three antiferromagnetic (AFM) states as illustrated in Fig. S3. This result demonstrates the superior energetic favorability of the FM configuration over the AFM configuration in the germane/MnI₂ HTS. Specifically, FM configuration exhibits a 1.4 meV per formula unit (f.u.⁻¹) lower energy compared to AFM1 and a 1.3 meV⋅f.u.⁻¹ lower energy compared to both AFM2 and AFM3 configurations, highlighting the enhanced preference for FM coupling. The FM state of germanene/MnI₂ HTS is closely linked to its geometric structure. The estimated bond angle between I–Mn–I is 88.6°, which is close to 90.0°. According to the Goodenough–Kanamori–Anderson rules,^[42] this structure would favor FM coupling due to super-exchange interactions mediated by neighboring Mn atoms through the I atom. When germanene/MnI₂ HTS is formed, two electrons are transferred from one Mn atom to the six neighboring I atoms. As a result, Mn²⁺ adopts an electronic state of 3d⁵. This leads to the half-filled 3d⁵ orbitals contributing to a formal magnetic moment of 5μ_B on each Mn atom. Additionally, our DFT calculations validate the presence of a spontaneous spin-polarized state in germanene/MnI₂ HTS, exhibiting a magnetic moment of 5μ_B per unit cell. Furthermore, the magnetism in the FM ground state is predominantly contributed by the Mn atoms.

In light of the Mermin–Wagner theorem,^[43] strong thermal fluctuations can easily disrupt long-range ferromagnetic ordering in low-dimensional systems at finite temperatures. However, the recent discovery of 2D magnetic material Cr₂Ge₂Te₆^[44] reveals that magnetic anisotropy energy (MAE), primarily influenced by the SOC effect, plays an important role in maintaining the magnetic ordering stability. A larger MAE indicates a stronger resistance to thermal fluctuations in magnetic ordering, which can enhance data storage performance.^[10] The MAE, defined as the energy difference E_MAE = E_in-plane – E_out-of-plane between in-plane (IP) and out-of-plane (OOP) magnetization directions, incorporates the effect of SOC. A positive or negative MAE signifies an easy magnetization axis aligned with either OOP or IP orientation. In this work, the MAE of germanene/MnI₂ HTS is calculated to be E_MAE = −14 μeV. Hence, this system can be regarded as a 2D XY magnet due to the above elucidation, wherein manipulation of the magnetization from in-plane to OOP orientation is achieved by surpassing an energy barrier of 14 μeV, thereby resulting in a valley polarization.

In order to elucidate the charge transfer mechanism between the ML-germanene and MnI₂ layers, we further perform the work functions, Bader charge analysis and differential charge density of germanene/MnI₂ HTS. The work function is defined as the energy difference between the vacuum level and the Fermi level (E_F). As shown in Figs. 1(d)–1(f), the work functions of ML-germanene, ML-MnI₂, germanene/MnI₂ HTS are 4.48 eV, 5.84 eV, and 4.63 eV, respectively. The lower work function of ML-germanene, compared with that of ML-MnI₂, shows the possibility of electron flow from ML-germanene to ML-MnI₂ upon their contact in Fig. 1(c). Consequently, the ML-MnI₂ acquires positive charges while the ML-germanene gathers negative charges. Additionally, the work function of the germanene/MnI₂ FM HTS lies between those of ML-germanene and ML-MnI₂ owing to spontaneous charge transfer between the two layers. This results in a balanced work function achieved by lifting E_F of ML-germanene and ML-MnI₂ up and down, respectively. In Fig. 1(f), we present the 3D differential charge density of HTS to directly observe the charge transfer occurring at its interface. The cyan and yellow regions represent electron depletion and accumulation, respectively, indicating the spontaneous flow of electrons from the ML-germanene layer to the adjacent ML-MnI₂ layer. Consequently, a vertical built-in electric field forms at the interface, pointing from the germanene layer towards the MnI₂ layer. The presence of the built-in electric field impedes the spontaneous diffusion of holes and electrons yielded by the work function difference, which results in a dynamic balance at the interface. Besides, Bader charge analysis is conducted in germanene/MnI₂ FM HTS, revealing that 0.122 electrons are transferred from the germanene layer to the MnI₂ layer. Aligned with theoretical results from other HTSs, such as stanene/CrI₃^[41] and germanene/NiI₂,^[45] these findings confirm the interlayer charge transfer in the germanene/MnI₂ HTS through various methods, thus supporting the accuracy of our results.

Next, we mainly focus on the valley polarization properties of germanene/MnI₂ FM vdW HTS. The germanene/MnI₂ HTS shows semiconducting characteristics with a direct bandgap of 44.3 meV in Fig. 2(a). Upon inclusion of SOC, the removal of valley degeneracy is observed due to broken T -symmetry. It can be seen that obvious valley polarization occurs at the valence band maximum (VBM) and the conduction band minimum (CBM). As shown in Fig. 2(b), the band structures of the valley-polarized states reveal that the K/K′ valleys in VBM exhibit the opposite spin channels and unequal energy values. Specifically, the presence of the magnetic substrate disrupts T-symmetry by means of the magnetic proximity effect, thereby inducing valley polarization similar to that observed in van der Waals heterostructures based on TMD and FM substrates. For the sake of convenience, the degree of valley polarization can be denoted as ΔE_val = |E_K –E_K′| and ΔE_con = |E_K – E_K′|, where they represent the energy level difference between K/K′ points at the VBM and CBM. If the magnetization is oriented towards the +z direction, the valley polarization originating from the spin-down channel can reach up to 12.4 meV in CBM and 5.8 meV in VBM. By examining the orbital-resolved band structures with SOC in Fig. 2(c), we observe that the states near the E_F are predominantly contributed by Ge₁-p_z and Ge₂-p_z orbitals. This suggests the existence of hybridization and charge transfer between the germanene and MnI₂ layers.

To verify this conjecture, we then establish the Wannier-function-based tight-binding (WFTB) model by utilizing the generated Wannier functions including SOC. The Hamiltonian of WFTB with SOC can be written as^[46,47]

where |ψ_k〉 and R represent the Bloch states over the BZ and the Bravais lattice vector. The hopping parameters t_αβ (R–0) represent the transition from orbital β located at site s_β in the home cell at R = 0 to orbital α situated at sites s_α in the unit cell positioned at R. Next, the Zeeman exchange energy (E_exc) can be expressed as E_exc = gμ_BB, where g is the effective g factor and B is the magnetic field strength. It should be noted that the Peierls phase in the hopping parameters t_αβ or Landau levels is disregarded here. The method is deemed reliable and effective for investigating the topological states induced by an FM substrate.

A vital characteristic for distinguishing between K/K′ valleys in germanene/MnI₂ FM HTS without inversion symmetry is the presence of valley-contrasting Berry curvature, which can be computed as^[48,49]

Here, the sum is over all occupied contributions. f_n is a function of Fermi–Dirac distribution, υ_x(y) represents the velocity operator in the x (y) direction. 〈ψ_nk| and |ψ_mk〉 are the Bloch wave functions with eigenvalues E_n and E_m. Next, we validate the feasibility of realizing VQAHE in the germanene/MnI₂ HTS by calculating the Berry curvature. As illustrated in Fig. 2(e), it is observed that the Berry curvatures of the K valley and K′ valley exhibit unequal magnitudes and opposite signs, indicating that the flow of electrons occurs within a curled field and contributes to the quantized Hall conductance. The Chern number C is calculated to be −1, which is also obtained by integrating the Berry curvature in the whole BZ. It is noteworthy that the C solely originates from the K′ valley, that is, C_K′ = 1 and C_K = 0, resulting in a valley Chern number C_υ = C_K – C_K′ = −1 and so indicating the realization of VQAHE. The non-zero valley Chern number C_υ of germanene/MnI₂ HTS confirms the existence of topologically nontrivial edge states and quantized Hall conductivity. Thus, we utilize the iterative Green’s function method based on the WFTB Hamiltonian to compute the edge states, which is plotted in Fig. 2(f). It is evident that a topological chiral edge state connects the valence and conduction bands, indicating the presence of a quantum anomalous Hall phase. Next, we employ the WFTB Hamiltonian based on Kubo formula to calculate the anomalous Hall conductance (AHC) σ_xy as depicted in Fig. 2(d). σ_xy shows a quantized plateau of σ_xy = −e²/h inside the nontrivial band gap.

Nowadays, electrical manipulation of magnetism is a topic of significant interest for spintronic applications that prioritize low energy consumption. To validate the aforementioned concept, an additional layer of In₂Se₃ is introduced at the bottom of germanene/MnI₂ HTS, resulting in a multiferroic vdW HTS term as germanene/MnI₂/In₂Se₃. Here, two distinct polar configurations are considered for the In₂Se₃ layer: either both ferroelectric layers are oriented upwards (P↑) or downwards (P↓). Figures 3(a) and 3(c) illustrate the geometric arrangements of germanene/MnI₂/In₂Se₃ in the P ↑ and P ↓ configurations. For P↑ cases, characterized by a direct band gap of 0.57 eV, the germane/MnI₂/In₂Se₃ HTS keeps a semiconducting nature as depicted in Fig. 3(b). By inducing a flip in the electrical polarization of In₂Se₃, we observe an upward shift in the bands associated with the germanene/MnI₂ HTS, resulting in their intersection at the E_F. This indicates that fascinating metallic properties are generated by the germane/MnI₂/In₂Se₃ HTS, as depicted in Fig. 3(d). From the band structure, it can be seen that the electronic states of the E_F layer are predominantly contributed by the germanene/MnI₂ layer, thereby imparting metallic properties to germanene/MnI₂/In₂Se₃ HTS. Consequently, when subjected to an upward ferroelectric polarization of In₂Se₃, it becomes feasible to generate a complete polarization current. Therefore, by reversing the polarization orientation of the ML-In₂Se₃ from upward to downward, a transition can be induced in the germanene/MnI₂ system, transforming it from a semiconductor into a metal. Notably, this process leads to a significant change in the material’s properties. The energy barrier for the transition from P ↑ configuration to P ↓ configuration is 44.8 eV. Therefore, the germanene/MnI₂/In₂Se₃ HTS offers sufficient bistability for the device application.

In conclusion, we explain a mechanism of VQAHE in FM hexagonal lattice based on first-principles calculations. We further systematically study topological phase transitions and valley-based Hall effects in germanene/MnI₂ HTS. More interesting, we have designed vdW HTS comprising FM germanene/MnI₂ HTS and ferroelectric In₂Se₃ monolayers with multiferroic properties. Reversing the direction of ferroelectric polarization of In₂Se₃ induces a transition in germanene/MnI₂/In₂Se₃ HTS, transforming it from a semiconductor into a metal. The intriguing phenomenon can be primarily ascribed to the charge transfer occurring between the germanene/MnI₂ HTS and ML-In₂Se₃. The present work aims to achieve nonvolatile electrical control of metallicity, thereby paving the way for the development of interfacial magnetoelectric physics and applications.