Quantum Anomalous Hall Effect in Ti/Zr/Hf–Bi/Sb Honeycomb Monolayers: First‑Principles Prediction

Background

Two‑dimensional topological insulators (2D TIs) exhibit spin‑polarized, gapless edge states while maintaining an insulating bulk, a hallmark of the quantum spin Hall (QSH) effect [1]. When time‑reversal symmetry (TRS) is broken—either by intrinsic magnetism or external perturbations—helical edge channels are converted into chiral ones, giving rise to the quantum anomalous Hall effect (QAHE) [3]. The dissipationless, magnet‑field‑free charge transport of QAHE makes it attractive for ultra‑low‑power spintronics [4], yet experimental realizations have been limited to diluted magnetic topological insulators [8]. Theoretical work indicates that introducing ferromagnetism and strong spin‑orbit coupling (SOC) into a QSH material can open a topological band gap and generate a non‑zero Chern number [9,10]. While many QSH systems—such as group IV (Sn) and group V (Bi, Sb) monolayers—have been extensively studied, the potential of transition‑metal‑based honeycombs for QAHE remains largely unexplored. Here, we fill this gap by exploring MPn (M = Ti, Zr, Hf; Pn = Bi, Sb) honeycombs, building on recent predictions that III‑V honeycombs can host QSH phases and that transition‑metal doping can induce magnetism [17,25–27]. Our work employs first‑principles calculations to assess whether Ti, Zr, or Hf substitution on Bi/Sb honeycombs can stabilize an intrinsic QAH phase.

Results and Discussions

Replacing half of the Bi atoms in a pristine honeycomb lattice with a transition metal (Ti, Zr, or Hf) yields MPn monolayers that can adopt either low‑buckled or planar geometries. Figure 1a displays the top view of the M‑Pn lattice, while Figures 1b and 1c illustrate the side views of the buckled and planar configurations, respectively. The corresponding first Brillouin zone (BZ) with high‑symmetry points is shown in Figure 1d.

a Crystal structure of M‑Sb/Bi honeycomb. b, c Side views of buckled and planar structures, respectively. d The first Brillouin zone (BZ) with high‑symmetry points

We evaluated structural stability and strain effects by varying the lattice constant and relaxing atomic positions for both buckled and planar phases. The total‑energy curves for TiBi, ZrBi, and HfBi are plotted in Figure 2a–c. Low‑buckled MPn structures consistently possess lower energies than their planar counterparts. Nonetheless, the QAHE appears predominantly in the strained planar regime, with only a narrow lattice‑constant window supporting a QAH phase in buckled HfBi (see Fig. 2c).

Phase diagram of a TiBi, b ZrBi, and c HfBi showing the total energy at different lattice constants. The diagram is divided into various regions labeled as QAH (quantum anomalous Hall phase), I (insulator), and SM (semi‑metal). Blue circles and red triangles represent buckled and planar cases, respectively

Table 1 lists the equilibrium lattice constants, magnetic moments, band gaps, and calculated Chern numbers for all MPn systems. Non‑zero integer Chern numbers identify QAH insulators: planar TiBi (C = +1, Δ = 15 meV) and planar HfBi (C = –1, Δ = 7 meV) are the most promising candidates. The phase transition in TiBi can be driven by reducing the buckling distance (Fig. 3), whereas tensile strain induces a transition in HfBi (Fig. 4).

Phase transition after varying the buckled distance. a Phase diagram of TiBi at a=4.6 Å. The arrow shows the path of the transition. b–f The band structure transition as the buckling distance (δ) was reduced from 0.44 to 0.4 Å. The transition occurs at δ=0.41 Å

Phase transition after varying the lattice constant. a Phase diagram of buckled HfBi. The arrow shows the path of the transition. b–h The band structure transition as the lattice constant was increased from 4.7 to 5.1 Å

Electronic band structures for the equilibrium geometries are presented in Figure 5. In the planar geometry, TiBi exhibits a clear band inversion and a 15 meV gap, whereas planar HfBi shows a 7 meV gap with C = –1. The buckled HfBi, however, is a semi‑metal with a high Chern number (C = –3). Other configurations, such as buckled ZrBi and TiSb, remain trivial insulators.

Electronic band structures of M‑Pn (M=Ti, Zr, and Hf; Pn=Sb and Bi) at their equilibrium lattice constants for a planar and b buckled cases. Red and blue circles indicate +s_z and −s_z contributions, respectively

To dissect the role of SOC and magnetism, we focus on planar TiBi (a = 4.76 Å). Figure 6 shows that the non‑magnetic calculation yields a metallic state, while ferromagnetic ordering (magnetic moment ≈ 1.05 µ_B per cell, dominated by Ti) produces a half‑metallic spectrum. Inclusion of SOC opens a 15 meV gap, confirming that band inversion is SOC‑driven and responsible for the QAHE.

Electronic band structures of planar TiBi film at a=4.76 Å for non‑magnetic calculations (a) without SOC and (c) with SOC as well as ferromagnetic calculations (b) without SOC and (d) with SOC. Red and blue circles indicate +s_z and −s_z contributions, respectively, for (c) non‑magnetic (d) ferromagnetic calculations with SOC

Edge states are essential fingerprints of a topological phase. Using Wannier‑derived tight‑binding Hamiltonians, we constructed a 127 Å wide zigzag HfBi ribbon (Fig. 7). The band structure reveals chiral edge modes crossing the Fermi level an odd number of times, equal to |C| = 1, confirming the QAH nature of planar HfBi.

Band structure along the edge of buckled HfBi zigzag nanoribbon with a=4.9 Å and the width of 127 Å. Blue (red) circles indicate the contribution from the left (right) edges. The bulk bands are denoted by the orange‑filled region

Phonon calculations show small imaginary frequencies for the free‑standing monolayers, suggesting that a substrate may be required for experimental realization. Additionally, within a 2×2 supercell, the ferromagnetic configuration remains energetically favorable compared to antiferromagnetic order in buckled cases, while planar systems exhibit near‑degeneracy between FM and AFM states.

Conclusions

Our first‑principles study demonstrates that substituting Ti, Zr, or Hf into Bi or Sb honeycomb lattices can induce an intrinsic QAHE. While the buckled geometries are energetically preferred, modest tensile strain or a reduction of the buckling distance drives a transition to a planar configuration that hosts a non‑zero Chern number. Planar TiBi and HfBi emerge as robust QAH insulators with band gaps of 15 meV and 7 meV, respectively. These materials add to the growing roster of two‑dimensional QAH candidates and are promising platforms for low‑power spintronic devices.

Methods/Experimental

All calculations were carried out within the density‑functional theory framework using the generalized gradient approximation (GGA) and projector‑augmented‑wave (PAW) potentials as implemented in VASP 5.3. The plane‑wave cutoff was set to 350 eV, and structures were relaxed until residual forces were below 5×10⁻³ eV/Å. Convergence of the electronic self‑consistency loop was reached at 10⁻⁶ eV. A vacuum spacing of 20 Å along the z‑axis prevented interlayer interactions, and a 24×24×1 Gamma‑centered Monkhorst–Pack grid sampled the 2D Brillouin zone. Maximally localized Wannier functions were generated with WANNIER90, enabling the construction of tight‑binding models and the calculation of edge states. Topological invariants were obtained via the Z2Pack package, which tracks hybrid Wannier charge centers to yield the Chern number.