Introduction

Quantum anomalous Hall (QAH) effect is the most profound topological quantum state and triggers the flourishing of topological matters since proposed by Haldane1,2,3,4,5,6,7,8,9. Its quantized Hall conductivity \({\sigma }_{H}=C\frac{{e}^{2}}{h}\) is characterized by a finite Chern number C of the band structure, corresponding to the non-dissipative chiral edge states, which have great prospective applications in low-power-consumption electronics. However, the accomplishment of QAH effect so far requires extremely low temperature, i.e., ~30 mK in magnetically doped topological insulators (TI)10,11,12,13, ~1.4 K in the intrinsic magnetic TI MnBi2Te414, ~2.5 K in the moiré systems15,16, thus hinders the application in reality. Enhancing the working temperature is the key step for the future advancement of this field, which encounters cHallenges primarily from two fronts. The first challenge is to enhance the Curie temperatures (Tc) of magnetic materials, while the second is to enhance the topological gap of the band structure. The pursuit of room temperature QAH effect encourages many theoretical efforts on the ferromagnetic candidates with high Tc and sizable topological gaps17,18,19,20,21,22,23,24,25,26. To date, an experimentally accessible candidate for the room temperature QAH effect is still sought.

Recent studies on topological classification27,28,29,30 suggest that the time-reversal symmetry-breaking three-dimensional (3D) insulators could exhibit weak topological properties, featuring integer-quantized Hall conductivity per layer. This topological phase can be understood as a stacking of QAH layers by weak interlayer interaction, and it has been referred to as a “3D weak Chern insulator”31,32. When fabricated into a thin film, its Hall conductivity is quantized and proportional to the number of layers. Such a feature suggests a promising route to realize the QAH effect by exfoliating the 3D weak Chern insulator to obtain the QAH layer.

In this work, we predict the Lanthanide halide TbCl as such a prominent candidate based on first-principles calculations. The van der Waals (vdW) layered bulk TbCl is identified by our calculation as a ferromagnetic insulator with out-of-plane magnetic moments, which exhibits the 3D weak Chern insulator phase characterized by the Hall conductivity G = −e2/h per layer. Due to the small exfoliation energy ~0.24 J/m2, the bulk TbCl can be exfoliated easily into a monolayer, and the monolayer is confirmed to be dynamically stable by the phonon spectrum. Notably, the monolayer inherits the ferromagnetism (FM) of the bulk and exhibits the high Tc ≈ 457 K because of the sizable exchange and superexchange interactions originating from the extended 5d electrons. The band structure calculation identifies the monolayer TbCl as QAH insulator featured by Chern number C = −1 with a large gap 42.8 meV, which is obviously beyond the room temperature. These results strongly demonstrate that the monolayer TbCl is an ideal candidate for achieving the room temperature QAH effect, due to its small exfoliation energy, high Tc, and large topological gap.

Results

Electronic and topological properties of bulk TbCl

The experiment has synthesized the bulk TbCl33, which presents an ABC-type stacking pattern along the c direction with the \(R\bar{3}m\) (D3d) symmetry, as shown in Fig. 1a. The lattice constants are determined as a = 3.787 Å and c = 27.461 Å. Tb and Cl both sit at Wyckoff positions 6c with coordinates (0,0,0.116) and (0,0,0.385).

Fig. 1: Structure and orbital analysis of TbCl.
figure 1

a Crystal structure of the bulk TbCl and its magnetic structures in the primitive cell. b HSE06 calculated DOS (Right panel) and band structure (Left panel) for the bulk in the nonmagnetic state. Red, blue, and green represent the projection on \({d}_{{z}^{2}}\), \({d}_{xy/{x}^{2}-{y}^{2}}\) and s orbitals of Tb. c Splitting of the 5d and 6s orbitals due to the hybridization, crystal field, and spin polarization. d The HSE06 calculated FM band structure without SOC.

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We perform the HSE06 calculation to show the band and density of states (DOS) for the bulk nonmagnetic state in Fig. 1b. The 4f orbitals are pushed away from the Fermi level by the strong Coulomb interaction, and eight f orbitals on each Tb are occupied. Thus, the Fermi level is dominated by the \({d}_{{z}^{2}}\), \({d}_{xy/{x}^{2}-{y}^{2}}\) and s orbitals. These results are further confirmed by the DMFT calculation as shown in Supplementary Fig. 1 in the Supplemental Material. To better understand the 5d and 6s orbital bands, we further study the evolution of the orbitals at the Γ point as shown in Fig. 1c. The neighboring Tb atoms, due to the short distance (3.54 Å), hybridize with each other to form the lower bonding states s†/d† and higher antibonding states s*/d* (Stage I). Among these, the bonding orbital s† is fully occupied, leading to the 6s1/Tb, and the bonding orbitals d† near the Fermi level are further split by the trigonal crystal field into the occupied singlet \({a}_{1}^{\dagger }\) (\({d}_{{z}^{2}}^{\dagger }\)), fully empty doublets \({e}_{1}^{\dagger }\) (\({d}_{xy/{x}^{2}-{y}^{2}}^{\dagger }\)) and \({e}_{2}^{\dagger }\) (\({d}_{xz/yz}^{\dagger }\)) (Stage II). The resulting 4f85d16s1 configuration agrees with the previous observation of Tb+1 ion very well34. Subsequently, the spin splitting of 5d orbitals (Stage III) further pushes the \({a}_{1}^{\dagger }\uparrow\) branch away from the Fermi level to make it fully occupied, while moves \({a}_{1}^{\dagger }\downarrow\) and \({e}_{1}^{\dagger }\uparrow\) branch toward the Fermi level and cross with each other as shown in Fig. 1d, which is the HSE06 calculated band structure of FM configuration without spin-orbital coupling (SOC). We employ both HSE06 and DFT+U methods to capture the correlation effects of 5d and 4f electrons. As shown in Fig. S2, of Supplemental Material DFT+U with U = 6 eV yields electronic structures in good agreement with HSE06. Therefore, we adopt U = 6 eV for all subsequent DFT+U calculations unless otherwise noted.

Table 1 lists the calculated difference energy of four different magnetic structures (Fig.1a), among which FM-z is the ground state with 4.2 meV/f. u. lower energy due to magnetic anisotropy. The spin magnetic moment is found to be ≈6.36 μB/Tb, which is understood as follows. Firstly, the eight 4f electrons on each Tb occupy ↑7↓1 due to Hund’s coupling, and contribute 6 μB/Tb. Secondly, the partial occupation of \({d}_{{z}^{2}}^{\dagger }\downarrow\) reduces the contribution of the spin-up 5d orbitals. Thus, the total spin magnetic moment is finally reduced to about 6.36 μB/Tb.

Table 1 The magnetic space group, spin magnetic moment, and total energies of bulk TbCl in different magnetic configurations
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We further discuss the topological properties of the bulk TbCl. Figure 1d shows the band inversion between \({d}_{{z}^{2}}^{\dagger }\downarrow\) and \({d}_{xy/{x}^{2}-{y}^{2}}^{\dagger }\uparrow\) around F point. The SOC can open a topologically nontrivial gap, as shown in Fig. 2a. We calculate the evolution of the Wannier charge centers to abtain the Chern number in kz = 0 and kz = π planes, which are found to be both C = −1 as demonstrated in Fig. 2b. Such results mean that the bulk TbCl is a 3D weak Chern insulator stacked by the QAH layers with Hall conductivity G = −e2/h per layer, which can be also confirmed by the chiral edge state in the (010) surface along \(\bar{\Gamma }-\bar{X}\) and \(\bar{Z}-{\bar{X}}^{{\prime} }\) as shown in Fig. 2c, d, respectively.

Fig. 2: Band structure and topological properties of bulk TbCl.
figure 2

a is the DFT+U band structure with SOC in FM-z state at U = 6 eV. b The evolution of Wannier charge center (WCC) along kx at kz = 0 (blue dots) and kz = π (red dots) plane, respectively. The inset is the Brillouin zone and the projected surface Brillouin zones of (010) plane. c, d are the surface states on (010) plane at kz = 0 and kz = π, respectively.

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Room temperature QAH effect in monolayer TbCl

The topological properties of the 3D vdW bulk TbCl with the Hall conductivity G = −e2/h per layer of the crystal suggest it as a perfect candidate to fabricate the 2D QAH system26,27,28,29. The exfoliation energy of monolayer TbCl, determined using a five-layer slab model as shown in Fig. 3a, is 0.24 J/m2, which is noticeably lower than the experimental value for graphite (~0.32 J/m2) and H-MoS2 (~0.29 J/m2)35,36. Such a low exfoliation energy suggests that monolayer TbCl can be more easily exfoliated from its bulk counterpart, indicating its excellent experimental accessibility. The monolayer TbCl belongs to space group \(P\bar{3}m1\) (no.164), where the Tb and Cl atoms are located at 2c and 2a Wyckoff positions, respectively, as shown in Fig. 3c. The calculated phonon spectrum demonstrates that the monolayer TbCl is dynamically stable, as shown in Fig. 3b. Additionally, the formation energy of monolayer TbCl is calculated to be -2.8908 eV/atom, and ab initio molecular dynamics simulations further confirm its thermal stability at both 300 K and 600 K (see Supplemental Material).

Fig. 3: The stability and magnetism of the TbCl monolayer.
figure 3

a Calculated exfoliation energy vs. separation distance (d−d0). The inset shows the side view of the five-layer slab model. b Phonon spectra of the TbCl monolayer. The inset shows the Brillouin zone. c Side view of monolayer TbCl. J1 and J2 are the nearest and next-nearest neighboring exchange couplings, respectively. d–h The magnetic configuration of Zigzag-z, Neel-z, Zigzag-x, Neel-x, and 120∘ type antiferromagnetic structure. i Illustration of the local coordinates \({x}^{{\prime} }-{y}^{{\prime} }-{z}^{{\prime} }\) along the Tb-Cl bonds. The light red, blue, and green colors label the \({x}^{{\prime} }{y}^{{\prime} }\), \({x}^{{\prime} }{z}^{{\prime} }\), \({y}^{{\prime} }{z}^{{\prime} }\) planes, respectively. The global \({d}_{{z}^{2}}\) orbital is represented by the orange ball. The inset shows the side view of the local coordinates. j The magnetization (blue dot-line) and specific heat (red dot-line) depending on the temperature.

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To determine whether the bulk magnetism is inherited by the monolayer, we compare the energies of several representative magnetic structures, including the FM, Neel, Zigzag, and the 120∘-AFM, as illustrated in Fig. 3d–h. The magnetic anisotropic energy 3.8 meV/f. u. is obtained (see Table. S1 in SM), consistent with the bulk magnetic anisotropic energy. Thus, we only list the z-oriented configurations and 120∘-AFM in Table 2, which identifies FM-z as the ground state, consistent with the bulk magnetism. Based on these results, we estimate the magnetic transition temperature by employing the Heisenberg model up to the next nearest exchange coupling:

$$H=-{J}_{1}\sum _{\langle i,j\rangle }{{\boldsymbol{S}}}_{i}\cdot {{\boldsymbol{S}}}_{j}-{J}_{2}\sum _{\langle \langle i,j\rangle \rangle }{{\boldsymbol{S}}}_{i}\cdot {{\boldsymbol{S}}}_{j}-D\sum _{i}{\left({S}_{i}^{z}\right)}^{2}.$$
(3)

where J1 and J2 are the nearest (interlayer) and next nearest neighboring (intralayer) exchange coupling, as labeled in Fig. 3c, 〈i, j〉 and 〈〈i, j〉〉 represent the summation over nearest and next nearest neighbors. S (∣S∣ = 3.5) is the spin angular momentum of Tb atom and \({S}_{i}^{z}\) is the z component. D represents the anisotropy energy. Using the energy of different configurations, we can immediately extract the values of the J1, J2, and D as listed in Table 2. With these parameters, the Curie temperature is evaluated by Monte Carlo simulation as Tc ≈ 457 K (see details in the Supplemental Material), which is obtained by plotting the temperature-dependent magnetization and specific heat in Fig. 3j. Such significant high Tc, even exceeding room temperature, is because that both J1 and J2 are FM and sizable that are originated from the more extended 5d orbitals as analyzed in the next paragraph, which usually leads to an order of magnitude larger exchange coupling comparing to that of the 4f orbital-dominated rare earth magnets like GdI337 and EuS38,39, and gives rise to transition temperatures in the hundreds of Kelvin, as reported in LaCl (~260 K)23 and GdI2 (~241 K)31,40,41.

Table 2 The relative total energy of four different magnetic structures (meV per unit cell), and the estimated exchange coupling J1, J2 (meV), anisotropy energy D (meV/Tb), and Curie temperature Tc (K)
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The FM mechanism of J1 and J2 can be understood qualitatively in the following way. The FM between the nearest Tb is attributed to their short distance (3.54 Å), which is comparable to that of Tb single crystal (3.525 Å), and such a short distance leads to the positive exchange integral42,43. The FM between the next nearest Tb is understood by the superexchange mediated via the Cl-p orbitals. As sketched in Fig. 3i, the occupied \({d}_{{z}^{2}}\) orbital (orange orbit in the sketch) can be decomposed into \({t}_{2g}^{{\prime} }\) orbitals in the local coordinate as \(\vert {d}_{{z}^{2}}\rangle =\frac{1}{\sqrt{3}}\vert {d}_{{x}^{{\prime} }{y}^{{\prime} }}\rangle +\frac{1}{\sqrt{3}}\vert {d}_{{x}^{{\prime} }{z}^{{\prime} }}\rangle +\frac{1}{\sqrt{3}}\vert {d}_{{y}^{{\prime} }{z}^{{\prime} }}\rangle\). Then one can analyze the superechange across the ~90∘ Tb-Cl-Tb bonds in \({x}^{{\prime} }{y}^{{\prime} }\), \({y}^{{\prime} }{z}^{{\prime} }\) and \({z}^{{\prime} }{x}^{{\prime} }\) plane, respectively. Without lose the generality, we take the Tb1-Cl-Tb2 in the \({x}^{{\prime} }{y}^{{\prime} }\) plane as an example, as labeled in Fig. 3i and also in Fig. S4 of Supplemental Material. The virtual hopping between \({p}_{{x}^{{\prime} }}\) (\({p}_{{z}^{{\prime} }}\)) of Cl and \({d}_{{x}^{{\prime} }{y}^{{\prime} }}\) (\({d}_{{y}^{{\prime} }{z}^{{\prime} }}\)) of Tb1 is allowed, which occurs in such a way that the spins on Tb1 are mutually antiparallel. Meanwhile, the remaining \({p}_{{x}^{{\prime} }}\) (\({p}_{{z}^{{\prime} }}\)) electron of Cl is orthogonal to all the \({t}_{2g}^{{\prime} }\) (\({d}_{{y}^{{\prime} }{z}^{{\prime} }/{x}^{{\prime} }{y}^{{\prime} }}\)) orbitals on Tb2, which leads to the FM exchange integral. Therefore, according the Goodenough-Kanamori-Anderson (GKA) rules44,45,46, the superexchange coupling between Tb1 and Tb2 is dominated by an FM interaction. A similar analysis is applied to \({y}^{{\prime} }{z}^{{\prime} }\) and \({z}^{{\prime} }{x}^{{\prime} }\) plane as shown in Fig. S5 of Supplemental Material, which presents the same results yielding the C3z symmetry, and finally gives rise to a sizable FM J2 comparing to J1.

Additionally, we would like to highlight the significant anisotropy energy D = 0.31 meV in monolayer TbCl, which is considerably higher than that of 3d compound CrI3 (~0.2 meV)47,48. This difference is attributed to the stronger SOC in 5d orbitals compared to 3d orbitals. The large anisotropy energy can suppress the in-plane magnetic fluctuation, and guarantees the robustly stable out-of-plane magnetic order in the monolayer TbCl. As a result, the sizable FM exchange coupling J1, J2, and anisotropy energy D strongly suggest that the monolayer TbCl is an ideal room temperature FM material.

Finally, we determine the band topology and QAH effect of the monolayer TbCl. Based on HSE06 functional method with SOC, the band structure of FM-z monolayer is calculated and demonstrates a large global band gap ~42.8 meV, as shown in Fig. 4a. Such large gap is obviously above the room temperature, and is characterized by the Chern number C = −1, which is obtained by integrating the Berry curvature of all the occupied bands49,50. The anomalous Hall conductivity as a function of chemical potential is calculated in the left panel of Fig. 4b, in which a quantized plateau with \({\sigma }_{xy}=-\frac{{e}^{2}}{h}\) appears in the topological band gap, corresponding to the chiral edge state as shown in the right panel. Thus, based on the significant band gap and high Curie temperature mentioned earlier, the room temperature QAH effect can be achieved in a monolayer of TbCl.

Fig. 4: The topological properties of the TbCl monolayer.
figure 4

a The band structure of monolayer TbCl with SOC. The band gap and the Chern number are marked. b The left panel is the anomalous Hall conductivity σxy as a function of Fermi energy. The right panel is the chiral edge state on (110) edge.

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Discussion

In summary, we predict the monolayer TbCl as the experimental candidate to realize the room temperature QAH effect. The bulk TbCl is found to be an FM insulator, supporting the Hall conductivity G = −e2/h per layer of the crystal. The weak van der Waals interactions allow TbCl to be easily exfoliated into a monolayer, which retains the ferromagnetic properties of the bulk material. And also, the QAH effect is inherited in the monolayer, corresponding to the Chern number C = −1. We estimate the Curie temperature as ~457 K, and the topological gap as ~42.8 meV, which are both beyond the room temperature. Therefore, the monolayer TbCl has great potential to be a platform to observe and apply the topological quantum phenomena at room temperature. Compared to the previously theoretically proposed high-temperature QAH insulators, monolayer TbCl presents clear advantages. Unlike most proposals, which are based on the 3d orbitals, TbCl is featured by the extended 5d orbitals, which induce strong FM and a high Curie temperature. Furthermore, its vdW layered structure and low exfoliation energy (0.24 J/m2) make it experimentally accessible, similar to CrI3 and FePS3.

Methods

Methods for first-principles calculations

The first-principles calculations are carried out based on density functional theory (DFT), which is implemented in the Vienna ab initio Simulation Package51,52,53. The plane wave energy cutoff is set as 400 eV, and a Monkhorst-Pack k-point mesh of 10 × 10 × 10 and 20 × 20 × 1 for bulk and monolayer unit cells is used, respectively. To accurately capture the interlayer vdW interactions in layered TbCl, we adopted the DFT-D3 scheme, which yields lattice constants and interatomic distances closest to experimental values among the tested vdW correction methods. The vacuum layer used for monolayer TbCl is 30 Å. For the correlation effects of the Tb-4f electrons, the Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional and DFT+U approach are adopted in this work54,55,56. The calculations of edge state and anomalous Hall conductance are carried out using maximally localized Wannier functions through the Wannier9057 and WannierTools packages58. The phonon dispersion spectrum is calculated using a 4 × 4 × 1 supercell for the monolayer and PHONOPY code with the finite displacement59. The Curie temperature of the monolayer is estimated by Monte Carlo simulations MCsolver code60.