Introduction

Since the discovery of graphene, two-dimensional (2D) materials have been one of the most intensively studied systems owing to their intriguing physical properties and the chance of applications to atomistic low-power devices1,2,3. Transition metal chalcogenides (TMC) with layered structures is a representative example of such 2D materials4,5. One intensively pursued subject recently in TMC is magnetism originating from the presence of localized magnetic moments in d orbitals at TM ions, where hybridization between TM and chalcogen anions give rise to interesting magnetic behaviors in XPS3 (X = Mn, Fe, Ni) and CrBTe3 (B = Si, Ge)6,7,8,9,10. Systems like CrBTe3 or CrI3 are reported to show Ising-type magnetism and exhibit magnetic orders down to single-layer limit11 at finite temperatures12,13, defying the Mermin-Wagner theorem that prohibits long-range order in systems with d ≤ 2 dimensions via thermal fluctuations14. Successful exfoliations of atomistically-thin CrI3 and CrGeTe3 layers present promising magnetic 2D system with potential device applications12,13.

The presence of robust magnetism in two-dimensional system like CrBTe3 becomes even more interesting in the study of topological properties15,16,17,18,19,20,21,22,23,24, where TMC materials has been recently considered as one of good platform of researching topological phases of matters25,26 and magnetism can be employed as a control knob to tune topological properties in such systems27,28,29. Specifically, quantum anomalous Hall effect (QAHE) can be observed in materials with magnetic ordering and sizable band gap, showing quantized integer Hall conductivity in the absence of external magnetic fields. Chern insulator is one of topological phases of matter showing QAHE characterized by Z topological invariant, i.e., Chern numbers, unlike time-reversal-symmetric conventional topological insulators classified by Z2-invariants such as Bi2Se3, Bi2Te3, Sb2Te330, or HgTe31. Recent studies suggest Chern insulating phases with high Chern numbers and wide band gap energy25,26 can be applied for low-power dissipationless device applications, but exploring suitable candidates is a nontrivial task. CrSiTe3, originally reported to be a 2D ferromagnetic semiconductors with trivial band topology8,32,33,34, can be an ideal candidate in the presence of charge doping because of the robust magnetism and the insulating behavior down to the single-layer limit, in addition, due to the presence of strong SOC from Te sites.

In this work, by performing first-principles density-functional theory (DFT) calculations, we find Chern insulating phases in CrSiTe3 monolayer in the presence of electron doping. We find crossing points within the Cr eg conduction band manifolds, where the crossings are removed as spin-orbit-coupling (SOC) is introduced and leading to topologically nontrivial bands with Chern number up to 8. We also find quantized anomalous Hall conductivity (AHC) at one- and two-electron-doping per unit cell, which can be further varied as spin orientation angle changes. Additionally, it is found that the magnetic anisotropy energy (MAE) is suppressed as electron doping is introduced via electrostatic gating, consistently with recent findings35,36,37, which makes tuning of AHC via external magnetic field feasible. Finally, we construct a tight-binding (TB) hamiltonian for a deeper understanding of our DFT results and to further pursue the possibility of realizing distinct Chern insulating phases via external stimuli such as strain. Our result reveals that single-layers of magnetic CrSiTe3 and its structural siblings are promising platforms to realize Chern insulator materials with high Chern numbers.

Results

Electronic and magnetic properties of stoichimetric CrSiTe3

CrSiTe3 is one of MAX3-type TMTC compounds. It has a \(R\overline{3}\)-type stacking of neighboring CrSiTe3 monolayers, where neighboring layers are coupled by vdW interaction. Crystal structure of a CrSiTe3 monolayer is depicted in Fig. 1, showing edge-sharing honeycomb network of CrTe6 octahedra and Si-dimers located at the centers of Cr-hexagons. In undoped CrSiTe3, Cr cations are in the d3 (Cr3+) high-spin configuration (S = 1.5) with fully occupied t2g orbitals. In addition, strong dpσ hybridization between Cr d- and Te p-orbitals gives rise to additional magnetic moment contributions, so that the magnitude of Cr spin moment is 3.87 μB38. The ferromagnetic ground state can be attributed to the Goodenough–Kanamori–Anderson (GKA) superexchange mechanism39,40. Our results are in agreement with previous reports8,32,41 where choice of Hubbard U parameters is crucial for determining magnetic ground state of CrSiTe338.

Fig. 1: Crystal structure of monolayer CrSiTe3 (top view).
figure 1

The unit cell features the Cr2Si2Te6 formula unit. Edge sharing CrTe6 octahedron consists of Cr atom (gray sphere) and surrounding six Te atoms (yellow sphere). Dimerized two Si atoms (blue sphere) are sitting at center of Cr honeycomb lattice. a, b, c is unit vector of hexagonal unit cell, where x, y, z indicates local axis.

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When electron doping is introduced, orbital occupation of CrSiTe3 changes so that magnitude of the magnetic moment of Cr d orbitals, as well as lattice constant, is also affected. With two-electron doping per formula unit cell, Cr eg orbital of spin majority channel becomes half-filled so that the high-spin Cr2+ (S = 2) configuration is obtained (additional moment of 0.51 μB further induced due to dpσ-hybridization). With this electron doping, the in-plane lattice parameter a expands to be 7.26 Å, 5.7% larger than that of undoped case6,32,38. We check that the ferromagnetic ground state with insulating properties and also topological phases remain unchanged within the Hubbard U parameters and lattice constant range of 0.5 eV ≤ U ≤ 1.5 eV and 6.87 Å ≤ a ≤ 7.26 Å, respectively.

Before taking a closer look at anomalous Hall responses of CrSiTe3, we note in passing on our choice of the value of Hubbard U. Although a typical value of U parameters used in DFT+U calculations would be 3.0 eV for Cr 3d orbital states, ARPES measurements of CrGeTe3, a structural sibling of CrSiTe3 with similar electronic and magnetic properties, does not support the DFT+U calculation results with the choice of U = 3.0 eV38. Instead, the photoemission results are in good agreement with the U = 1.1 eV result42. Furthermore, the U value larger than 2.5 eV gives a metallic band structure for CrSiTe342, which contradicts the experiment’s insulating behavior33,41,43 In addition, we confirmed that the ferromagnetic ground state and topological properties of CrSiTe3 remain unchanged in the range of 0.5 eV ≤ U ≤ 1.5 eV. Therefore our choice of U = 1.5 eV seems appropriate.

Band crossings in CrSiTe3 e g conduction bands

To study magnetic and topological properties of electron-doped monolayer CrSiTe3 in the FM state, we focus on the electronic structure of undoped CrSiTe3. Figure 2 describes band structures and projected density-of-states of ferromagnetic single layer CrSiTe3. Majority- and minority-spin components show exchange splitting where the minority-spin parts have large band gap energy compared to the majority-spin parts. Right above the Fermi level, four majority-spin bands consisting of Cr eg- and Te p-orbitals exist and are separated from other bands (except Cr t2g bands in the minority-spin channel). Interestingly, band crossings are found within the eg bands at Γ, K points and on the Γ–M, Γ–K high-symmetry lines. This observation suggests the possibility for the monolayer CrSiTe3 to host nontrivial band topology in the presence of electron doping and gap opening via SOC, as discussed in the following section.

Fig. 2: Band structure and pDOS plot.
figure 2

a Band structure with majority spin (red line) and minority-spin channel (blue line), respectively. b Total density of states (black line), projected density-of-states (pDOS) of Cr eg orbital (red line) and t2g orbital (blue line), Si p orbital (purple line) and Te p orbital (sky blue line), respectively. In the pDOS plot, plus and minus signs correspond to the majority and minority-spin components, respectively. Fermi level is set to be zero for both panels.

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Berry curvature plot, Chern numbers, and chiral edge states

Figure 3 shows magnified Cr eg band structures with Chern numbers assigned to each band, plots of Berry curvature in the momentum space, and edge spectra in the presence of integer electron doping, SOC, and out-of-plane ferromagnetic spin orientation. We consider the cases of one and two electron doping per formula unit cell, where both systems are insulating as shown in Fig. 3a and d. For each band, we find unusually high Chern numbers of up to 8, with about 10 meV order of SOC-induced band gap mainly originating from Te p orbital contributions.

Fig. 3: Chern number, Berry curvature, and chiral edge modes.
figure 3

Cr eg bands manifold calculated by using DFT (black line) and Wannier function interpolation (red circle) under a one electron doping and d two electron doping per Cr2Si2Te6 formula unit, respectively. Chern numbers for each band are also remarked. Berry curvature plot of the occupied Cr eg band in momentum space under b one electron doping and e two electron doping per formula unit, respectively. Unit of Berry curvature is Å2. Plot of chiral edge states along armchair direction under c one electron doping and f two electron doping per formula unit, respectively. Red lines indicate edge state components. Pink regions are bulk states, while the blue area corresponds to vacuum. All panels are drawn using data calculated by introducing SOC effect with ferromagnetic ordering to out-of-plane direction.

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Figure 3b and e presents Berry curvature of occupied Cr eg band in the presence of one and two electron doping per formula unit cell, respectively. In the presence of one electron doping per unit cell (Fig. 3b), peaks of positive Berry curvature for the lowest Cr eg band appear around Γ point and on the Γ–M lines, while negative peaks appear close to K and K’ points, yielding the Chern number of 2 for the lowest eg band. In the case of half-filled eg (Fig. 3e), two peaks of negative Berry curvature are located at K and K’ points. In addition, six negative peaks are located between six Γ–K lines, resulting in a total Chern number of −4. Figure 3c and f show edge spectra from the one and two-electron doping (per formula unit cell) results, respectively. In agreement with the Chern number calculation results, one can find two and four chiral edge states near the Fermi level in Fig. 3c and f, respectively. The emergence of multiple band crossings in Cr eg bands, as shown in Fig. 2a, gives rise to nontrivial topology with high Chern numbers. Explanation of microscopic origin of multiple Dirac cones in previous theoretical work25 can also be applied in our case. Specific description of additional crossings in band structure, visualization of MLWF, hopping integrals calculation results of CrSiTe3 are organized in Supplementary Figure 1, Supplementary Figure 2 and Supplementary Note 1, respectively. In addition, explanation for band gap closing at the midpoint of the Γ–K line are also discussed. See more details in Supplementary Note 2 and Supplementary Figure 3.

AHC under electron doping

We further examine the AHC of Cr eg bands manifold, the total density-of-states of CrSiTe3 and MAE illustrated in Fig. 4. Total AHC shows a quantized plateau at an integer number of electron doping, consistent again with the Chern number calculation results in Fig. 3a and d. In addition, the total density-of-states vanishes at one- and two-electron doping as depicted in Fig. 4b. Therefore, Cr eg bands are separated from each other together with nonzero Chern numbers so that Chern insulating phases will be realized under one- or two-electron doping in the formula unit cell.

Fig. 4: Electron doping dependent AHC, DOS, and MAE.
figure 4

a AHC contributions for each band of Cr eg band manifolds from the lowest band ‘b1’ (red line), the second-lowest band ‘b2’ (blue line), and the sum of two bands `Total' (black line), respectively. b The total density-of-states of Cr eg bands. Note that the results of panels (a) and (b) are obtained in the presence of the out-of-plane spin orientation. c The energy difference between easy plane and easy axis configurations of ferromagnetic monolayer CrSiTe3.

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Magnetic anisotropy and its doping dependence

So far, our DFT calculations with SOC effect consider spin aligned to the out-of-plane direction. However, this out-of-plane spin configuration can be suppressed or become unstable as the electron doping is introduced because magnetic anisotropy may depend on eg occupation. Figure 4c shows MAE as a function of electron doping concentrations, which oscillate between easy-plane and easy-axis anisotropies. For example, easy-axis anisotropy occurs at one-electron doping, while two-electron doping per formula unit cell favors easy-plane anisotropy. It is worth mentioning that MAE may vanish as doping is introduced, which may enable tuning the direction of FM moments and the resulting electronic structure via external magnetic fields.

Since spins are magnetic dipoles, magnetic long-range dipole-dipole interactions between local moments may change magnetic anisotropy. Because these inter-dipole interactions are not captured within DFT, we employ Ewald’s lattice summation technique to compute dipolar energy and estimate its effect on magnetic anisotropy44,45,46,47. Table 1 lists dipolar interaction and anisotropy energies from DFT calculations as a function of electron doping. It is shown that dipolar interactions favor the easy-plane configuration over the easy-axis one in FM states (as shown in the negative values of D-MAE in Table 1). Combining anisotropy energies from DFT (MAE) and dipolar interactions (D-MAE), it can be seen that easy-plane spin configurations are more favored except for undoped and one electron doping conditions. Interestingly, total magnetic anisotropy energy (MAE + D-MAE) at Δn = 1 is reduced to 0.289 meV per atom, equivalent to 1.18 Tesla of magnetic field strength. Hence controlling spin alignment and the resulting topological properties of one-electron-doped (per formula unit cell) ferromagnetic monolayer CrSiTe3 via applying an external magnetic field become achievable, which will be discussed further (see Section “Switching AHC via external magnetic fields”).

Table 1 Magnetic moment and anisotropy energies for each number of doping electrons per atom.
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To investigate more details about the suppression of magnetic anisotropy energy under electron doping as depicted in Fig. 4c, we calculate total energy changes induced by varying spin directions as illustrated in Fig. 5a. Without the change in the electronic structure (i.e., when the local spin moment picture is robust), total energy as a function of spin directions should have a form of (M ⋅ z)2 ~ cos2θ (see Fig. 5b for the angle definition). Such trend is maintained until θ is increased up to 30∘. Beyond that angle, the anisotropy energy deviates from the simple local moment picture. The result implies an onset of additional terms that comes into play to favor the easy-plane configuration around θ = 30∘, which turns out to originate from the evolution of the electronic structure for θ (see below the discussion on Fig. 6 for more detailed discussion). As a result, shape of total energy shows local minimum near θ ~ 50∘ and magnitude of magnetic anisotropy energy is suppressed about 1 meV at θ = 90∘ in Fig. 5a.

Fig. 5: Total energy and AHC as a function of spin-orientation angle.
figure 5

a Total energy at spin-orientation angle θ (black line and marker), where total energy of θ = 0∘ is set to be zero. By interpolating the calculated total energy of 0∘ ≤ θ ≤ 30∘, we obtain fitting function indicated as blue line. Difference between total energy and the fitting function are depicted as red line and marker. b Azimuthal spin angle (ϕ) averaged AHC as a function of polar angle (θ). Upper right panel shows definition of polar angle (θ) and azimuthal spin angle (ϕ) of magnetization vector M together with CrSiTe3 monolayer described as gray slab. Black line with square marker and blue errorbar in AHC graph correspond to average and standard deviation of AHC as a function of ϕ for each θ.

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Fig. 6: Spin orientation angle dependent Berry curvature plot.
figure 6

Berry curvatures of occupied Cr eg states in the momentum space (unit in Å2) for spin-orientation angle (θ) a θ = 0∘, c θ = 30∘, e θ = 45∘, g θ = 60∘, i θ = 75∘, and k θ = 90∘, respectively. Berry-curvature-projected band structures of Cr eg bands near Fermi level for b θ = 0∘, d θ = 30∘, f θ = 45∘, h θ = 60∘, j θ = 75∘, and l θ = 90∘, respectively. In panel of Berry curvature plots (a, c, e, g, i, k), dotted and dashed lines correspond to Fermi surfaces of lowest and second lowest bands of Cr eg bands, respectively. High symmetry points are marked in panel (a). Data in all panels is for one electron doping per unit cell and azimuthal angle ϕ = 0∘.

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Magnetic Hamiltonian and critical temperature

To understand the evolution of magnetic configuration and critical temperature as a function of doping, we construct a first-principles-based Heisenberg model on the two-dimensional honeycomb lattice,

$$H=\mathop{\sum}\limits_{\langle ij\rangle }{J}_{1}{{{{\bf{S}}}}}_{i}\cdot {{{{\bf{S}}}}}_{j}+\mathop{\sum}\limits_{\langle \langle ij\rangle \rangle }{J}_{2}{{{{\bf{S}}}}}_{i}\cdot {{{{\bf{S}}}}}_{j}+\mathop{\sum}\limits_{\langle \langle \langle ij\rangle \rangle \rangle }{J}_{3}{{{{\bf{S}}}}}_{i}\cdot {{{{\bf{S}}}}}_{j}+\mathop{\sum}\limits_{i}\lambda {({{{{\bf{S}}}}}_{i}\cdot {{{\bf{z}}}})}^{2}$$
(1)

where J1, J2, and J3 are nearest-, next-nearest-, and third-nearest-neighbor exchange interactions, respectively, extracted from DFT+U total energy calculations employing four different spin configurations6. The last term represents the magnetic anisotropy energy discussed in the previous section. In this model, we set ∣S∣ = 1. Table 2 shows the calculated values of J1,2,3 and corresponding Tc for given electron doping. For the undoped case, we get Tc = 46 K, which is consistent with previous reports48,49. For the one-electron doping case, we observe a massive enhancement of Tc, reaching 227 K. We also observe that the third-nearest neighbor exchange parameter J3 is significantly enhanced, contributing to the large FM coupling. Upon one-electron doping, however, the system becomes metallic so that the type of magnetic exchange interactions changes from weak ferromagnetic super-exchange to metallic double-exchange. Similar behavior has also been reported in CrGeTe3 with ion intercalation, where Tc is enhanced from 67 K to 208 K50. This observation is consistent with our findings.

Table 2 Magnetic exchange parameters and critical temperature for each number of doping electrons in the formula unit cell.
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Switching AHC via external magnetic fields

Because the magnetic anisotropy energy of Cr eg bands in the presence of one electron doping conditions is small enough to tune the FM via external fields, we investigate the behavior of AHC as the spin orientation is changed from the out-of-plane to in-plane direction. Figure 5b shows the magnitude of AHC, averaged over the azimuthal angle ϕ of the net magnetization, as a function of polar angle θ of the FM spin orientation to the layer-normal direction. From θ = 0∘ to θ = 45∘, AHC remains quantized at \(\frac{{2e}^{2}}{h}\) where there are deviations with respect to azimuthal spin angle ϕ. Surprisingly, as θ increases, AHC begins to reduce and goes to almost zero at θ = 90∘. We also confirm the anti-symmetric behavior of AHC as a function of spin polar angle θ. One can imagine that spin directions for θ = 180∘ is opposite to θ = 0∘ so that rotation of corresponding chiral edge modes are reversed, i.e., clockwise to counterclockwise. Because the direction of the FM moment can be switched between the out-of-plane and in-plane direction via external magnetic fields at one-electron doping per formula unit cell, the quantum anomalous Hall phase at this doping can also be switched on-off via external magnetic fields of about 1.18 Tesla estimated above (see Section “Magnetic anisotropy and its doping dependence”).

To understand the changes of band features and AHC at one-electron doping as the net magnetization is tilted, we plot Berry curvatures in the BZ and along with the eg band dispersion with tilting the spin orientation direction as summarized in Fig. 6. Here we set spin azimuthal angle ϕ = 0∘ to plot Berry curvatures and band structures. As θ increases, electron and hole pockets start to develop, close to θ = 45∘, in the middle of the Γ–M1,2,3 lines and around M1,3 points (see Fig. 6e and f). Especially, the presence of hole pockets and their expansion contribute to the reduction of the AHC as θ increases beyond 30∘~45∘ (compare with Fig. 5b), while the AHC contributions from electron pockets around M1,3 points are almost vanishing.

As θ is further increased beyond 60∘, sign of Berry curvature distribution around M1 and M3 points are flipped (compare Fig. 6g with i and k). This behavior is attributed to the band touching below the Fermi level and the resulting sign reversal of Berry curvature of occupied bands. Comparing panel Fig. 6h, j, and l, it can be noticed that band crossings occur on Γ–M1 and Γ–M3 lines, slightly below the Fermi level, around θ = 75∘. The crossings give rise to sign flipping of the Berry curvature of involved bands in the vicinity of the crossing point, which leads to cancellation of net Berry curvature and vanishing AHC at θ = 90∘.

In addition, note that the band gap between highest-occupied and lowest-unoccupied bands at Γ is suppressed as θ is increased, so that at θ = 90∘ the quadratic band touching is restored. This occurs because (i) the bands close to the Γ point consist mostly of Te px,y-orbitals, and (ii) SOC in the presence of FM behaves as an orbital Zeeman fields \({\lambda }_{{{{\rm{SO}}}}}{{{\bf{m}}}}\cdot \hat{{{{\bf{L}}}}}\) (λSO and m being SOC strength and net magnetization, respectively). When \({{{\bf{m}}}}={m}_{{{{\rm{z}}}}}\hat{{{{\bf{z}}}}}\) the λSOmzLz splits the px,y doublet into Lz eigenstates (px ± ipy), but when m is in-plane the splitting cannot occur due to the absence of pz character. Hence tilting the spin direction leads to the quenching of SOC in the vicinity of Γ point.

Tight-binding model

To explore possibilities of further tuning the topological nature of this compound and obtain a deeper understanding of high Chern numbers, we construct a model Hamiltonian that correctly captures characteristics of the conduction bands. As seen in the Fig. 2, the four conduction bands are composed in Cr-eg and Te-p orbitals. We can integrate out the Te-p orbitals and construct the eg effective Hamiltonian as follow:

$$H=\mathop{\sum}\limits_{i,j,\alpha ,\beta }{t}_{i\alpha ,j\beta }{d}_{i,\alpha }^{{\dagger} }{d}_{j,\beta }+{{{\rm{i}}}}{t}_{i\alpha ,j\beta }^{{{{\rm{soc}}}}}{d}_{i,\alpha }^{{\dagger} }{d}_{j,\beta },$$
(2)

where the first and second term represent the spin-independent and and spin-dependent hopping, respectively. The detailed derivation can be found in Supplementary Note 3.

Figure 7 shows the TB band structure without SOC effect. The hopping parameters are obtained from the Wannier function analysis. Since the TB model is constructed in the perfect octahedron system without trigonal distortion for simplicity, there exists a slight difference in the energy dispersion compared to the DFT conduction band in Fig. 4, but the overall features agree well. When we include the tsoc parameters in the model, the band gap opens, and each band shows [−2, 6, 8, −4] of Chern number, which also verifies the validity of the model Hamiltonian.

Fig. 7: Band structures from the tight-binding model on four conduction bands above the Fermi level.
figure 7

a Without SOC effect b with SOC effect and magnetization along [001] direction, and c with SOC effect and magnetization direction along [100] direction.

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In this model, the SOC-dependent Hamiltonian has quiet an interesting form. For instance, the NN SOC Hamiltonian (\({H}_{1}^{{{{\rm{soc}}}}}\)) simply looks as

$${H}_{1}^{{{{\rm{soc}}}}}=\left(\begin{array}{cccc}0&0&0&1\\ 0&0&-1&0\\ 0&1&0&0\\ -1&0&0&0\end{array}\right)\cdot {{{\rm{i}}}}{t}_{1}^{{{{\rm{soc}}}}}({{{\bf{k}}}})$$
(3)
$${t}_{1}^{{{{\rm{soc}}}}}({{{\bf{k}}}})={t}_{1}^{{{{\rm{soc}}}}}\hat{{{{\bf{M}}}}}\cdot (\hat{{{{\bf{z}}}}}+\hat{{{{\bf{x}}}}}{{{{\rm{e}}}}}^{-{{{\rm{2\pi i}}}}{{k}}_{1}}+\hat{{{{\bf{y}}}}}{{{{\rm{e}}}}}^{-{{{\rm{2\pi i}}}}{{k}}_{2}}),$$
(4)

where \(\hat{{{{\bf{M}}}}}\) denotes the unit vector for magnetization direction. In this equation, each vector \(\hat{{{{\bf{x}}}}},\hat{{{{\bf{y}}}}}\), and \(\hat{{{{\bf{z}}}}}\) represents the local coordinates denoted in Fig. 1. This magnetization-dependent hopping gives a different effect on each k point. Since the gap closing is observed at some high symmetry k points, let us examine how these parameter changes on each k point.

At Γ point, the parameter is proportional to \(\hat{{{{\bf{M}}}}}\cdot {{{\bf{c}}}}\), where c denotes the [001] direction. Thus, the SOC effect becomes maximum when the spins point out-of-plane direction, while it vanishes when the spin lies on the plane. However, we have slightly different behavior on \({{{\bf{k}}}}=(\frac{2}{3},\frac{1}{3})\) point. In this case, the matrix element vanishes when the magnetization points out-of-plane direction (\({t}_{1}^{{{{\rm{soc}}}}}({{{\bf{k}}}}) \sim \hat{{{{\bf{c}}}}}\cdot (\hat{{{{\bf{z}}}}}+\hat{{{{\bf{x}}}}}{{{{\rm{e}}}}}^{\frac{-4\pi {{{\rm{i}}}}}{3}}+\hat{{{{\bf{y}}}}}{{{{\rm{e}}}}}^{\frac{-2\pi {{{\rm{i}}}}}{3}})=0\)). For the general k points, this term does not vanishes. However, due to the dependence on the local axis, these terms cancel each other and become relatively small. Thus, we can observe that the band structure from the full SOC Hamiltonian when the magnetization lies in-plane looks very similar to the FM band structure without SOC effect.

To examine the relation between the parameter strength and topological character, we also examined the change of Chern number of each band by varying some parameters. Figure 8 shows the evolution of the Chern number while varying two parameters t3 (3rd NN hopping) and \({t}_{{{{\rm{onsite}}}}}^{{{{\rm{soc}}}}}\) (onsite SOC term). Other parameters such as NN and 2nd NN terms are extracted from the Wannier function analysis. In this figure, the four integer pair (i1, i2, i3, i4) represent the Chern number of each band calculated with model Hamiltonian. We can observe that the Chern number set of (2, −6, 8, −4), the actual Chern number of the CrSiTe3 system, is observed on the lower-left parameter region. With two-electron doping, the total Chern number equals −4, which does not change even if we change t3. It change from 4 to 2 with \({t}_{{{{\rm{onsite}}}}}^{{{{\rm{soc}}}}}\) being 0 ~ −4 meV.

Fig. 8: Change of the Chern number of each band with respect to hopping parameters in tight-binding model.
figure 8

t3 and \({t}_{{{{\rm{onsite}}}}}^{{{{\rm{soc}}}}}\) represent 3rd NN hopping and onsite SOC term, respectively. The four integer pair (i1, i2, i3, i4) represent the Chern number of each band. DFT calculations with three different structures which have distinct Cr–Te–Cr angles (X∘) was also performed, and the corresponding parameter values extracted from the Wannier function analysis are makerd with red triangle.

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Interestingly, this figure shows that structure distortion can achieve in each phase. The fully optimized structure of CrSiTe3 suffers trigonal distortion. The angle between Cr–Te–Cr is about 86∘. In this case, t3 and \({t}_{{{{\rm{onsite}}}}}^{{{{\rm{soc}}}}}\) becomes about 180 and −1 meV as marked with red triangle in the figure. If we perform the DFT calculation in the cubic structure where the local octahedron has no distortion (Cr–Te–Cr angle being 90∘), the Chern number changes from (2,−6,8,−4) to (2,0,2,−4). The hopping parameters corresponding to this case is about 240 and −3 meV for t3 and \({t}_{{{{\rm{onsite}}}}}^{{{{\rm{soc}}}}}\), respectively.

To determine the effect of structure distortion on the change of the Chern number, we performed the DFT calculations on the various structures where Cr–Te–Cr angles were set to be between 86∘ (fully optimized structure) and 90∘ (no distortion). On each step, we extracted t3 and \({t}_{{{{\rm{onsite}}}}}^{{{{\rm{soc}}}}}\) parameters. Other parameters also vary, but their changes were minimal compared to these two parameters, so we fixed them. Each parameter increased linearly from the fully optimized structure marked with a dotted line. During this transition, the total Chern number changes from −4 to 2. Even though it is not an optimized structure, we can further distort the local octahedron to achieve the Cr-Te-Cr angle to be 93∘ and monitor the change of hopping parameters. In this case, the Chern number becomes (−4,6,2,−4), and the corresponding 3rd NN hopping parameter becomes about 300 meV. During this process, the Chern number changes when Cr–Te–Cr angles become about 91.8∘ (t3 being about 270 meV). These observation implies that the local distortion near metal site (Cr) on CrSiTe3 induces large modification of t3 and \({t}_{{{{\rm{onsite}}}}}^{{{{\rm{soc}}}}}\) parameters and phase transition. The significant modification of t3 comes from the modification of bond length between Te p orbitals.

Table 3 summarizes the structure information on each calculation. We can see that the Te–Te bond length decreases from 4.23 (fully optimized) to 3.90 Å (trigonally compressed distortion). As discussed, the 3rd NN hopping contains Te → Te hopping channel. This considerable reduction of bond length between Te sites increases the t3 parameter. We can also observe that the Cr–Te distance decreases. This parameter is related to \({t}_{{{{\rm{onsite}}}}}^{{{{\rm{soc}}}}}\), and we also observe that this value slightly increases. Other parameters such as t2 or t5 remains almost the same. The local distortion of the octahedron can be achieved by applying external strain, and it induces a change in the hopping parameters. Thus, we can effectively map the effect of distortion to the change of hopping parameters, although this Hamiltonian assumes a cubic structure. Since the Chern number depends on the hopping parameters and the optimized structure lies near the phase boundary, we expect it to be possible to tune the Chern number phase by tuning external strain or change of ligand hopping strength.

Table 3 Structure and hopping parameters of fully optimized, cubic, trigonally compressed structure.
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Discussion

We study the electronic and topological properties of the electron-doped single-layer structure of CrSiTe3 by performing first-principle calculations based on density functional theory (DFT). The lattice constant and Hubbard U parameters are fixed by confirming that FM ground state and topological characters are invariant under certain conditions. We construct MLWF, the Fourier transform of Kohn-Sham wavefunctions of each band, and calculate Berry curvatures to analyze topological characters using converged MLWF. Nontrivial topology appears within Cr eg band manifold which can lead to Chern insulating phases in CrSiTe3. Chiral edge modes are also calculated, consistent with total Chern number calculations.

Furthermore, at one electron doping per unit cell, we find that AHC can be switched on and off via controlling direction of FM magnetization and that magnetic anisotropy is suppressed. Hence AHC can be controlled via an external magnetic field at one electron doping per unit cell, corresponding to 2.36 × 1014 cm−2 of electron density. Thanks to recent advancements in ionic liquid gating and chemical intercalations, such a doping concentration is now within reach of state-of-the-art experimental techniques. For example, ionic liquid gating can introduce up to 1014 cm−2 order of carrier doping in a 2D system51,52,53. A recent experimental study reports 4 × 1014 cm−2 of electron doping at the top two layers on the surface of CrGeTe335, exceeding our theoretical estimate of electron doping density (2.36 × 1014 cm−2) necessary to achieve the quantized AHC in CrSiTe3. In another experimental study, one electron doping per unit cell in CrGeTe3 was achieved through organic ion intercalation50. Because CrGeTe3 and CrSiTe3 are isostructural, we expect doping CrSiTe3 up to one electron per unit cell can be challenging but achievable in experiments.

By performing tight-binding (TB) model analysis, we confirm that DFT calculation results of Chern number for Cr eg orbitals are reproduced. We also find distinct Chern insulating phases by adjusting hopping parameters in our constructed TB model and drawing relevant phase diagrams. We expect that CrSiTe3 can be a Chern insulator controlled by the external magnetic field, which can be realized by electron doping.

Methods

Density functional theory calculations

To obtain band structures and projected density-of-states, we perform ab initio electronic structure calculations based on density-functional theory (DFT) using OpenMX54 code, which employs linear-combination-of-pseudo-atomic-orbital basis with norm-conserving pseudopotentials. We choose generalized gradient approximation (GGA) exchange-correlation functional in the parameterization of Perdew, Burke and Enzerhof (PBE)55 with Hubbard U parameters chosen to be 1.5 eV for Cr d orbital. SOC effects are included in the calculations via fully-relativistic pseudopotentials implemented in OpenMX54. Pseudo-atomic orbitals are set to be s3p2d2 for Cr, s2p2d1 for Si, s3p3d2 for Te, respectively. To simulate two-dimensionality, we insert 20 Å of vacuum in the unit cell. 10 × 10 × 1 of k-space mesh is adopted for the momentum space integration. Energy cutoff for choosing real-space grid is set to be 700 Ry (96 × 96 × 360 real space grid). 10−6 Hartree per Bohr of force criterion is chosen for the optimization of internal coordinates while keeping C3 rotation, inversion, and mirror symmetries. SOC effects are excluded in the process of structural relaxation. We fix Bravais lattice as hexagonal and determine lattice constant by performing total energy minimization calculation as a function of unit cell size.

Analysis of topological characteristics

Berry curvature is given by

$${B}_{n}({{{\bf{k}}}})={{{\rm{i}}}}\mathop{\sum}\limits_{{n}^{{\prime} }\ne n}\frac{\left\langle n\left\vert \frac{\partial H}{\partial {k}_{{{{\rm{x}}}}}}\right\vert {n}^{{\prime} }\right\rangle \left\langle {n}^{{\prime} }\left\vert \frac{\partial H}{\partial {k}_{{{{\rm{y}}}}}}\right\vert n\right\rangle -({k}_{{{{\rm{x}}}}}\leftrightarrow {k}_{{{{\rm{y}}}}})}{{({E}_{n}-{E}_{{n}^{{\prime} }})}^{2}}$$
(5)

where integrating Berry curvature in BZ gives Chern number

$${C}_{n}=\frac{1}{2\pi }{\iint }_{{{{\rm{BZ}}}}}{B}_{n}({{{\bf{k}}}})d{k}_{{{{\rm{x}}}}}d{k}_{{{{\rm{y}}}}}$$
(6)

En is Kohn-Sham eigen energy of band index n at certain k = (kx, ky) point and H is Hamiltonian of system. Anomalous Hall conductivity is calculated by integrating Berry curvatures over BZ up to arbitrary energy level which is given by

$${\sigma }_{{{{\rm{H}}}}}(E)=\frac{{e}^{2}}{h}{\iint }_{{{{\rm{BZ}}}}}{B}_{n}({{{\bf{k}}}}){f}_{{{{\rm{FD}}}}}(E,{{{\bf{k}}}})d{k}_{{{{\rm{x}}}}}d{k}_{{{{\rm{y}}}}}$$
(7)

where e is electron charge, h is Planck constant and fFD(E, k) is Fermi-Dirac distribution function, respectively. To compute Berry curvatures, anomalous Hall conductivity, Chern numbers, and chiral edge states, the maximally localized Wannier function (MLWF) method as implemented in OpenMX is employed56,57. \({d}_{{{{{\rm{z}}}}}^{2}}\) and \({d}_{{{{{\rm{x}}}}}^{2}-{y}^{2}}\) orbitals at Cr sites are chosen as initial projectors. 10 × 10 × 1 k-space grid is chosen for the construction of MLWFs. From the MLWF tight-binding Hamiltonian obtained, Wannier Tools code is used to compute Berry curvature, Chern numbers, and edge states58. Specifically, we employ Fukui-Hatsugai formalism59 to compute Berry curvature. Moreover, edge state calculations utilize the iterative Green’s functions method in the semi-infinite geometry.

Generalized tight-binding model for e g manifold

In addition to the MLWF tight-binding model, we further introduce a Slater-Koster-type tight-binding model for a deeper understanding of SOC and lattice distortion effects.

We start from including both Te p orbitals and Cr d orbitals, then project out Te p orbitals to obtain an effective four-band Hamiltonian.