Enhanced Itinerant d0 Ferromagnetism and Tunable Topological Properties of a Ca2C/Ca2N Heterobilayer

Recent studies have shown that a rich variety of exotic physical phenomena can emerge from the presence of electronic flat bands. Using first‐principles calculations, here a monolayered Ca2C possesses a relatively flat band near the Fermi level, and across the whole Brillouin zone is predicted. Next, the weak d0 ferromagnetism in Ca2C is substantially enhanced by a proximity‐coupled electride monolayer of Ca2N is shown. The enhanced magnetism can be attributed to the straightening of the flat band caused by the charge transfer to the electride, while the flattening is tied to the magic ratio of three between the σ and π channels of the effective C─C coupling strength. More surprisingly, even though each of the constituent systems is topologically trivial, the Ca2C/Ca2N heterobilayer is empowered with nontrivial band topology, characterized by a high Chern number tunable by the magnetization orientation or lateral strain. These findings provide a new platform for realizing intriguing magnetic topological metals in 2D materials.


Introduction
Magnetism in solids is usually related to the presence of transition metals with partially filled d or f shells, which develop highly localized magnetic moments from strong onsite Coulomb repulsion and Hund's coupling.However, there have been credible experimental and theoretical works demonstrating that distinctive magnetism can also emerge in materials free of d/f-block elements, via a subtle interplay of kinetic energy and correlation of itinerant electrons. [1]1a,b] In DOI: 10.1002/aelm.202300733itinerant systems, spontaneous ferromagnetism occurs when the gain of exchange interaction is greater than the loss of the kinetic energy, as is usually justified by the "Stoner criterion" of N(E F )I > 1, with N(E F ) and I being the density of states (DOS) at the Fermi level (E F ) and strength of exchange interaction, respectively. [2]Therefore, the van Hove singularities (VHSs) in the DOS, which can result in ultrahigh or even divergent N(E F ), are likely to induce electronic instability and time-reversal symmetry breaking.Such characteristics can be utilized as design principles for band engineering toward d 0 ferromagnetism, by shifting the chemical potential to band extrema or saddle points. [3,4]he concept of d 0 magnetism was originally coined to describe unconven tional magnetic properties of 3D metal oxides with metal elements containing no unpaired d/f electrons, [1b,d,j,2,5] but a thorough understanding is still lacking.5f] As a representative system, graphene nanoribbons [6] or nanoflakes [7] have been demonstrated to exhibit edge magnetism, typically with antiferromagnetic ordering following the Lieb theorem. [8]Another intriguing example is twisted graphene, which can give rise to a set of low-energy flat bands and thus can achieve high N(E F ). [9] The flattened bands magnify the electron correlation effect and can harbor fascinating collective phenomena [10] such as superconductivity [10] and magnetism. [11]11c,12] Such rich physics highlights the crucial role of interlayer coupling in manipulating electron hoppings for band flattening and mediating itinerant d 0 ferromagnetism.
In this study, using first-principles calculations based on density functional theory (DFT), we propose an MXene/electride heterobilayer of Ca 2 C/Ca 2 N to achieve d 0 ferromagnetism by altering the electron hoppings via strong interlayer coupling and induce strikingly tunable topological properties.We first show that very weak itinerant d 0 ferromagnetism can be achieved in a Ca 2 C monolayer hosting a relatively flat band, and can further be substantially enhanced in the heterobilayer of Ca 2 C/Ca 2 N.Such an enhancement can be attributed to the straightening of the flat band caused by the charge transfer from Ca 2 C to the electride, while the flattening is tied to the magic ratio of three between the  and  channels of the effective C─C coupling strength, consistent with the recently proposed orbital design principle for nonline-graph flat bands. [13]More surprisingly, even though each of the constituent systems is topologically trivial, the Ca 2 C/Ca 2 N heterobilayer is empowered with nontrivial band topology, as characterized by a high Chern number of −4, which is readily tunable by the magnetization orientation or lateral strain.Our findings provide a new platform for realizing intriguing magnetic topological metals in 2D materials.

Crystal Structure and Stabilities
The crystal structure of the heterobilayer of Ca 2 C/Ca 2 N is shown in Figure 1a,c, with an AB stacking that is subject to the space group P3m1 and energetically favored (see Figure S1, Supporting Information).The calculation details can be found in the Experimental Section; in particular, the vdW-DF2 functional [14] was adopted to treat van der Waals (vdW) corrections, as it yielded the best accuracy in our benchmark calculations.The optimized lattice constant of the hetero-bilayer is a = 3.744 Å, lying between that of the two constituent monolayers.To check the energetic stability, we calculate the formation energy (E f ), defined as ), with the terms denoting the total energies of the heterobilayer and two constituent monolayers, respectively.The yielded negative E f of −0.299 eV per unit cell validates that the formation of such a heterobilayer is energetically feasible.The vertical charge mapping within the range of 0.1 eV below E F shows a significant spatial overlap of interfacial charges (Figure 1d), implying strong interlayer interactions.The phonon spectrum with negligible imaginary frequencies (Figure S2, Supporting Information) also suggests that Ca 2 C/Ca 2 N is dynamically stable.Furthermore, we perform ab initio molecular dynamics simulations at room temperature for up to 20 ps.The evolutions of the total potential energy and Ca─N/Ca─C bond lengths confirm that the ground-state geometry is well preserved upon heating (Figure S3, Supporting Information), demonstrating the thermal stability of the heterobilayer.These results collectively verify that Ca 2 C/Ca 2 N is a stable heterobilayer for exploring intriguing properties.Notably, the MoTe 2 /Ca 2 N [15] and MoS 2 /Ca 2 N [16] vdW heterostructures have been successfully prepared in the experiment, wherein the Ca 2 N electride could be acquired through scotch-tape-assisted exfoliation from the bulk material that was synthesized via a solid-state chemical reaction.A similar method may also be employed to synthesize the isostructural Ca 2 C and assemble the two into a vdW heterostructure.Given the air/water sensitivity of Ca 2 N, high-vacuum or inert-gas protective conditions should be crucial for subsequent measurements of the relevant properties.

Electronic and Ferromagnetic Properties
Before investigating the electronic and magnetic properties of Ca 2 C/Ca 2 N, we briefly revisit the band structures of the two constituent monolayers.Ca 2 N is the first experimentally realized layered electride [17] and each layer contains anionic electrons (AEs) floating on both surfaces, which can serve as an "electronic repository".Figure 2a,b plots the band structures of a freestanding Ca 2 N monolayer, showing an ultra flat dispersion along the Γ-M path and below the well-studied AE-dominated metallic bands.Since this flat band is deeply situated at −2 eV below the E F , it is unlikely to induce any intriguing property.A Ca 2 C monolayer, which has been proposed as a dynamically stable MXene material, [18] holds the same structure but with one valence electron less than that of Ca 2 N.Such a valency difference endows Ca 2 C with great potential for effectively modulating the position of the flat band.As shown in Figure 2c,d S1, Supporting Information), Ca 2 C is likely to transfer charge to Ca 2 N when forming the heterobilayer, as illustrated in Figure 1b.Indeed, the differential charge density and the corresponding ab-plane-averaged charge difference (Figure 1e,f) confirm that the charge transfer (≈0.081 e/unit cell) mainly occurs at the interface, from Ca 2 C to Ca 2 N. It is striking that such transfer of a small fraction of charge results in an optimal hole doping of Ca 2 C for adjusting band filling, which amounts to ≈6.67 × 10 13 cm −2 .More importantly, compared to the freestanding Ca 2 C monolayer (Figure 2c), we find that the interlayer interactions nonadiabatically modulate the band dispersions rather than rigidly shifting the chemical potential.As displayed in Figure 2e, the N-2p bands of Ca 2 N lie deep below the Fermi Sea, while the C-2p bands of Ca 2 C hybridize strongly with the parabolic bands (labeled by //) dominated by AEs or Ca-4s orbitals.Upon such hybridization, the effective hoppings between C-2p orbitals are subtly engineered, and the highest branch of the C-2p bands is significantly flattened along the Γ-M path, leading to a much more pronounced VHS at E F .
The enhanced VHS could induce strong Fermi surface instability, which is presumed to reduce the overall energy of the system via spin splitting.Our spin-polarized calculations indicate that the energy gain from time-reversal symmetry breaking reaches 50.4 meV per unit cell.Accordingly, we find giant spin splitting of electronic states near E F as exhibited in Figure 2f.The largest splitting occurs in the partially flattened bands near E F , where the spin-up and spin-down bands are separated by ≈0.57eV yet both remain flat.Spin splitting can also be observed in some electronic states below E F , such as the parabolic bands near the Γ point and the C-2p z -dominated flat bands (see Figure S4c, Supporting Information) along the K-M path.In real space, spin splitting also reduces the charge density corresponding to the states near E F (Figure S5, Supporting Information).The magnetism of the heterobilayer is mainly located at the C atoms, with a total magnetization of 0.622 μ B per unit cell, which is significantly enhanced compared to that of the Ca 2 C monolayer.Such a fractional feature of the magnetization is also consistent with the itinerant picture.It is thus established, at the first-principles level, that the combination of the isostructural Ca 2 N and Ca 2 C monolayers alters the effective hoppings and fillings of the C-2p bands, giving rise to flattened band dispersion and enhanced itinerant ferromagnetism.
To elucidate the underlying origin of the partially flat band and the associated ferromagnetism, we construct a tight-binding (TB) model to depict the primary electron hoppings and interactions.The model is based on a triangular lattice representing the C atomic layer, consisting of three p-orbitals, as shown in Figure 3a.We first consider the spinless and noninteracting case, Here, the onsite energy  i is set to a unified constant , and c † and c represent the electron creation and annihilation operators, respectively.The hopping (t  ij ) between two orbitals  and  on the nearest neighboring sites indexed by i and j is calculated following the Slater-Koster formalism, [19] using two parameters V  and V  as sketched in Figure 3a.The Hamiltonian in momentum space can be obtained via Fourier transformation, with ).The block-diagonalized form implies that the p x,y, and p z subspaces are decoupled, and we here demonstrate that the p x,y subspace is responsible for the partially flat band (Figure 2e).Along a representative Γ-M path (0, k y ), the structure factors reduce to F 2 = 1, F 4 = cos( √ 3k y /2), and G 4 = 0; thus, the two bands are One can immediately deduce that, when 3V  + V  = 0 (3V  + V  = 0), the band E 1 =  + 2V  (E 2 =  + 2V  ) becomes perfectly flat.The former case with stronger -type hopping is more realistic.The bands subject to various V  /V  ratios are displayed in Figure 3b, illustrating that the magic ratio of V  /V  = −3 gives rise to a perfectly flat band on the Γ-M path.It has been well perceived that the topological band is usually tied to a "compact localized state," [20] wherein the destructive interferences of wave functions lead to effectively quenched kinetic energy. [21]Here, the underlying mechanism of the "partially" flat band can be understood as a unique "directional" compact localized state, namely, the propagation of Bloch wave packets is forbidden due to the delicate cancellation of -and -type hoppings.Fundamentally, the magic ratio responsible for the partially flat band can also be understood in line with the recently orbital design principle, which maps well-studied linegraph lattices to non-line-graph ones while preserving the destructive quantum interference. [13]n connection to the studied monolayers or heterobilayer, we note that the nearest C─C spacing is much larger than the typical C─C bond length, signifying that V  and V  should be viewed as effective hoppings accounting for spatial overlaps between p-orbitals as well as hoppings mediated by the Ca-s orbitals or AEs.It has also been noted that the latter may lead to extra couplings beyond the two-center Slater-Koster framework, based on which a more comprehensive model has been developed. [22]Nevertheless, in our studied system, the topmost p-band is mainly flattened on the Γ-M path and such features can basically be captured by the simplified V  -V  model.We first downfold the spin-unpolarized DFT band structures of the freestanding Ca 2 C monolayer and Ca 2 C/Ca 2 N heterobilayer into the C-2p subspace with the aid of Wannier90, [23] and then fit the downfolded bands with Equation (2) (see Figure 3c,d).The yielded V  /V  ratios are −1.97 and −2.34, respectively, indicating that the formation of the heterobilayer modulates the effective hoppings between nearest C-2p orbitals to make V  /V  closer to the magic ratio of −3 and further flattens the Γ-M bands.
Our TB model also provides valuable insights into the intriguing itinerant ferromagnetism, by extending Equation (1) to the spinful (s = ↑ / ↓ ) case and solving it at the mean-field level (see Experimental Section).The spin-polarized mean-field bands corresponding to the 7/8 and 4/5 fillings are plotted in Figure 3e,f.Our calculations indicate that the ferromagnetic solution is energetically more favored than the nonmagnetic one for each case, and the reduced filling gives rise to enhanced band splitting, in accordance with our DFT results.In the heterobilayer, charge transfer from Ca 2 C to Ca 2 N leads to hole doping of the flat bands dominated by C-2p orbitals, thereby inducing larger spin-splitting than that in the case of freestanding Ca 2 C. The enhanced ferromagnetism can be understood from the filling of the unique flat bands near E F .In the full-filling limit, spin polarization does not mitigate the onsite repulsion and thus the system remains nonmagnetic; in the low-filling limit, electron repulsion becomes insufficient to trigger ferromagnetic transition.Here, the filling of our proposed Ca 2 C/Ca 2 N heterobilayer falls into an intermediate optimal regime between these two limits due to the versatility of the Ca 2 N electride, which can serve as a nano-capacitor to bind excess AEs and transferred electrons.

Topological Properties
12b,24] Here, we explore the topology encoded in the partially flat band and ferromagnetism of the Ca 2 C/Ca 2 N heterobilayer.We first calculate the magnetic anisotropy energy of Ca 2 C/Ca 2 N, and find that it has a magnetic easy axis along the diagonal direction of the y and z axes (denoted as the yz direction), which is energetically favored over other directions by at least ≈4.6 μeV due to the small spin-orbital coupling (SOC).This implies that Ca 2 C/Ca 2 N is susceptible to an external magnetic field and may serve as a promising "soft" ferromagnet for applications in magnetic sensors, data storage devices, and spintronics.The band structures of Ca 2 C/Ca 2 N calculated with different magnetization orientations exhibit tiny differences even if the SOC is included (Figure 4a,c,e), yet the underlying symmetry is diverse.It is noted that even though these systems are metallic, there exists a well-defined "curved chemical potential" [25] that separates the "occupied" and "empty" states for each system (as illustrated by the yellow-shaded region and the blue dashed line in Figure 4a), below which the number of occupied states at each kpoint equals the number of valence electrons (49 e punit cell).The Chern number of such systems can be determined by calculating and analyzing the Wannier charge centers [26] corresponding to these "occupied" states.Taking the case magnetized along yz as an example, we construct a set of localized bases for the heterobilayer by using the maximally localized Wannier functions, [23] which is capable of faithfully reproducing the DFT-calculated band structure (Figure S6a, Supporting Information), and the resulting Wilson loops calculated from the Wannier charge centers (Figure S6b, Supporting Information) indicate that Ca 2 C/Ca 2 N is topologically nontrivial, with a high Chern number of −4.
To further reveal the underlying origin of the nontrivial band topology, we analyze the Berry curvature distribution in momentum space.It is found that the Berry curvatures mainly originate from the interactions between the two spin-split partially flat bands and the parabolic bands, which induce inverted mini gaps at the crossing points.The crossing points for the spin-down flat bands are located on the six Γ-M paths, while the ones for the spin-up flat band are near the Brillouin zone center.These two sets of momenta with Berry curvature accumulation, labeled as Γ and N in Figure 4b, give rise to integrals of −1 and −1/2 (per N point) within the selected circular regions enclosing the singular point (Figure 4h,i; see also Figure S7, Supporting Information for testing of numerical convergence), and collectively endow the yzmagnetized system with a high Chern number of C = −4.Switching the magnetization direction to the z (-z) direction preserves (reverses) the sign of the accumulated Berry curvature, resulting in C = −4 (4).When magnetized to the in-plane direction, e.g., along x or y, the heterobilayer becomes topologically trivial with C = 0 (Figure 4e,f).The magnetization-dependent topological properties can be manifested by measuring the anomalous Hall conductivity (AHC) experimentally.The AHC can be calculated by integrating the Berry curvature overall all the bands below a selected energy level (a "straight" chemical potential allowing actual experimental verification).The chemical potential is adjustable with respect to charge neutral point.As shown in Figure 4g, although not quantized, the AHC exhibits a signal on the order of 100 (Ωcm) −1 when the chemical potential is tuned to ≈0.04 eV above E F (E F is aligned to 0 eV), consistent with the position of the spin-down flat band and the revealed Berry curvature singularity.
In-plane strain engineering has been widely used as an efficient strategy for modulating the fundamental properties of 2D systems.Note that upon formation of the heterobilayer, the Ca 2 N (Ca 2 C) monolayer endures tensile (compressive) strain compared to the corresponding freestanding state, here we investigate the stabilities and evolutions of magnetic and topological properties of Ca 2 C/Ca 2 N under different biaxial strains.The calculated phonon spectra and ab initio molecular dynamics simulations at 300 K (Figure S8, Supporting Information) verify the dynamic and thermodynamic stabilities of the Ca 2 C/Ca 2 N heterobilayer under biaxial strains ranging from −5% to 7.5%, respectively.Figure S9 (Supporting Information) shows the evolution of the magnetic moment per unit cell, revealing a monotonic decreasing trend as the lattice constant expands.When the applied tensile strain reaches 7.5%, ferromagnetism is fully suppressed.Compared to the unstrained case, the highest branch of the C-2p bands is bent along the Γ-M path and moves down away from the Fermi level when the heterobilayer is under the 7.5% tensile strain, leading to a substantial decrease in the DOS at the Fermi level (see Figure 2e; Figure S10a, Supporting Information).Consequently, spin splitting is no longer favored due to the reduced VHS at the Fermi level, as observed in Figure S10 (Supporting Information).This tendency coincides with the fact that band flatness and d 0 ferromagnetism emerge when Ca 2 C is compressed upon forming the heterobilayer.The strain-dependent itinerant ferromagnetism also significantly influences the topological properties.For example, when the heterobilayer is magnetized in the z direction, a 2% compressive/tensile biaxial strain can tune its Chern number from −4 (Figure 4c,d) to −1/−5 (Figure S11, Supporting Information).These results suggest that strain can serve as a knob to effectively tune the magnetism and topological properties of the heterobilayer, which may also be manifested from anomalous Hall transport measurements.Given the coexistence of the anionic electrons, flat-band-induced itinerant ferromagnetism, and topological states with Berry curvature singularity at the Fermi level, the heterobilayer holds promising potential to be exploited for spintronics or topological catalysis. [27]

Conclusion
In summary, we have used first-principles methods to predict that the weak d°0 ferromagnetism in Ca 2 C can be substantially enhanced in the heterobilayer of Ca 2 C/Ca 2 N and that the heterobilayer possesses strikingly tunable topological properties.The enhanced magnetism can be attributed to the straightening of the flat band near the Fermi level caused by the charge transfer from Ca 2 C to Ca 2 N, while the flattening is tied to the magic ratio of three between the  and  channels of the effective C─C coupling strength.The active AE-dominated states of the electride play important roles in engineering both the effective hoppings and band-filling that are indispensable for itinerant ferromagnetism.More intriguingly, the spin-split partially flat bands intertwine with the parabolic bands, endowing the heterobilayer with nontrivial band topologies characterized by a high Chern number of −4.Distinct topological states can be realized by manipulating the magnetization orientations or applying biaxial strains, and may be detected by the anomalous Hall conductivity.These findings provide a new platform for realizing intriguing magnetic topological metals in 2D materials and highlight the potential of partially flat bands for exploiting exotic quantum many-body effects.

Experimental Section
First-Principles Calculations: All first-principles calculations were performed with DFT implemented in the Vienna ab initio simulation package (VASP), using the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE) as the exchange-correlation functional. [28]A 21 × 21 × 1 Monkhorst-Pack k-mesh was used to sample the Brillouin zone. [29]The core electrons were treated fully relativistically by the projector augmented wave method, while the valence electrons were processed in the scalar relativistic approximation, with a plane-wave cutoff of 600 eV.The vdW interactions were explicitly included in all calculations by adopting the vdW-DF2 functional. [30]This choice was based on extensive testing calculations and benchmark of bulk Ca 2 N (Table S2, Supporting Information), which indicated that the nonlocal vdW-DF functionals overall outperform the semi-empirical atom-pairwise methods and that the vdW-DF2 functional performs best among all according to the mean absolute percentage error of lattice constants. [14,31]The Ca 2 C/Ca 2 N heterobilayer was modeled within the unit cell of the two monolayers, and the lattice constant was varied to determine the optimal value, with all atomic positions fully relaxed until the maximum force component was less than 0.01 eVÅ −1 .The phonon spectrum was obtained by adopting the finite displacement method, with the atomic force calculated within a 3 × 3 × 1 supercell. [32]Spin-orbital coupling was included self-consistently by employing the second-variation technique using the scalar-relativistic eigenfunctions of the valence states. [33]The Chern number was calculated from the Wilson loop, and the Berry curvature was calculated invoking the Kubo formula, both implemented in the WannierTools package. [26,34]ean-field Treatment of the Hubbard Model: To reveal such novel physics associated with the flat band of the Ca 2 C/Ca 2 N heterobilayer, Equation (1) to the spinful (s = ↑ / ↓ ) and interacting case was extended.For simplicity, the p z subspace irrelevant to the partially flat band was eliminated.The two-orbital Fermionic Hubbard model thus reads with n i,↑ = c i ,↑ † c i ,↓ and U being the particle number operator and the onsite Coulomb repulsion.Here the onsite energy  i is set to a unified constant , and c † and c represent the electron creation and annihilation operators, respectively.t  ij is the hopping between two orbitals  and  on the nearest neighboring sites indexed by i and j.To investigate how the electron-electron interaction normalizes the bands and induces potential spin polarization, the Hubbard model (Equation 4) with the mean field (MF) approximation was treated The MF charge density 〈n i,↑ 〉 and 〈n i,↓ 〉 were determined by minimizing the total energy of the system in a self-consistent iterative procedure.The Fermi-Dirac function is used to describe the electron occupancy at a smearing temperature of 0.1 K.The Hubbard MF code developed by Bernard Field et al, [35] was modified so that it can be implemented for the multi-orbital case studied in the present work.

Figure 1 .
Figure 1.a) Top and c) Side views of the crystal structure of Ca 2 C/Ca 2 N, with a unit cell indicated by the black rhombus in (a).The charge centers of the "anionic electrons" in the Ca 2 N monolayer are marked by the empty circles with the label "X".b) Illustration of the alignment of the Fermi levels (E F ) of Ca 2 C, Ca 2 N, and Ca 2 C/Ca 2 N. "W" represents the work function, and the curved arrow indicates the charge transfer direction when forming the heterobilayer.d) Slice color-mapping of spin-unpolarized partial change density within the energy interval of [E F -0.1 eV, E F ]. e) Differential charge density of the heterobilayer.The yellow and blue isosurfaces represent charge accumulation and depletion, respectively, with the isosurface values of 3.5 × 10 −6 eÅ −3 .f) The corresponding ab-plane-averaged charge density difference.

Figure 2 .
Figure 2. Projected spin-unpolarized band structures and density of states (DOS) of monolayered a) Ca 2 N, c) Ca 2 C, and e) stacked Ca 2 C/Ca 2 N. The color and size of the circles represent the atom/orbital compositions and weights, respectively.The E F is aligned to 0 eV.In (c), three van Hove singularities (VHSs) near E F are indicated, where VHS 1 exhibits a Heaviside step function discontinuity, VHS 2 displays divergent behavior, and VHS 3 comes from the flat band on the K-M path.b,d,f) Same as (a,c,e), but with spin-polarized.
, Ca 2 C possesses a relatively flat band near E F dominated by C-2p orbitals, and spin splitting is induced by VHS 1 corresponding to the partial filling of the p-band maximum.The calculated N(E F ) is 0.94 states/eV per C atom and spin, and the Stoner parameter I is 1.41 eV, which yields N(E F )I = 1.33 > 1, sufficing to endow the system with a ferromagnetic ground state, albeit with a rather small magnetization of 0.098 μ B per unit cell.Now, we come to the Ca 2 C/Ca 2 N heterobilayer.Since the calculated work functions of monolayered Ca 2 N and Ca 2 C are 3.51 and 2.45 eV, respectively (Table

Figure 3 .
Figure 3. a) Schematic of a triangular lattice model with and -bond-like hoppings between the nearest sites illustrated.b) Band structures of the p x,y model with varying V  /V  .Inset: the first Brillouin zone and high-symmetry k-points.c,d) p-bands of the Ca 2 C monolayer and Ca 2 C/Ca 2 N heterobilayer, downfolded from the spin-unpolarized DFT band structures with the aid of Wannier90 (orange) and fitted by Equation (2) (green), respectively.The fitted effective V  /V  is also indicated.e, f) Spin-polarized mean-field bands of the p x,y Hubbard model, with (e) 7/8 and (f) 4/5 filling.Parameters: V  = −3V  = 4.5 eV,  = −0.9eV, and U = 1.4 eV.Notations:  =  ↑ +  ↓ =  ↑ - ↓ represent the total and spin-charge densities, respectively, while E s = E FM − E NM is the energy difference (per cell) between the ferromagnetic and nonmagnetic states.

Figure 4 .
Figure 4. Band structures and corresponding Berry curvature distributions of the Ca 2 C/Ca 2 N heterobilayer magnetized along the a,b) yz, c,d) -z, and e,f) x directions.The insets in (a) are the zoomed-in views.The Berry curvatures are calculated using the Kubo formula, with the bands below the "curved chemical potential" that is guided by a blue dashed line in the in (a).The bands below/above the yellow-shaded region in (a,c,e) are defined as occupied/empty states, where the yellow-shaded regions represent gaps.The two insets in (a) plot zoomed-in bands around the crossing points.The hexagons in (b,d,f) represent the first Brillouin zone, over which the integration of the Berry curvature yields the Chern number (C).g) Anomalous Hall conductivity as a function of chemical potential (μ) with different magnetization orientations.h,i) Zoomed-in plots of the Berry curvature distributions and their integrations (here also denoted as C) in the regions marked by pink and red circles in (b).
i, c † i, c i + n i,,↑ n i,,↓