Moiré magnetism and moiré excitons in twisted CrSBr bilayers

Significance

Moiré excitons in van der Waals (vdW) structures are of particular interest for both fundamental science and cutting-edge applications. The recent discovery of two-dimensional (2D) magnetic semiconductors has brought moiré magnetism as a new dimension in this exciting field. Despite rapid progress, there is no direct evidence of moiré excitons in magnetic vdW structures. Using a noncollinear time-dependent density functional theory (TDDFT) method, we predict a twist-induced quantum phase transition from antiferromagnetic to ferromagnetic interlayer coupling, which facilitates the formation of localized one-dimensional (1D) moiré excitons in twisted CrSBr bilayers. Moreover, vertical electric fields can induce net magnetic moments in the moiré excitons, greatly extending their lifetimes and paving the way for breakthroughs in quantum and optoelectronic technologies.

Abstract

Moiré excitons and moiré magnetism are essential to semiconducting van der Waals magnets. In this work, we perform a comprehensive first-principles study to elucidate the interplay of electronic excitation and magnetism in twisted magnetic CrSBr bilayers. We predict a twist-induced quantum phase transition for interlayer magnetic coupling and estimate the critical twist angle below which moiré magnetism with mixed ferromagnetic and antiferromagnetic domains could emerge. Localized one-dimensional moiré excitons are stable if the interlayer coupling is ferromagnetic and become unstable if the coupling turns to antiferromagnetic. Exciton energy modulation by magnons is estimated and dependence of exciton oscillator strength on the twist angle and interlayer coupling is analyzed. An orthogonally twisted bilayer is revealed to exhibit layer-dependent, anisotropic optical transitions. Electric field is shown to induce net magnetic moments in moiré excitons, endowing them with exceedingly long lifetimes. Our work lays the foundation for using magnetic moiré bilayers in spintronic, optoelectronic, and quantum information applications.

Moiré superlattices formed by stacking two-dimensional (2D) materials with a small angular or lattice mismatch provide a powerful and versatile platform to discover and engineer quantum matter (1–5). Localized around high-symmetry points in a semiconducting moiré superlattice, moiré excitons are correlated electron–hole pairs with large binding energies and distinctive selection rules (6–8). Observed in various 2D van der Waals (vdW) structures, moiré excitons have garnered significant interest in both fundamental science, such as Bose–Einstein condensation of moiré excitons (9–11) and novel applications, such as single-photon emitters, excitonic devices, etc (12, 13).

Recent discovery of 2D magnets has ignited renewed interest to explore exotic magnetic phases (14–17) and to elucidate the interplay of electron excitation and magnetism in magnetic vdW structures. For example, it has been reported that excitonic transitions in A-type magnetic CrSBr layers can be drastically changed when the interlayer magnetic order is switched from antiferromagnetic (AFM) to ferromagnetic (FM) (18), and strong magnon–exciton coupling has been observed in CrSBr layers (19, 20). Nontrivial magnetic orders and excitations, such as topological skyrmions (15, 21) and one-dimensional (1D) magnon channels (16), are predicted to emerge in magnetic vdW heterostructures due to the modulation of interlayer exchange interactions. In particular, coexistence of AFM and FM domains, separated by noncollinear spin texture, has been observed in twisted magnetic CrX₃ (X = Cl, Br, I) layers (22–25), and the observed moiré magnetism can be tuned by vertical electric fields (25). However, despite the rapid progress in this fledging field, there is no direct evidence, either from theory or experiments, on the existence of moiré excitons in magnetic vdW structures, and little is known of their properties. Moreover, a general understanding of how excitonic properties can be modulated by interlayer exchange interactions remains elusive.

In this work, we aim to fill the knowledge gap by conducting a comprehensive first-principles study of CrSBr magnetic bilayers. First-principles calculations can provide atomistic details and physical insights that may not be accessible to experiments and phenomenological theories, thus allowing us to make greater strides in this field. We have recently developed a first-principles method based on the linear-response time-dependent density functional theory (LR-TDDFT) with optimally tuned (OT), screened and range-separated hybrid (SRSH) exchange-correlation (XC) functionals. Formulated with spinor wavefunctions, the method can capture noncollinear magnetism and spin–orbit coupling (SOC) in 2D magnets. Compared to the conventional GW–Bethe–Salpeter equation (GW-BSE) method based on the many-body perturbation theory, the TDDFT method can treat much larger systems (~10³ atoms) with desirable accuracy. Using this method, we can study excitonic properties in CrSBr moiré bilayers with ~1,350 atoms in the unit cells. CrSBr is an A-type magnetic semiconductor with FM in-plane exchange interaction and AFM interlayer exchange interaction (26). Compared to other A-type magnets, such as CrI₃, CrSBr has a higher Néel temperature in both bulk (~132 K) and bilayers (~150 K) (27) with outstanding stability. In addition, there is high tunability in their magnetic properties under a uniaxial tensile strain (28), hydrostatic pressure (29), and electric field (30). However, no moiré magnetism or moiré exciton has been reported in twisted CrSBr bilayers thus far. In this work, we will provide first-principles evidence of moiré magnetism and moiré excitons in twisted CrSBr bilayers and shed light on the interplay of magnetic interaction and electronic excitation in magnetic vdW structures.

Results

Fig. 1A shows the atomic structure of a twisted CrSBr bilayer with a twist angle θ = 5.47°. The unit cell of the moiré superlattice is indicated by a dashed box, with lattice constants of 4.93 nm and 3.78 nm along a- and b-axes, respectively, and contains 1,350 atoms. Four high-symmetry stacking motifs, labeled as A, B, C, D, are highlighted in the unit cell and their atomic structures are displayed in Fig. 1B. A moiré superlattice or potential is typically defined in terms of the interlayer displacement vector d, shown in Fig. 1C, in the primitive unit cell of the pristine bilayer (θ = 0°). As d spans across the entire primitive unit cell, all possible stacking configurations in a moiré superlattice could be recovered. A low-symmetry stacking motif (M) is also included in Fig. 1A. Among these stacking motifs, A is the most stable one (31) and will be used to represent the pristine bilayer in this work.

Fig. 1.

(A) (Top) The Top view of the moiré superlattice and the unit cell (dashed box) for the twisted CrSBr bilayer with θ = 5.47°. (Bottom) The side view of the moiré superlattice. (B) The atomic structure and d vector for each of the five stacking motifs (Left: side view along the a-axis; Right: side view along the b-axis). d is given in terms of a and b. The blue, yellow, and brown spheres represent Cr, S, and Br atoms, respectively. (C) The definition of the in-plane interlayer displacement d in the primitive unit cell of the pristine CrSBr bilayer (Top view).

To elucidate magnetic interactions in a CrSBr bilayer, we perform first-principles LDA+U calculations to compute the total energy of the bilayer and fit the total energy to the following Heisenberg model:

$$\begin{array}{c}{E}_{\mathrm{ex}}=\frac{1}{2}{\sum }_{i,j}{J}_1{\mathit{S}}_i·{\mathit{S}}_j+\frac{1}{2}{\sum }_{i,j}{J}_2{\mathit{S}}_i·{\mathit{S}}_j+\frac{1}{2}{\sum }_{i,j}{J}_3{\mathit{S}}_i·{\mathit{S}}_j\hfill \\ +\frac{1}{2}{\sum }_{i,j}{\mathrm{J}}_{\mathrm{int}}{\mathit{S}}_i·{\mathit{S}}_j+{\sum }_i{J}_a({\mathit{S}}_i·\widehat{\mathit{a}}{)}^2+{\sum }_i{J}_c({\mathit{S}}_i·\widehat{\mathit{c}}{)}^2,\hfill \end{array}$$ [1]

where S is the spin operator on Cr ions with |S| = 3/2 (the magnetic moment on each Cr ion is computed as 3 µ_B); J₁, J₂, and J₃ are the in-plane exchange coupling parameters between the first nearest neighbor (NN), second NN, and third NN Cr ions, respectively. J_int denotes the NN interlayer exchange coupling parameter, and J_a and J_c are the magnetic anisotropy parameters with $\widehat{\mathit{a}}$ and $\widehat{\mathit{c}}$ as the unit vectors in a- and c-axes, respectively. For the pristine CrSBr bilayer with A stacking, the LDA+U (U = 1.0 eV) calculations with SOC yield J₁ = −5.22 meV/µ_B², J₂ = −6.44 meV/µ_B², J₃ = −5.32 meV/µ_B², and J_int = 0.018 meV/µ_B². A negative J value indicates FM coupling whereas a positive value implies AFM coupling. The b-axis is found as the easy axis and its magnetic anisotropy parameter is set to be zero; the c-axis is the hard axis with J_c = 0.033 meV/µ_B² and the a-axis is the intermediate axis with J_a = 0.021 meV/µ_B². These coupling parameters are in good agreement with previous results (20, 29) (e.g., J₁ = −3.17 meV/µ_B², J₂ = −6.50 meV/µ_B², J₃ = −5.70 meV/µ_B²; J_int = 0.02 ~ 0.04 meV/µ_B²). Similar calculations can be carried out for other stacking configurations as discussed below.

The atomic structure of a twisted moiré bilayer can be characterized by the in-plane interlayer displacement d and the interlayer distance h. In the following, we examine the dependence of the magnetic interactions on h and d. We first focus on the pristine bilayer with A stacking. As mentioned earlier, the pristine CrSBr bilayer is an A-type magnet with an in-plane FM coupling along the b-axis and AFM coupling between the layers. In this paper, we denote the total energy of the pristine CrSBr bilayer with the AFM interlayer coupling as E(AFM). The pristine bilayer could also adopt FM interactions both in-plane and out-of-plane, and we denote its total energy as E(FM). Thus, the energy difference $\mathrm{\Delta }E=E\left(\mathrm{FM}\right)-E\left(\mathrm{AFM}\right)$ is positive for the equilibrium structure of the pristine bilayer. In Fig. 2A, we display $\mathrm{\Delta }E$ as a function of h, where h is defined as the vertical Br–Br distance between the layers. It is found that for most h values, the interlayer coupling is AFM unless h is greater than 3.17 Å. In the twisted bilayer (θ = 5.47°) examined in this work, h is less than 3.17 Å (SI Appendix, Fig. S2), hence we conclude that the variation of h alone would not lead to AFM→FM phase transition in the twisted CrSBr bilayer. However, the phase transition could be induced by the interlayer displacement d along the b-axis as indicated in Fig. 2B. It is found that $\mathrm{\Delta }E$ changes sign when the displacement in the b-axis is between 0.1b and 0.9b, indicating a phase transition from AFM to FM for the interlayer magnetic order. As shown in (SI Appendix, Fig. S3), the displacement along the b-axis yields much stronger interlayer coupling in the FM phase than the AFM phase, which drives the phase transition. In addition, this phase transition is anisotropic, i.e., it cannot be triggered by the displacement along the a-axis. This is a unique feature of CrSBr due to its structural anisotropy that is absent in other 2D magnets (e.g., CrI₃) and has important consequences on its moiré magnetism.

Fig. 2.

(A) Variation of $\mathrm{\Delta }E$ as a function of the interlayer distance h in the pristine bilayer. (B) $\mathrm{\Delta }E$ as a function of the interlayer displacement d along a- and b-axes (in units of a and b), respectively. (C) Map of interlayer exchange coupling in the moiré superlattice (θ = 5.47°) where a blue (green) domain indicates preferred AFM (FM) interlayer coupling. (D) Color-coded map of the interlayer displacement vector d along the b-axis in the moiré superlattice. (E) Schematic picture of the phase transition from AFM interlayer coupling in the pristine bilayer (θ = 0°) to FM interlayer coupling in the twisted bilayer (θ = 5.47°). The red arrows indicate the spins on Cr ions.

A moiré superlattice can be considered as a collection of stacking motifs—each characterized by an in-plane displacement vector d. One can thus map out preferred interlayer exchange coupling in a moiré superlattice by computing $\mathrm{\Delta }E$ for each stacking motif therein (e.g., A, B, C, D, and M). In Fig. 2C, we display a map of such preferred interlayer coupling in the twisted bilayer (θ = 5.47°). The green (blue) domain indicates preferred FM (AFM) interlayer coupling. FM and AFM are found to coexist in A, B, and D regions, suggesting that a FM $\leftrightarrow$ AFM phase transition could be brought upon by a small in-plane displacement, underlying the experimental observation that the interlayer coupling in CrSBr can be switched by strains (28). In contrast, region C or M exhibits exclusive AFM or FM coupling. To understand these results, we plot in Fig. 2D the displacement vector d along the b-axis in the moiré superlattice. A and B regions are shown to have a displacement ~0.1b which according to Fig. 2B could trigger a phase transition from AFM (blue) to FM (green), thus explaining the results in Fig. 2C. Similarly, the displacement in regions M is ~0.1 to 0.4b, preferring FM coupling; and regions C and D have displacement around 0.4 to 0.5b, yielding AFM coupling. These results suggest that the interlayer exchange coupling depends sensitively on the in-plane displacement d, and moiré magnetism could emerge in twisted CrSBr bilayers. The above discussion only pertains to the interlayer coupling, but it is the competition between the in-plane and out-of-plane exchange interactions that determines moiré magnetism in these bilayers. To capture this competition, we perform spin dynamics simulations for the twisted CrSBr bilayer (θ = 5.47°) based on the parameterized Heisenberg model (Eq. 1) which includes both the in-plane and out-of-plane exchange couplings. The VAMPIRE package is used in the simulations where the time evolution of magnetization is described by the Landau–Lifshitz–Gilbert (LLG) equation (32). The simulation details and results can be found in SI Appendix. In particular, we find that the twisted bilayer (θ = 5.47°) exhibits an exclusive FM interlayer coupling as shown in (SI Appendix, Fig. S5). In other words, we predict a quantum phase transition which turns AFM interlayer coupling in the pristine bilayer to FM interlayer coupling in a twisted bilayer as shown schematically in Fig. 2E. Moreover, as discussed below, there is a critical angle θ_c below which moiré magnetism with mixed FM/AFM domains would appear and above which the exclusive FM interlayer coupling is the ground state configuration.

In the following, we elucidate the origin of the phase transition and estimate θ_c in CrSBr bilayers. To this end, we first determine the interlayer coupling parameter J_int in Eq. 1. Based on $\mathrm{\Delta }E$ and the area of each stacking configuration in Fig. 2C, we can obtain average values for the interlayer coupling parameters in the moiré superlattice (θ = 5.47°): $\overline{J_{\text{FM}}}=-0.340 \text{meV}/{\text{µ}}_{\text{B}}^2$ and $\overline{J_{\text{AFM}}}=0.262 \text{meV}/{\text{µ}}_{\text{B}}^2$. Since the strength of FM interlayer coupling is larger than AFM, the AFM→FM phase transition is expected in the moiré lattice. To estimate the critical angle, we need to consider the competition of the in-plane and out-of-plane exchange interactions. The formation of a mixed FM and AFM domain would increase the in-plane energy since the in-plane exchange interaction favors FM coupling, and this energy increase, α, scales with the length of the domain boundary. On the other hand, the formation of a mixed domain would lower the interlayer coupling energy β (relative to the pure FM phase) because the formation of an AFM domain (e.g., in C, D regions) is energetically preferred as shown in Fig. 2C, and β scales with the area of the AFM domain. Therefore, there is a critical twist angle θ_c below which β is greater than α and the formation of a mixed magnetic domain is energetically favorable (see SI Appendix for more detail). For CrSBr bilayer, θ_c is estimated as 2.2°, which is smaller than that of CrI₃ bilayer (θ_c ~ 3-5°) (17, 23, 24). This is because the in-plane exchange coupling (5 to 6 meV/µ_B²) in CrSBr is stronger than that in CrI₃ (2 to 3 meV/µ_B²) while the interlayer exchange coupling (0.2 to 0.3 meV/µ_B²) is weaker than that in CrI₃ (0.4 to 0.8 meV/µ_B²) (21, 33). Thus, it is more difficult (i.e., smaller θ_c) for β to overcome α in CrSBr. In the pristine bilayer, the energy of the AFM phase is 0.16 meV/unit cell lower than that of the FM phase, whereas in the twisted bilayer (θ = 5.47°), the FM phase is 0.14 meV/unit cell lower than the AFM phase.

We subsequently examine the ground state electronic structure of the twisted bilayer (θ = 5.47°) with FM interlayer coupling. In Fig. 3A, we present the single-particle band structure of the moiré superlattice. Along Γ-X and X-S directions in the Brillouin zone, the formation of a nearly flat valence band maximum (VBM) (~0.1 meV) and conduction band minimum (CBM) (~4 meV) band is observed, in contrast to dispersive bands along Γ-Y and Y-S directions, as a consequence of strong anisotropy in CrSBr. We also calculate wavefunctions of VBM and CBM in the pristine bilayer, shown in Fig. 3B. The VBM wavefunction comprises primarily p-orbitals of Br atoms with strong interlayer hybridization. In contrast, the CBM is mainly contributed by Cr dz²-orbitals and S s-orbitals, exhibiting weak interlayer interaction. Thus, VBM can be modulated by the moiré potential while CBM cannot. Note that the flat CBM band does not stem from the moiré potential, but rather from localized Cr dz²-orbitals. In Fig. 3C, we compare the band structures of the pristine bilayer with AFM and FM interlayer coupling. It is found that with AFM coupling, both VBM and CBM (and their adjacent states) are twofold degenerate; these degenerate states reside at separate layers, resulting in negligible layer hybridization. In contrast, the degeneracy is broken in FM coupling, leading to stronger interlayer hybridization and energy splitting. Therefore, the low-energy excitations or excitons (S1 and S2) in the AFM phase are expected to be degenerate and layer-resolved whereas such degeneracy is broken in the FM phase.

Fig. 3.

(A) Band structure of the moiré superlattice (θ = 5.47°) with FM interlayer coupling calculated with the LDA+U method. (B) Wavefunctions of the VBM and CBM for the pristine bilayer; cyan and purple isosurface indicates positive and negative phase of the wavefunctions, respectively. (C) Band structures of the pristine bilayer with AFM (Left and Middle) and FM (Right) interlayer coupling. Two copies of the degenerate states in the AFM phase are shown in the Left and Middle panels. The degeneracy is broken in the FM phase, as indicated by dashed circles. The projection of the wavefunction onto the top and bottom layer is colored in red and blue, respectively. (D) Schematic diagram of the doubly degenerate excitations in the AFM phase and the degeneracy is broken in the FM phase.

To elucidate how electron excitation may be affected by the interlayer exchange interaction, we compute the fundamental gap (E_g) and optical gap (E_opt) of the pristine bilayer as a function of spin relative angle in the separate layers. Experimentally, such a spin twist could be generated by exciting coherent magnons in the pristine bilayer (19, 20). Computationally, we let the bilayer start in the collinear AFM phase and then decrease the angle between the spins in the layers from 180° to 160° (red curves) or conversely, let the bilayer start in the collinear FM phase and then increase the angle from 0° to 20° (blue curves). The variations of the bandgaps as a function of the relative angle are shown in Fig. 4 A and B. In semiconductors, it is generally expected that enhanced (or reduced) electronic interaction decreases (or increases) the bandgap. As indicated in Fig. 3C, there is strong interlayer hybridization in the FM phase, and increasing the spin angle (from the FM phase) would reduce the interlayer interaction, thus increasing the bandgap. Conversely, there is negligible interlayer coupling in the AFM phase, and decreasing the angle (from the AFM phase) would increase the interlayer coupling, thus decreasing the bandgap. The same reasoning can explain the trends for both E_g and E_opt. Note that E_opt represents the energy of the lowest exciton, which is a many-body state involving single-particle excitations at all K-points, thus E_opt is a weighted average of single-particle gaps from all K-points. E_g, on the other hand, is the single-particle gap at the Γ point in pristine CrSBr. As shown in Fig. 3C, the band splitting at CBM and VBM could lead to reduced E_g in the AFM phase, and the splitting is maximal at the Γ point. Hence E_g decreases faster than E_opt as a function of the spin angle in the AFM phase. In particular, as shown in Fig. 4B, E_opt can change by ~3 meV when the spin angle varies by 20°. This is in line with the experimental finding that coherent magnons in CrSBr could modulate the exciton energy by ~4 meV (19). The exciton binding energy, defined as E_b = E_g − E_opt, can also be modulated by the magnons. For example, E_b in the AFM phase can be lowered by 24 meV; this large reduction in E_b is attributed to the direct electron–hole Coulomb interaction because the electron–hole exchange interaction varies only about 1 meV (SI Appendix, Fig. S6).

Fig. 4.

Variation of fundamental gap E_g (A) and optical gap E_opt (B) as a function of the relative spin angle between the top and bottom layer in the pristine bilayer. The spatial variation of the fundamental gap (δE_g) in the moiré superlattice (θ = 5.47°) with (C) AFM interlayer coupling and (D) FM interlayer coupling; various local stacking motifs are indicated.

We now turn to exciton properties in the twisted moiré bilayer (θ = 5.47°). In Fig. 5A, we display the charge density of the lowest-energy exciton (1.195 eV) in the twisted bilayer. The moiré exciton is localized in A and B regions of the supercell with a slight charge separation across the layers. The moiré potentials for the twisted bilayer in the AFM and FM phases are presented in Fig. 4 C and D, respectively. The moiré potential is defined as the spatial variation of the bandgap δE_g = E_g − ⟨E_g⟩ in the moiré superlattice. At each point of the moiré superlattice, we first determine the in-plane displacement vector d. We then shear the top layer relative to the bottom layer with the displacement d in the pristine bilayer and calculate E_g. We can thus obtain a spatial distribution of E_g in the moiré superlattice and ⟨E_g⟩ is its average value. The moiré potential, which is the amplitude of E_g variation, is found to be 16 meV in the AFM phase. Such a small moiré potential can hardly localize the exciton, as confirmed by its charge density (SI Appendix, Fig. S7). For the FM phase, the moiré potential is 160 meV, comparable to that in a TMD moiré bilayer (34). The moiré potential has its troughs in A and B regions and its ridges in M, C, and D regions, giving rise to 1D moiré excitons shown in Fig. 5A. Thus, we predict that localized moiré excitons can be formed in twisted CrSBr bilayers with FM interlayer interaction, but they are not stable with AFM interlayer coupling. In other words, the formation of moiré excitons in CrSBr depends on the twist angle. If θ < θ_c, localized moiré excitons are not expected to form.

Fig. 5.

Charge density of the lowest energy exciton in the twisted bilayer with θ = 5.47° (A) and θ = 90° (B); both the top and side views are shown with the cyan and purple color representing the electron and hole, respectively. The moiré supercell (dashed box) along with the top and bottom layers (side view) and the stacking motifs are indicated. (C) and (D) are the corresponding charge density of the lowest energy exciton in the twist bilayers under a vertical electric field ε. The isosurface value is set as 5 × 10⁻⁵ Å⁻³.

In Fig. 6, we display the exciton oscillator strength in the pristine and twisted bilayers. In the pristine bilayer with AFM coupling (AFM-0°), two main absorption peaks are observed, consistent with the experimental finding (18). The lowest energy exciton is bright and doubly degenerate, agreeing with previous results (30, 31). These degenerate excitons originate from the degenerate single-particle excitations S₁ and S₂ shown in Fig. 3D. As interlayer coupling turns to FM (FM-0°), the exciton energy redshifts because the FM phase has stronger interlayer interaction, thus a smaller bandgap (Table 1). The interlayer interaction breaks the double degeneracy in the AFM phase and mixes the two states. As a result, the doubly degenerate excitons in the AFM phase split into a pair of spin-correlated excitons, corresponding to the first and fourth peaks encircled in Fig. 6. The energy splitting depends on the interlayer interaction, which can be tuned by the spin twist angle via magnons (19, 20). In the twisted CrSBr bilayer (FM-5.47°), the interlayer coupling is FM and the exciton energy blueshifts as compared to the pristine bilayer with FM coupling. This result is consistent with the fact that E_opt increases as the spin twist angle increases. The moiré superlattice comprises distinct stacking motifs, and each could host excitons with different oscillator strengths. As shown in (SI Appendix, Fig. S8), the lowest energy exciton is dark in C and D regions and bright in A, B, and M regions. Since the lowest exciton in the moiré bilayer is trapped in A and B regions, it is bright. However, bright and dark excitons from different stacking motifs could mix to create a multitude of excitons with relatively lower oscillator strengths in the moiré superlattice. Therefore, the moiré bilayer exhibits broad and relatively low absorption. These results clearly demonstrate the potential and tunability of CrSBr bilayers in magneto-optic and opto-spintronic applications.

Fig. 6.

Exciton oscillator strength in CrSBr bilayers with different twist angle and interlayer exchange coupling; the blue bars represent the exciton energy levels.

Table 1.

The summary of fundamental gap (E_g), optical gap (E_opt), radiative lifetime in CrSBr bilayers with different twist angle, interlayer exchange coupling, and vertical electric field (ε)

	Eg (eV)			Eopt (eV)			Lifetime
ε (V/nm)	0	0.3	0.6	0	0.3	0.6	0	0.3	0.6
AFM-0°	1.690			1.251			53 fs
FM-0°	1.534			1.108			40 fs
FM-5.47°	1.635	1.381		1.195	1.013		142 fs	9 ms
OM-90°	1.693	1.638	1.580	1.282	1.269	1.226	62 fs	73 fs	45 ms

Furthermore, we examine an orthogonally twisted bilayer (θ = 90°), which has been recently fabricated in experiments (35). The spins in each layer are oriented along its respective easy axis, thus are orthogonal to each other, and we label this bilayer as OM-90°. Compared to the pristine bilayer, OM-90° has larger interlayer distances and weaker interlayer interaction, which result in a larger bandgap than AFM-0° (and FM-5.47°), as shown in (SI Appendix, Fig. S9) and Table 1. The exciton energy of OM-90° blueshifts as shown in Fig. 6, and its lowest energy excitons are bright and doubly degenerate. Interestingly, these are delocalized, intralayer excitons, residing on separate layers as shown in Fig. 5C, due to the weak interlayer interaction. Fig. 7A displays the polarized oscillator strength for the lowest energy exciton in OM-90°. The exciton residing at the top/bottom layer shows the maximum oscillator strength at 0°/90°. Hence, OM-90° bilayer enables layer-dependent, anisotropic optical transitions, and one could use linearly polarized light to selectively excite the top or bottom layer in OM-90°. The ability to address spatially separate and degenerate exciton states is of particular importance to quantum information and spintronic applications (36, 37).

Fig. 7.

(A) Polarized oscillator strength for the doubly degenerate, lowest energy exciton in OM-90°. The cyan and purple colors represent the oscillator strength at the top and bottom layer, respectively. (B) The VBM and CBM wavefunctions for the pristine bilayer under a vertical electric field; the cyan and purple isosurface indicate the positive and negative phases of the wavefunction, respectively. (C) Schematic illustration of spin directions for the electron (blue arrow) and hole (red arrow) in FM-5.47° under a vertical electric field ε.

We have examined how a vertical electric field could modulate the magnetic and optical properties of the twisted CrSBr bilayer (FM-5.47°). We find that the electric field not only can change the exciton energy at different layers due to Stark effect, but also can modulate the interlayer exchange interaction. For example, the interlayer exchange energy (i.e., the energy difference between the AFM and FM phases) in FM-5.47° is lowered by 0.06 eV with ε = 0.3 V/nm. More importantly, the electric field can also change the spatial distribution of the frontier orbitals. In the absence of the field, the wavefunctions of CBM and VBM in FM-5.47° are strongly mixed and spread on both layers. Thus, as illustrated in Fig. 7C, the spin directions of CBM and VBM coincide, at 2.74° relative to the easy axes of both layers. Interestingly, the strongly mixed CBM and VBM can be decoupled by the electric field so that their wavefunctions separate onto different layers, with CBM at the top layer and VBM at the bottom layer. As a result, their spin directions diverge, turning to their respective easy axis. As CBM and VBM have distinct spin directions, the bilayer can be considered as a tunable bipolar magnetic semiconductor (38) with interesting opto-spintronic applications (39, 40).

The electric field can also modulate excitonic properties. As shown in Fig. 5 B and D, the electric field can separate the electron and hole to different layers, forming an interlayer exciton. A different stacking region has a different interlayer distance h, thus a different energy E according to E = eεh (ε is the electric field and e is electron charge). Hence, the electric field could shift the in-plane position of moiré excitons. In addition, the electric field could switch the polarization of an interlayer exciton, with its electron and hole exchanging layers (33). More importantly, the internal spin configuration of an exciton could be tuned by the electric field. For example, the magnetic moment of the lowest exciton in FM-5.47° vanishes with ε = 0 because the spin directions of the electron and hole are the same thanks to its strong interlayer coupling. With ε = 0.3 V/nm, the net magnetic moment of the exciton becomes 0.09 µ_B as the spin directions of the electron and the hole have a small angle of 5.47°. In OM-90°, the lowest energy exciton is intralayer with zero net magnetic moment. With ε = 0.6 V/nm, the exciton changes to interlayer and acquires a magnetic moment of 1.41 µ_B. Therefore, the exciton magnetic moment in twisted CrSBr bilayers is highly tunable with the electric field. Crucially, interlayer excitons with nonzero magnetic moments are expected to have exceedingly long lifetimes. For example, the radiative lifetime of the lowest energy exciton in FM-5.47° and OM-90° bilayers under the electric field is estimated as 9 ms and 45 ms, respectively, similar to the lifetime of dark excitons in TMDs (41, 42). Here, we focused on the radiative lifetimes (τ_r) of the excitons, but phonon-assisted nonradiative recombination (τ_nr) could also be important. The overall lifetime (τ) of an exciton includes both contributions and can be estimated as $\frac{1}{\mathrm{\tau }}=\frac{1}{{\mathrm{\tau }}_{\mathrm{r}}}+\frac{1}{{\mathrm{\tau }}_{\text{nr}}}$. Typically, τ_nr is in the order of ns (43–45), much less than τ_r (~ms) reported here, thus the overall lifetime τ of these excitons should be dominated by τ_nr with τ_r dropping out the equation. It will be of interest to determine precise values of τ_nr from first-principles in future work. Nevertheless, moiré excitons with long radiative lifetimes and tunable magnetic moments are expected to have fascinating properties, leading to high-temperature Bose–Einstein condensation (41, 46) and robust single-photon emitters.

Discussion

Based on a noncollinear first-principles TDDFT method, we carried out a comprehensive study of moiré magnetism and moiré excitons in twisted CrSBr bilayers. A first-principles parameterized Heisenberg model is used to characterize moiré magnetism in CrSBr bilayers. The magnetic coupling depends sensitively on the interlayer displacement, responsible for the twist-induced quantum phase transition, i.e., AFM→FM for the interlayer coupling. We estimate the critical twist angle below which mixed FM/AFM domains could emerge in CrSBr bilayers. Dependences of exciton binding energy and bandgap on the relative spin angle between the layers are examined and the exciton energy modulation by magnons is determined, agreeing well with the experimental result. The interlayer coupling in the FM phase is much stronger than the AFM phase, leading to a much deeper moiré potential. As a result, localized 1D moiré excitons are observed in FM-5.47° bilayer. The lowest energy excitons in OM-90° bilayer, on the other hand, are delocalized and doubly degenerate. Separated at opposing layers, these excitons can be selectively excited by linearly polarized light. The dependence of exciton oscillator strength on the twist angle and interlayer coupling is analyzed. The vertical electric field can modify the properties of the moiré excitons, including their energies, in-plane positions, polarity, and internal spin configurations. In particular, the electric field can induce net magnetic moments in the moiré excitons, endowing them with long lifetimes. Our work provides direct evidence of moiré exciton formation in twisted CrSBr bilayers of FM interlayer coupling, elucidates the interplay of electron excitation and magnetism, and lays the theoretical foundation for using magnetic moiré bilayers in spintronic, optoelectronic, and quantum information applications.

Materials and Methods

First-Principles Ground State Calculations.

First-principles ground state calculations were carried out to determine the equilibrium atomic structures of various bilayers, their magnetic structures, electronic and single-particle band structures based on the DFT, and projector-augmented-wave (PAW) pseudopotentials (47, 48) as implemented in Vienna Ab initio Simulation Package (49, 50). The LDA+U method (51) was used to determine the atomic structures and magnetic properties of the bilayers. The energy cutoff for the planewave basis set is 400 eV. The atomic structures are fully relaxed until the residual force on each atom is less than 0.01 eV Å⁻¹. The energy convergence criterion is set as 1 × 10⁻⁵ eV. A vacuum layer of 15 Å is included in the calculations to eliminate spurious interactions between the periodic images. A Monkhorst-Pack k-point mesh of 11 × 15 × 1 is used to represent the reciprocal space of the primitive unit cell, and the calculation in the moiré supercell is at Γ point. SOC is taken into consideration in the band structure calculations.

Atomic Spin Simulations.

The magnetic simulations for the twisted CrSBr bilayers (5.47°) were performed using VAMPIRE package in which the time evolution of magnetization is modeled by the LLG equation and numerically integrated with the Heun method (32, 52). The simulation box size is 50 nm × 50 nm along x and y axes with periodic boundary conditions and the box thickness in the z axis is 3 nm, which contains 69,300 magnetic atoms in the twisted CrSBr bilayer. The system is cooled from 200 K (its Néel temperature is ~150 K) to 0 K, and the total evolution time is 60 ns with a time step of 0.1 fs.

First-Principles Excited-State Calculations.

The conventional first-principles approach for determining excitonic properties in semiconductors is the GW-BSE method (53–55) based on the many-body perturbation theory. However, the GW-BSE method is highly expensive for studying moiré excitons due to substantial numbers of atoms in moiré supercells. Inclusion of SOC could further increase its computational cost. In this work, we used a first-principles method that can offer a reliable description of excitons while significantly reducing the computational cost. This approach is based on the linear-response TDDFT (56, 57) with OT, SRSH XC functionals (58–61). Unlike the traditional TDDFT methods with local and semilocal XC functionals, the TDDFT-OT-SRSH method can capture the long-range electron–electron and electron–hole interactions in solids correctly by choosing appropriate parameters, and has been extensively used to study optical and excitonic properties in solids and vdW bilayers (34, 41, 46, 62–68). Formulated with spinor wavefunctions (69), the TDDFT-OT-SRSH method can also capture noncollinear magnetism, including SOC, in magnetic materials. In this method, the following non-Hermitian eigenvalue equation (70) is solved to determine the exciton energies and wavefunctions:

$$\left(\begin{array}{cc}A& B\\ B^{\ast }& A^{\ast }\end{array}\right)\left(\begin{array}{c}{X}_I\\ Y_I\end{array}\right)={\omega }_I\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\left(\begin{array}{c}{X}_I\\ Y_I\end{array}\right),$$ [2]

where the pseudoeigenvalue ω_I represents the Ith exciton energy. The matrix elements of A and B in the basis of two-component spinor orbitals (ijσ) are given by

$$A_{ij\sigma ,kl\tau }={\delta }_{i,k}{\delta }_{j,l}{\delta }_{\sigma ,\tau }({\epsilon }_{j\sigma }-{\epsilon }_{i\sigma })+K_{ij\sigma ,kl\tau },$$ [3]

$$B_{ij\sigma ,kl\tau }=K_{ij\sigma ,lk\tau }.$$ [4]

Here, K is the coupling matrix where indices i and k indicate the occupied Kohn–Sham (KS) orbitals, and j and l represent the virtual KS orbitals. According to the assignment ansatz of Casida, the many-body wavefunction of an excited state I can be written as

$${\mathrm{\Phi }}_I\approx {\displaystyle \sum _{ij\sigma }}\frac{X_{I,ij\sigma }+Y_{I,ij\sigma }}{\sqrt{{\omega }_I}}{\widehat{a}}_{j\sigma }^{\mathrm{†}}{\widehat{a}}_{i\sigma }{\mathrm{Ф}}_0={\displaystyle \sum _{ij\sigma }}{Z}_{I,ij}{\widehat{a}}_{j\sigma }^{\mathrm{†}}{\widehat{a}}_{i\sigma }{\mathrm{Ф}}_0,$$ [5]

where $Z_{I,ij}=(X_{I,ij}+Y_{I,ij})/\sqrt{{\omega }_I}$; ${\widehat{a}}_{i\sigma }$ is the annihilation operator acting on the ith KS orbital with spin $\sigma$, and ${\mathrm{Ф}}_0$ is the ground-state many-body wavefunction taken to be the single-Slater determinant of the occupied KS orbitals. In order to reduce the computational cost associated with the Fock-like exchange on large systems, the first-order perturbation theory (71, 72) to the range-separated hybrid KS Hamiltonian is used.

In this noncollinear (TD)DFT-OT-SRSH method, there are three parameters α, β, and γ that need to be specified. α determines the contribution from the exact exchange and β controls the contribution from the long-range exchange terms. γ is the range-separation parameter. α and β satisfy the requirement of α+β=1/ε₀ where ε₀ is the scalar dielectric constant of the solid. The optimal set of the parameters (α = 0.087, β = 0.246, and γ = 0.12) was determined by fitting the quasiparticle bandgap (E_g) of the pristine CrSBr bilayer (with A stacking) computed from this method to that computed from the GW calculations. With these parameters, the calculated E_g and E_opt of the pristine CrSBr are 1.69 eV and 1.25 eV, respectively. The binding energy (E_b) is 0.44 eV. These values agree well with the GW-BSE results (E_g = 1.69 eV, E_opt = 1.23 eV, E_b = 0.46 eV) (18).

The Oscillator Strength of Exciton.

To obtain the optical dipole moment for an exciton, we first determine the single-particle electron–hole transitions involved in the exciton. As mentioned above, the many-body wavefunction of the exciton Φ_I is expressed as a linear combination of the single-particle electron–hole transitions, and Z_I,ij represents the corresponding electron–hole transition amplitude. The polarization-dependent optical dipole moment (μ_I) of the exciton I is calculated as

$${\mathrm{\mu }}_I={\sum }_{i,j,\sigma }{Z}_{I,ij}{P}_{\mathit{ij}},$$ [6]

$$P_{\mathit{ij}}=⟨{\mathrm{\varphi }}_i\left|\widehat{\mathit{\epsilon }}\mathit{r}\right|{\mathrm{\varphi }}_j⟩,$$ [7]

where P_ij is the transition dipole moment between the KS occupied state ϕ_i and unoccupied state ϕ_j. $\widehat{\mathit{\epsilon }}$ is the unit vector of the electric field of the polarized light, and r is the position operator of the electron. Since the position operator r is ill-defined in an extended solid, we employ the momentum operator (73, 74) to compute the transition dipole moment via the relation $i\left[\mathit{H},\mathit{r}\right]=\mathit{p}+i\left[V_{\mathrm{N}\mathrm{L}},\mathit{r}\right]$, where H is the KS Hamiltonian, p is the momentum operator and $V_{\mathrm{N}\mathrm{L}}$ is the nonlocal pseudopotential. Considering that the errors by neglecting the commutator of the nonlocal pseudopotential can be significantly reduced by including d-projectors in the PAW potential (75, 76), we omit the commutator and only consider the momentum operator in the calculation of the transition dipole moment. The oscillator strength is then determined by

$$f=\frac{2m_e{\omega }_I}{3{\hslash }^2}{{\mu }_I}^2,$$ [8]

where ω_I is the exciton energy, m_e is the mass of the electron and ћ is the reduced Planck constant. The radiative recombination rate of the exciton I is given by

$${\gamma }_I=\frac{e^2{\omega }_I{{\mu }_I}^2}{{\epsilon }_0{\hslash }^{2}c{A}_{\mathit{uc}}},$$ [9]

where $A_{\mathit{uc}}$ represents the area of the in-plane unit cell, e is the electron charge, ${\epsilon }_0$ is the vacuum permittivity, and c is the speed of light. The radiative lifetime of the exciton at 0 K is given as

$${\tau }_0={{\gamma }_I}^{-1}.$$ [10]

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

All study data are included in the article and/or SI Appendix.