Abstract
The recent discovery of magnetism in atomically thin layers of van der Waals crystals has created great opportunities for exploring light–matter interactions and magnetooptical phenomena in the twodimensional limit. Optical and magnetooptical experiments have provided insights into these topics, revealing strong magnetic circular dichroism and giant Kerr signals in atomically thin ferromagnetic insulators. However, the nature of the giant magnetooptical responses and their microscopic mechanism remain unclear. Here, by performing firstprinciples GW and BetheSalpeter equation calculations, we show that excitonic effects dominate the optical and magnetooptical responses in the prototypical twodimensional ferromagnetic insulator, CrI_{3}. We simulate the Kerr and Faraday effects in realistic experimental setups, and based on which we predict the sensitive frequency and substratedependence of magnetooptical responses. These findings provide physical understanding of the phenomena as well as potential design principles for engineering magnetooptical and optoelectronic devices using twodimensional magnets.
Introduction
The magnetooptical (MO) effects, such as the magnetooptical Kerr effect (MOKE) and the Faraday effect (FE), have been intensively investigated experimentally in a variety of magnetic materials, serving as a highly sensitive probe for electronic and magnetic properties. Recent measurements using MOKE have led to the discovery of twodimensional (2D) magnets, and demonstrated their rich magnetic behaviors^{1,2}. In particular, a giant Kerr response has been measured in ferromagnetic mono and fewlayer CrI_{3}^{2}. Magnetic circular dichroism (MCD) in photo absorption has also been measured in ferromagnetic monolayer CrI_{3} ^{3}. However, the exact microscopic origin of such large MO signals and MCD responses in 2D materials is still unclear, because treating accurately sizable spin−orbit coupling (SOC) and excitonic effects that are essential for such an understanding in these systems has been beyond the capability of existing theoretical methods.
CrI_{3}, in its monolayer and fewlayer form, is a prototypical 2D ferromagnetic insulator with an Isinglike magnetic behavior and a Curie temperature of about 45 K, exhibiting tremendous outofplane magnetic anisotropy^{2}. Within one layer, the chromium atoms form a honeycomb structure, with each chromium atom surrounded by six iodine atoms arranged in an octahedron (Fig. 1a, b), and the point group of the structure is S_{6}. The crystal field therefore splits the Cr 3d and I 5p ligand states into t_{2g} and e_{g} manifolds; the spin degeneracy of which are further lifted by the exchange interaction. Although the majorspin e_{g} states are delocalized due to strong p−d hybridization, the magnetic moment is approximately 3μ_{B} at each Cr site, in accordance with an atomic picture from the first Hund’s rule^{4}.
With our recently developed fullspinor GW and BetheSalpeter equation (BSE) methods, we show from first principles that the exceedingly large optical and MO responses in ferromagnetic monolayer CrI_{3} arise per se from strongly bound exciton states consisting of spinpolarized electron−hole pairs that extend over several atoms. These exciton states are shown to have distinct characteristics compared with either the Frenkel excitons in ionic crystals and polymers, or Wannier excitons in other 2D semiconductors. By simulating realistic experimental setups, we further find that substrate configuration and excitation frequency of the photon strongly shape the MO signals. Our results provide the conceptual mechanism for the giant optical and MO responses, explaining quantitatively the recent experiments on CrI_{3} ^{2,3}. In addition, comparison between bulk and monolayer CrI_{3} reveals the pivotal role of quantum confinement in enhancing the MO signals.
Results
Quasiparticle band structure
An accurate firstprinciples calculation of the electronic structure of CrI_{3} should account for both the dielectric polarization from the ligand groups and the onsite Coulomb interactions among the localized spinpolarized electrons. We adopt the following approach. The firstprinciples GW method has become a de facto stateoftheart approach to describe dielectric screening and quasiparticle excitations in many real materials^{5}. In practical calculations here, the G_{0}W_{0} approximation^{5} is employed where the GW selfenergy is treated as a firstorder correction, and the singleparticle Green’s function G as well as the screened Coulomb interaction W are calculated using eigenvalues and eigenfunctions from densityfunctional theory (DFT). Through the screened Coulomb interaction W, the nonlocal and dynamical screening effects as well as the selfenergy effects beyond the DFT Kohn−Sham orbital energies (within the localspindensity approximation (LSDA)) are captured. Also, in previous studies, the method of LSDA with an onsite Hubbard potential (LSDA + U) has served as a reasonable meanfield starting point for G_{0}W_{0} calculations in correlated systems to avoid the spurious p−d hybridization^{6,7}. In this work, we adopt an onsite Hubbard potential in the rotationally invariant formulation^{8} with U = 1.5 eV and J = 0.5 eV, with fully relativistic pseudopotentials and a plane wave basis set. The validity of this specific set of U and J has been carefully tested (see Supplementary Figs. 1 and 2). Throughout this work, the magnetization of ferromagnetic monolayer CrI_{3} is taken to be along the +z direction (Fig. 1b). As shown in Fig. 1c, our calculations reveal a strong selfenergy correction to the quasiparticle bandgaps, due to the weak dielectric screening in reduced dimensions and the localized nature of the d states. The direct bandgap is 0.82 eV at the Γ point at the LSDA + U level, whereas the direct G_{0}W_{0} quasiparticle bandgap including the selfenergy effect is 2.59 eV, as shown in Fig. 1c. Throughout the calculations, we incorporate the SOC effect from the outset by employing full twocomponent spinor wave functions.
Excitondominant optical responses
The strong SOC strength and the ligand states strongly hybridizing with Cr d orbitals (see Supplementary Fig. 3) have decisive influences on the electronic structure and optical responses of ferromagnetic monolayer CrI_{3}. SOC significantly modifies the bandgap and band dispersion near the valence band maximum^{4}. Figure 2a shows the G_{0}W_{0} band structure together with each state’s degree of spin polarization (with an outofplane quantization axis), of which the orbital and spin degeneracy are consistent with the above discussions. After solving the firstprinciples BSE, which describes the electron−hole interaction^{9}, with spinor wave functions, we find a series of strongly bound dark (optically inactive) and bright (optically active) exciton states with excitation energies (Ω_{S}) below the quasiparticle bandgap, as shown in the plot of the exciton energy levels (Fig. 2b). As seen in Fig. 2c, the calculated linearly polarized absorption spectrum including electron−hole interactions (i.e., with excitonic effects, solid red curve labeled GWBSE) features three peaks at around 1.50, 1.85 and 2.35 eV (below the quasiparticle gap of 2.59 eV), which are composed of several bright exciton states in each peak and denoted as A, B and C, respectively. This is in contrast to the calculated stepfunctionlike noninteracting absorption spectrum (i.e., without excitonic effects, dashed blue curve labeled GWRPA). The magnitude of the absorbance peak around 1.50 eV is deduced to be 0.7% from a previous differential reflectivity measurement (Fig. 2c, inset)^{3}, while our calculated absorbance with a broadening factor of 80 meV is around 0.6% at 1.50 eV. From our calculation (Fig. 2b), there are also two dark states (excitons D) with enormous binding energy of larger than 1.7 eV. The existence of two states of nearly the same energy comes from the fact that there are two Cr atoms in a unit cell. We plot the realspace exciton wave functions of these states, with the hole fixed on a Cr atom, in Fig. 2d–k. Unlike monolayer transition metal dichalcogenides where the bound excitons are of Wannier type with a diameter of several nanometers^{10,11}, ferromagnetic monolayer CrI_{3} hosts dark Frenkellike excitons localized on a single Cr atom (Fig. 2d, e) and bright chargetransfer or Wannier excitons with wave functions extending over one to several primitive cells (Fig. 2f–k). These plots are consistent with the intuition that a smaller exciton binding energy is related to a larger exciton radius^{11,12}. Numerical calculations of the exciton radius further corroborate this conclusion (see Supplementary Table 1).
In addition, ferromagnetism and broken timereversal symmetry (TRS) play vital roles in determining the internal structure of the exciton states in ferromagnetic monolayer CrI_{3}, in contrast to the Frenkel/chargetransfer excitons determined solely by flatband transitions in boron nitride systems^{13,14}, organic materials^{15,16} or alkali halides^{9}. The eigenstate of an exciton is a coherent superposition of free electron−hole pairs at different k points (cv, k〉), and may be written as \(S\rangle = \mathop {\sum}\nolimits_{cv{\bf{k}}} {A_{cv{\bf{k}}}^S\left {cv,{\bf{k}}} \right\rangle }\), where \(A_{cv{\mathbf{k}}}^S\) is the exciton envelope function in kspace^{9}. Here c denotes conduction (electron) states and v denotes valence (hole) states. In Fig. 3a–d we plot the module square of the exciton envelope function in kspace. As expected of highly localized Frenkel excitons in real space, the lowestlying dark state D in Fig. 3a shows a uniform envelope function in kspace, whereas the bright states A (Ω_{S} = 1.50 eV) and B^{+} (at Ω_{S} = 1.82 eV) in Fig. 3b, c have the envelope function localized around Γ and have s characters. From Fig. 3d, an interesting hexagonal petal pattern with a node at Γ can be found for exciton B^{−} (Ω_{S} = 1.92 eV). In Fig. 3e–h, we plot the distribution of the constituent free electron−hole pairs specified by (E_{v}, E_{c}) for selected exciton states, weighted by the module squared exciton envelope function for each specific interband transition. It is obvious that the electron−hole composition of exciton D is distinct from those of the bright states (A and B).
Because of broken TRS and strong SOC effect^{9}, the electron (hole) states that compose a given exciton in this system are from Bloch wave functions with spin polarization along different directions, giving rise to rich excitonic spin configurations. In fact, the lowestlying bound exciton states are all formed by Kohn−Sham orbitals with particular spinpolarization. Our calculations verify that the dark excitons D are dominated by (>99.5%) transitions between the majorspin valence bands and minorspin conduction bands. The bright states (forming peaks A, B and C) in Figs. 2f–k and 3b–d, f–h, however, are all dominated by (>96%) transitions between the majorspin valence bands and majorspin conduction bands (see Supplementary Table 2). Ligand field theory can provide a qualitative understanding of the lowestlying D and A exciton states of which the optical transitions mainly occur among the localized Cr d orbitals^{3,17}. However, ligand field theory is insufficient to evaluate the oscillator strength of the excitons quantitatively. In addition, the coexistence of Frenkel and Wannier excitons in our system poses significant challenges to ligand field theory, while this excitonic physics can be fully captured by the firstprinciples GWBSE method.
MO effects from first principles
The abovementioned internal structures of the exciton states are essential for a deeper understanding of the MO responses. Note that all the irreducible representations of the double group \(S_6^{\mathrm{D}}\) are onedimensional, which facilitates our analysis of optical selection rules for circularly polarized lights around the Γ point, as shown in Fig. 3i. For 1slike bright states A and B^{+} wherein the transition mainly happens between the topmost valence band and the majorspin e_{g} manifold near the Γ point, only one σ^{+} circularly polarized transition is allowed among all the transitions, e.g., \(\langle c_3\hat p_ + v_1\rangle \ne 0\). Here v_{1}〉 denotes the first valence state, c_{3}〉 denotes the third conduction state and \(\hat p_ + = \frac{{  1}}{{\sqrt 2 }}( {\hat p_x + i\hat p_y})\) is the momentum operator in spherical basis with an angular momentum equal to 1; σ^{±} denotes the circularly polarized light with the complex electric field amplitude along the direction of the spherical basis: \({\bf{e}}_ \pm = \frac{ \mp }{{\sqrt 2 }}( {{\bf{e}}_x \pm i{\bf{e}}_y})\), where \({\bf{e}}_x\) \(( {{\bf{e}}_y})\) is the unit vector along the +x (+y) direction. This conclusion is further confirmed by our firstprinciples circularly polarized absorption shown in Fig. 3j. The 2slike exciton B^{−}, unlike A and B^{+}, is dominated by σ^{−} circularly polarized transitions. We quantify the MCD of absorbance by calculating the contrast, \(\eta = \frac{{{\mathrm{Abs}}\left( {\sigma ^ + } \right)  {\mathrm{Abs}}\left( {\sigma ^  } \right)}}{{{\mathrm{Abs}}\left( {\sigma ^ + } \right) + {\mathrm{Abs}}\left( {\sigma ^  } \right)}}\), where Abs(σ^{±}) denotes the absorbance of σ^{+} and σ^{−} circularly polarized light, respectively. η is dominated by σ^{+} circularly polarized light below 1.8 eV (Fig. 3k). If we flip the magnetization direction, η will also flip sign at all frequencies, which agrees with the measured MCD of photoluminescence signals^{3}.
In the following, we investigate the MO Kerr and Faraday effects of ferromagnetic monolayer CrI_{3}. Previous studies have shown that both SOC and the exchange splitting should be present to ensure nonzero MO effects in ferromagnets^{18,19,20,21,22}, and recent calculations within an independentparticle picture using DFT have been carried out for the MO responses of monolayer CrI_{3}^{23}. The essence of a theoretical modeling of the MO effects lies in accurately accounting for the diagonal and offdiagonal frequencydependent macroscopic dielectric functions, which are readily available from our GWBSE calculations with electron−hole interaction included. We find that the abovediscussed giant excitonic effects in ferromagnetic monolayer CrI_{3} strongly modify its MO responses, leading to significantly different behaviors going beyond those from a treatment considering only transitions between noninteracting Kohn−Sham orbitals^{23}. Here we shall only consider the most physically relevant measurement for 2D ferromagnets, namely, polar MOKE (PMOKE) and polar FE (PFE), where both the sample magnetization and the wave vectors of light are along the normal of the surface. In accordance to typical, realistic experimental setup, we consider a device of ferromagnetic monolayer CrI_{3} on top of a SiO_{2}/Si substrate (the thickness of SiO_{2} layer is set to 285 nm, and Si is treated as semiinfinitely thick)^{2}, as shown in Fig. 4a. For insulating SiO_{2} with a large bandgap (8.9 eV), we use its dielectric constant \(\varepsilon _{{\mathrm{SiO}}_2} = 3.9\)^{24}. For silicon, we perform firstprinciples GW (at the G_{0}W_{0} level) and GWBSE calculations, and incorporate the frequencydependence of the complex dielectric function \(\varepsilon _{{\mathrm{Si}}}(\omega )\) (see Supplementary Fig. 4). Assuming an incident linearly polarized light, we calculate the Kerr (Faraday) signals by analyzing the polarization plane of the reflection (transmission) light, which is in general elliptically polarized with a rotation angle θ_{K}(θ_{F}) and an ellipticity χ_{K} (χ_{F}) (see Supplementary Fig. 5). Here we adopt the sign convention that θ_{K} and θ_{F} are chosen to be positive if the rotation vector of the polarization plane is parallel to the magnetization vector, which is along the +z direction.
We find that the MO signals are very sensitive to the thickness of SiO_{2} and to the photon frequency. As shown in Fig. 4c, d and Supplementary Fig. 6, the thickness of SiO_{2} layer will strongly affect the MO signals, due to the interference of reflection lights from multiple interfaces^{2}. Such interference has been accounted for with our threeinterface setup in Fig. 4a. To analyze the relation between MO signals and dielectric functions, we also consider a simpler twointerface setup. For a twointerface setup with semiinfinitely thick SiO_{2} layer, the Kerr angle θ_{K} (Fig. 4d, solid blue curve) is related to Im[ε_{xy}] (Fig. 4b, dashed blue curve) and therefore resonant with the exciton excitation energies; the Kerr ellipticity χ_{K} (Fig. 4d, dashed red curve), on the other hand, is proportional to Re[ε_{xy}] (Fig. 4b, solid blue curve). For a twointerface model, θ_{K} is also found to be proportional to \(n_0/(n_2^2  n_0^2)\), where n_{0} (n_{2}) is the refractive index for the upper (lower) semiinfinitely thick medium. Moreover, the θ_{K} and χ_{K} are connected through an approximate Kramers−Kronig relations, as expected from previous works^{22,25}. Because of this, close attention should be paid in interpreting MOKE experiments on 2D ferromagnets, where the substrate configuration significantly changes the behavior of the MOKE signals. The existing experimental data of θ_{K}, however, only have a few excitation frequencies of photons available, e.g., 5 ± 2 mrad at 1.96 eV for HeNe laser^{2}. As shown in Fig. 4c, our simulations with a 285 nm SiO_{2} layer in the threeinterface setup achieve the same order of magnitude for θ_{K} around the MO resonance at ~1.85 eV, in good agreement with experiment. Based on the simulations, we also predict a sign change of θ_{K} around 1.5 eV. For photon energies higher than the quasiparticle bandgap, the plasmon resonance along with a vanishing ε_{xx} will nullify our assumptions of continuous waves^{25,26}. It is also possible to achieve an inplane ferromagnetic structure with an external magnetic field^{27,28}. However, due to the broken C_{3} symmetry therein, we expect the system to have diminished values of MO signals (in the same polar configurations) but to remain having excitons with large binding energies, as confirmed by our firstprinciples calculations (see Supplementary Fig. 7).
Effects of quantum confinement
To further understand the effects of quantum confinement in 2D magnets, we compare the MO properties of ferromagnetic bulk and monolayer CrI_{3}. Interestingly, the calculated optical properties of bulk CrI_{3} are also dominated by strongly bound excitons with optical absorption edge starting from 1.5 eV (in good agreement with experiment^{3}), while the quasiparticle indirect bandgap is 1.89 eV and the direct bandgap at Γ is 2.13 eV (see Supplementary Fig. 8). Within a oneinterface model of semiinfinitely thick bulk CrI_{3}, θ_{K} reaches a magnitude of 60 mrad at the resonances at around 1.7 and 2.0 eV (Fig. 4g), proportional to Re[ε_{xy}] shown in Fig. 4f. To study the quantum confinement effect, we employ the PFE setup shown in Fig. 4e, because PFE in this setup is almost linear with respect to the ferromagnetic sample thickness and free from the substrate effects. Our calculated magnitude of the specific Faraday angle (θ_{F}) of bulk CrI_{3} is (1.3 ± 0.3) × 10^{3} rad cm^{−1} at the excitation frequency of 1.28 eV, in agreement with the experimental value of 1.9 × 10^{3} rad cm^{−1} at the same excitation frequency^{29}. By extrapolating the bulk θ_{F} to the monolayer thickness^{30}, and comparing with that of suspended ferromagnetic monolayer CrI_{3} as shown in Fig. 4h, we find that quantum confinement significantly enhances the MO response by a factor of 2.5 near 2.0 eV and introduces a redshift of 0.2 eV.
Discussion
In summary, from our firstprinciples calculations, we discover that the optical and MO properties of ferromagnetic monolayer CrI_{3} are dominated by strongly bound excitons of chargetransfer or Wannier characters. A systematic modeling framework for PMOKE and PFE experiments is also developed, where we have shown that the MO signals exhibit a sensitive dependence on photon frequency and substrate configuration. These findings of the exciton physics in 2D magnets should shed light on design principles for future magnetooptical and optoelectronic devices, such as photospinvoltaic devices^{31} and spininjecting electroluminescence^{32,33}. As a prototypical monolayer Ising magnetic insulator with a bandgap in an easily accessible optical range, ferromagnetic monolayer CrI_{3} is also expected to be useful in highspeed and highdensity flexible MO drives using van der Waals homostructures or heterostructures^{27,34}.
Methods
Firstprinciples GW and GWBSE calculations
Firstprinciples calculations of the electronic structure of ferromagnetic monolayer CrI_{3} (as the meanfield starting point of the G_{0}W_{0} and BSE studies) were performed at the DFTLSDA level, as implemented in the Quantum ESPRESSO package^{35}, with parameters for the onsite Hubbard interaction U = 1.5 eV and Hund’s exchange interaction J = 0.5 eV^{8}. A slab model with a 16 Å vacuum thickness was adopted to avoid interactions between periodic images. We employed optimized normconserving Vanderbilt pseudopotentials including Cr 3s and 3p semicore states^{36,37}. The Kohn−Sham orbitals were constructed with planewave energy cutoff of 80 Ry. Experimental structure was used in the calculations for both the bulk and monolayer CrI_{3}, with the lattice constants: a = 6.867 Å^{30} (see Supplementary Table 3). SOC was fully incorporated in our calculations. The GW (at G_{0}W_{0} level) and GWBSE calculations, for the quasiparticle and optical properties, respectively, were performed using the BerkeleyGW package^{38}. The dielectric cutoff was set to 40 Ry. We adopted a 6 × 6 × 1 grid with six subsampling points for calculating the dielectric function in ferromagnetic monolayer CrI_{3}^{39}. An 18 × 18 × 1 grid was then used for calculating the selfenergy corrections. We treated the dynamical screening effect through the Hybertsen−Louie generalized plasmonpole model^{5}, and the quasiparticle bandgap was converged to within 0.05 eV. The resulting quasiparticle band structure was interpolated with spinor Wannier functions, using the Wannier90 package^{40}. Within our GWBSE calculations, the exciton interaction kernel was interpolated from an 18 × 18 × 1 grid to a 30 × 30 × 1 grid using a linear interpolation scheme^{9}, and the transitions between 21 valence bands and 14 conduction bands were considered in order to converge the calculation of the transverse dielectric functions from the GWBSE results. The GW (at G_{0}W_{0} level) and GWBSE calculations of ferromagnetic bulk CrI_{3} used identical energy cutoffs and convergence thresholds as of monolayer CrI_{3}, and we adopted a 4 × 4 × 4 grid for calculating the dielectric function and selfenergy corrections in bulk CrI_{3}. The GWBSE calculations of bulk CrI_{3} employed a coarse grid of 6 × 6 × 6 which was further interpolated to a fine grid of 10 × 10 × 10. In this work, we obtained the calculated dielectric function of a ferromagnetic monolayer in a supercell slab model by using a thickness of a monolayer CrI_{3} of d = c_{bulk}/3 = 6.6 Å (see Supplementary Figs. 9 and 10). Our calculations were performed for suspended CrI_{3} in vacuum. Addition of an insulating substrate, such as fused silica or hexagonal boron nitride (hBN), introduces a small redshift of the exciton energies (estimated to be less than 0.1 eV for an hBN substrate, see Supplementary Fig. 11), while the strong excitonic effects still dominate the optical and MO responses. Effects of the onsite Hubbard potential on singleparticle energies were systematically investigated to reveal the strong p−d hybridization of the majorspin e_{g} states (see Supplementary Fig. 12 and Supplementary Table 4).
Group theory analysis
We analyzed the symmetry of wave functions in ferromagnetic monolayer CrI_{3}. Ferromagnetic monolayer CrI_{3} has point group symmetry \(S_6 = C_3 \otimes C_i\). The irreducible representations labeled in Fig. 3i are for the double group \(S_6^{\mathrm{D}}\) due to the presence of strong spin−orbit coupling. The notation of the irreducible representations follows previous works^{41,42}.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The BerkeleyGW package is available from berkeleygw.org.
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Acknowledgements
The work was supported by the Theory Program at the Lawrence Berkeley National Lab (LBNL) through the Office of Basic Energy Sciences, U.S. Department of Energy under Contract No. DEAC0205CH11231, which provided the GW & GWBSE calculations and simulations and by the National Science Foundation under Grant No. DMR1508412 and Grant No. EFMA1542741, which provided for theoretical formulation and analysis of the MOKE simulations. Advanced codes were provided by the Center for Computational Study of ExcitedState Phenomena in Energy Materials (C2SEPEM) at LBNL, which is funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division under Contract No. DEAC0205CH11231, as part of the Computational Materials Sciences Program. Computational resources were provided by the DOE at Lawrence Berkeley National Laboratory’s NERSC facility and the NSF through XSEDE resources at NICS.
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S.G.L. conceived the research direction and M.W. proposed the project. M.W. developed the fullspinor methods/codes, carried out computations and wrote the manuscript; M.W., Z.L. and T.C. analyzed the data; S.G.L. directed the research, proposed analyses and interpreted results. All authors discussed the results and edited this manuscript.
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Wu, M., Li, Z., Cao, T. et al. Physical origin of giant excitonic and magnetooptical responses in twodimensional ferromagnetic insulators. Nat Commun 10, 2371 (2019). https://doi.org/10.1038/s41467019103257
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