**2.2 Magnetic properties**

**CrX3 (X = Cl, Br, I)** exhibit multiple magnetic phases. The superexchange coupling occurs between two adjacent Cr3+ sites and is mediated through the halide ions, which serves as the predominant exchange pathways in the chromium trihalides, as shown in **Figure 1c**. The interlayer exchange interaction between Cr atoms in different layers. **Figure 1d** presents the details of the magnetic order of atomically thin CrX3. Within a single layer, the chromium moments of CrX3 are ferromagnetically coupled as intuited by the Goodenough-Kanamori-Anderson rules [29, 52, 53]. In the case of monolayer CrI3 and CrBr3, the magnetic moments aligned perpendicular to the crystal plane [30, 54], while they aligned parallel to the crystal plane for monolayer CrCl3 [33, 55–57]. In the bulk, two adjacent layers of CrI3 and CrBr3 show FM interlayer exchange [30, 54], and AFM interlayer exchange in CrCl3 [33, 55–57]. However, for bilayer CrI3, two adjacent layers show AFM interlayer exchange [17]. Moreover, the corresponding magnetic ordering temperatures are 61 K for bulk CrI3 [30], 37 K for bulk CrBr3 [58], and 17 K for bulk CrCl3 [33]. **MPS3 (M = Fe, Ni, Mn)** are a class of vdW stacking AFM insulator, which possess different AFM ordering with the different transition metal due to their different spin dimensionalities. The magnetic ground states of MPS3 are summarized in **Figure 2b**. All of the compounds are AFM, but only MnPS3 acquires the Néel state where each magnetic site is anti-aligned with all nearest neighbors [59, 60]. In contrast, the ground states of FePS3 and NiPS3 exhibit Zigzag-type ordering, where each transition metal atom is aligned with two nearest neighbors and anti-aligned with one nearest neighbor [61, 62]. Based on the Mermin-Wagner theorem, FePS3 is predicted to have an easy axis, whereas NiPS3 and

#### **Figure 1.**

*a, Crystalline structure of CrX3 in top view showing the honeycomb arrangement of the Cr atoms [24] Copyright 2022, American Physical Society. b, Top view of the CrX3 layers in the monoclinic (left) and rhombohedral (right) phases. Two CrX3 layers without the X***−** *ions to show the relative stacking order of the bottom (dark blue) and top (blue) Cr3+ layers in two phases [25] Copyright 2021, American Chemical Society. c, Interlayer exchange of CrX3: interlayer Cr nearest-neighbor (J1*⊥*, in blue) and the second-neighbor (J2*⊥*, in red) in AB-stacking (left) and the nearest-neighbor (J'1*⊥*, in green) in AB'-stacking (right) [26] Copyright 2018, American Chemical Society. d, Illustrations showing the magnetic ground states of bilayer CrI3 (left), CrBr3 (middle), and CrCl3 (right). e, Molecular orbital energy diagram for CrBr3. The d-d LF transitions arise from the ground and excited configurations of the d3 electrons in the t2g \* and eg \* orbitals. The dashed orange box shows d-d transitions and the blue arrows show possible LMCT transitions. f, Exciton energy levels of monolayer CrBr3 calculated using the first-principles GW-BSE method (left). Top view of excitonic wave function in real space for selected states that are responsible for the exciton bands of CT1, exciton1 (EX1) and exciton2 (EX2) (right) [27] Copyright 2022, American Chemical Society.*

MnPS3 both have easy planes coinciding with the atomic planes [62]. The Néel temperatures are 118 K for bulk FePS3 [63–65], 155 K for bulk NiPS3 [66], and 78 K for bulk MnPS3 [61]. **CrSBr** is an AFM semiconductor with the A-type antiferromagnetism, which is ascribed to the halogen-mediated (Cr-Br-Cr) and chalcogen-mediated

#### **Figure 2.**

*a, Crystal structure of MPS3 viewed along the c and b axis [34] Copyright 2020, American Physical Society. b, The magnetic ground states of bilayer MnPS3, FePS3 and NiPS3. Arrows denote spin orientation. Star denotes an inversion center of the AFM structure [34] Copyright 2020, American Physical Society. c, Calculated electronic band structure of bulk NiPS3 with the red and blue circles corresponding to projected bands from Ni d-orbitals and S atoms. [35] Copyright 2022, AAAS.*

(Cr-S-Cr) strong superexchange interactions and weak interlayer coupling [40, 67]. As a result, each rectangular layer exhibits in-plane anisotropic FM order, and these FM layers couple antiferromagnetically along the stacking direction, as shown in **Figure 3b** [41, 43, 44]. The AFM Néel temperature is 132 K for bulk CrSBr and FM Curie temperature is 180 K [40, 43]. **CrPS4** is a promising vdW AFM semiconductor with the A-type antiferromagnetism, consisting of out-of-plane FM monolayers coupled antiferromagnetically, as shown in **Figure 4b** [48, 68]. The Néel temperature is ~36 K for bulk CrPS4, below which the spin-flip transition from AFM to FM orders occurs [69, 70].

#### **Figure 3.**

*a, Side and top views of the crystal structure of CrSBr [40] Copyright 2022, Wiley-VCH GmbH. b, The magnetic order of CrSBr, FM in-plane but layered in alternating directions for a bulk antiferromagnetism [41] Copyright 2022, Wiley-VCH GmbH. c, Splitting of d orbitals of the Cr atom under the octahedral crystal field of the CrSBr monolayer and the spin-polarized band structure of FM CrSBr [42] Copyright 2018, American Chemical Society.*

#### **2.3 Electronic band structures**

**CrX3** (**X = Cl, Br, I**)**.** In this crystal field geometry, the Cr *d* splits into a t2g\* triplet and an eg\* doublet. Cr3+ has a valence of three electrons, which fill the t2g majorityspin band according to Hund's first rule, leaving all other *d* bands empty. Since all three CrX3 compounds have FM order down to the monolayer below Curie temperatures, we use CrBr3 as an example to present the molecular orbital energy diagram based on ligand-field (LF) theory, as shown in **Figure 1e**. In the dashed orange box allows *d*-*d* transitions and the blue arrows show possible ligand-to-metal charge transfer (LMCT) transitions [71]. However, an accurate first-principles calculation

*Novel Light-Matter Interactions in 2D Magnets DOI: http://dx.doi.org/10.5772/intechopen.112163*

#### **Figure 4.**

*a, Crystal structure of CrPS4 with the C2/m space group [46] Copyright 2020, Springer Nature. b, Magnetic structure of CrPS4, the back and red arrows point to the direction of magnetic moments [48] Copyright 2020, WILEY-VCH Verlag GmbH and Co. KGaA. c, Molecular orbital energy diagram in the optical transitions of CrPS4 (top) and configuration diagram of Cr3+ for the d-d transitions (bottom) [46] Copyright 2020, Springer Nature. d, Spin-dependent band structures for bilayer CrPS4 in the AFM and FM states [49] Copyright 2021, American Physical Society.*

of the electronic structure of CrX3 should account for both the dielectric polarization from the ligand groups and the on-site Coulomb interactions among the localized spin-polarized electrons in 2D limit [24, 27, 72]. Thus, the recent work has achieved the accurate electronic band structures of CrBr3 monolayer with first-principles GW-BSE calculations including the excitonic effect, as shown in **Figure 1f** [27]. The calculation yields a series of bright exciton energy levels with energies of 1.68, 2.14, and 2.72 eV, which coincides with the measured absorption spectrum. The real-space exciton wave functions corresponding to LMCT, EX1 and EX2 states are shown in **Figure 1f** [27]. The electron and hole are localized separately on the anion and cation for LMCT state, which confirms the CT characteristics of higher energy absorptions in CrBr3. While the excitonic wave functions predominantly occupy the Cr atoms for EX1 and EX2 states, confirming the *d*-*d* origin. **MPS3** (**M = Fe, Ni, Mn**)**.** Here, we mainly discuss the electronic band structure of NiPS3, since the band-edge excitons have been observed in few-layer NiPS3 with interesting optical phenomena. NiPS3 exhibits a unique electronic structure with a highly localized electronic band composed by *d* orbitals. A recent work calculated the electronic band structure and density of states for bulk NiPS3 using first-principles density functional theory + *U* method, as shown in **Figure 2c** [35, 39, 73, 74]. The electronic bands near the conduction band minimum are predominantly contributed by Ni *d* orbitals, the small dispersion of these bands is due to the localization of the Ni *d* orbitals. In contrast, the valence bands are mostly from S *p* orbitals. The calculation indicates an indirect band gap of ~1.6 eV, consistent with the previous results [75, 76]. However, the most recently published work demonstrates that Zhang-Rice triplet (ZRT) state and Zhang-Rice singlet (ZRS) state are formed from the spin-orbital coupling between Ni *p* orbital and S *p* orbitals in APM NiPS3. The transition from ZRT to ZRS states will host a spin-orbit-entangled exciton state [77]. **CrSBr** combines a direct electronic band gap with layered A-type AFM order. Initially, due to the symmetry breaking under the octahedral crystal field, five degenerate *d* orbitals of the Cr atom are split into eg levels (dz 2 and dxy) with higher energy and t2g levels (dx 2 -y 2 , dyz, and dxz) with lower energy, as shown in **Figure 2c** (left) [42]. In the monolayer CrSBr structure, distortion of octahedron leads to further splitting of eg and t2g levels, consequently, the five d orbitals are no longer degenerate. With the coupling with atomic orbitals of S atoms, the spin-polarized band structures of CrSBr are formed, as shown in **Figure 2c** (right) [42]. Furthermore, the more accurate electronic structure calculations of monolayer CrSBr in its FM ground state within the GW approximation demonstrate a semiconducting band gap of ~1.8 eV and highly anisotropic band dispersion [78]. First-principles GW-BSE calculations unveil the different exciton wave functions for AFM and FM CrBrS bilayers. In the AFM bilayer, the electron is localized in the same layer as the hole. In the FM bilayer, the electron wave function is delocalized across both layers for a hole fixed in the bottom layer [78]. Thus, the optical transitions dominant by these excitonic transitions will be tuned during the AFM to FM phase transition. According to the crystal structure of **CrPS4**, each Cr atom is located in a distorted octahedral interstice formed by six S atoms, thus inducing an octahedral ligand perturbation field that splits that 3*d* orbitals of Cr3+ into t2g and eg orbitals. The ground state (4 A2g) and three lowest excited states (2 Eg, <sup>4</sup> T2g, <sup>4</sup> T1g) emerge from the ground term (4 F) of d3 , as shown in **Figure 4c** [46, 79]. The optical transitions should be from the spin-allowed *d*-*d* transition of the Cr3+ ion. For more realistic scenarios, spin is considered in the electronic band structure calculations, as shown in **Figure 4d**, the band gap for bilayer CrPS4 is 0.83 eV (AFM state), 0.72 eV (spin

up in the FM state) and 1.43 eV (spin down in the FM state), respectively [49]. However, for more accurate electronic band structures of CrPS4, the excitonic effect should be considered.
