The Defect Charge Effect on Magnetic Anisotropy Energy and Dzyaloshinskii–Moriya Interaction of the I Vacancy and 3d Transition Metal Co-Doped Monolayer CrI₃

Abstract

Recently, significant effort has been devoted to enhancing magnetic anisotropy energy (MAE) and the Dzyaloshinskii–Moriya interaction (DMI) in two-dimensional (2D) ferromagnetic materials through various tuning approaches. Among these methods, defect engineering is one of the most effective strategies. However, the influence of these charged defects on the MAE and DMI is unclear. Therefore, we systematically investigate the defect effect on the MAE and DMI of I vacancy-doped (v_I-CrI₃), 3d-transition-metal-doped (TM = Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) (3d-TM_i@CrI₃), and v_I-TM co-doped (3d-TM_i@v_I-CrI₃) monolayer CrI₃ using first-principles calculations. Our results indicate that Cr-rich conditions can promote the defect formation of v_I-CrI₃, 3d-TM_i@CrI₃, and 3d-TM_i@v_I-CrI₃ systems and demonstrate that 49 types of charged systems are stable. Among these systems, the Cu_i@v_I-CrI₃ in the +1 charge state (Cu_i^•@v_I-CrI₃) system has a smaller defect formation energy, exhibiting a large MAE exceeding 30 meV, and the ratio (D/J) of the antisymmetric magnetic exchange parameter (D) to the Heisenberg exchange parameter (J) reaches 1.04. The large MAE originates from the transition from single-ion anisotropy (SIA) to covalent interaction anisotropy (CIA) due to the coupling variation between the p_y and p_x orbitals of I atoms near the Fermi level caused by charge states. The enhancement of the DMI is due to the electrostatic potential differences between the I-top and I-bottom layers, which are conducive to forming stable chiral spin textures. This study provides insight into the defect charge state modulating the magnetism of 2D magnetic materials.

1. Introduction

Spintronic devices have inspired extensive interest in next-generation data storage because of their non-volatility, low energy consumption, faster processing speeds, and high storage density [1,2,3]. Two-dimensional (2D) magnetic materials are one of the most promising candidates for spintronic devices with a high specific surface area, layer-dependent magnetism, and efficient spin transport properties [4,5,6]. However, as the temperature increases, spins of 2D magnetic materials tend to misalign due to the thermal fluctuations, which causes the stored information to be irrevocably destroyed [7,8]. A large magnetic anisotropy energy (MAE) can effectively suppress thermal fluctuations and enhance the robustness of chiral spin textures [9,10]. Moreover, the Dzyaloshinskii–Moriya interaction (DMI) is also the most widely used for stabilizing chiral spin structures in 2D magnets with broken inversion symmetry [11], which can be accomplished by introducing vacancy defects [12]. The previous reports suggest that doping 2D materials with transition metal (TM) can enhance the MAE [13,14] and DMI [15,16]. Therefore, regulating 2D materials through vacancy defects and TM doping engineering, leading to significant MAE and DMI, has received extensive attention.

As a 2D intrinsic ferromagnetic material, the monolayer chromium triiodide (CrI₃) was first synthesized by Huang et al. [17], using the mechanical exfoliation method. It is a promising candidate for spintronic devices due to its intrinsic ferromagnetism, layer-by-layer antiferromagnetic ordering, and strong perpendicular magnetic anisotropy [18,19]. Moreover, previous studies have reported that point defects caused by the competition between energy and entropy are inevitable during the preparation processes [20,21,22]. For example, experimentally, the magnetism of monolayer CrI₃ was regulated by introducing defects (e.g., I, Cr, and CrI₃ cluster vacancies; V/Mn at the Cr site) [23,24,25]. Zhang et al. [23] discovered that in monolayer CrI₃, the concentrations of Cr vacancies remain largely unchanged as the temperature increases. This is because the rise in the growth temperature promotes the migration of vacancies, leading to the formation of more CrI₃ cluster vacancies, which counteracts the increase in Cr vacancies. Furthermore, single I and Cr vacancies enhance the ferromagnetism and Curie temperature of the system, while CrI₃ cluster vacancies induce a transition from ferromagnetism (FM) to antiferromagnetism (AFM). Compared to the CrI₃ system, the coercivity and MAE of the V-doped CrI₃ system (Cr_0.5V_0.5I₃) significantly increased, making it a hard magnet [24]. However, the nature of the magnetic changes in these defect systems is still unclear, and a significant amount of work remains to be done.

As one of the most commonly used theoretical methods for studying 2D materials, density functional theory (DFT) has been the basis for numerous studies reporting on the defect effects in monolayer CrI₃ [26,27,28,29]. For example, Wang et al. [26] reported that the reversal between FM and AFM can be achieved at a high double I atom vacancy concentration in monolayer CrI₃. Zhao et al. [30] reported that I vacancies in monolayer CrI₃ enhance FM by reducing the crystal field splitting energy between the e_g and t_2g orbitals and inducing switchable electric polarization in the system. Although many studies have focused on the effect of defects on the magnetic properties of CrI₃, these studies have only considered the neutral state of the systems. It is worth noting that these defects could result in a charge imbalance and that these defect systems are usually charged [31,32,33,34,35]. Therefore, it is necessary to understand the impact of vacancy defects and TM impurities on the magnetic properties of CrI₃ under different charge states.

In this paper, we systematically investigate the effects of defect charge on the MAE and DMI of I-vacancy-doped (v_I-CrI₃), 3d-TM-doped (TM = Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) (3d-TM_i@CrI₃), and v_I-TM co-doped (3d-TM_i@v_I-CrI₃) monolayer CrI₃ using DFT calculations. We find that the charge states greatly regulate the MAE and DMI of defect systems. The MAE exceeds 30 meV for the following systems: Co_i@CrI₃ in the +2 charge state (Co_i^••@CrI₃), 3d-TM_i@v_I-CrI₃ (TM = V and Cu) in the +1 charge state (3d-TM_i^•@v_I-CrI₃), and 3d-TM_i^••@v_I-CrI₃ (TM = Ti, V, Mn, Fe, and Zn). The increased MAE is derived from the p_y/p_x orbitals coupling change for I atoms near the Fermi level under different charge states. Interestingly, the D/J ratio of Cu_i^•@v_I-CrI₃ is notably large, reaching 1.04, due to the electrostatic potential differences between the I-top and I-bottom layers, which are conducive to forming stable chiral spin textures. Moreover, I vacancies and doping with TMs enhance the ferromagnetic stability of CrI₃, owing to an impurity band near the Fermi level. This study provides theoretical guidance for controlling the MAE and DMI through defect charge states.

2. Results and Discussion

As shown in Figure 1a,b, the monolayer CrI₃ is an I-Cr-I sandwiched layer structure. The magnetic Cr³⁺ ions form a honeycomb network in the octahedral coordination, edge-sharing with six I⁻ ions. The optimized lattice constant of the CrI₃ is 7.11 Å, and the bond length between Cr and I atoms is 2.79 Å, which is in good agreement with previous reports [36]. The band gap of monolayer CrI₃ at the PBE + U level is 1.01 eV, as shown in Figure 1c, slightly lower than the experimental value of 1.20 eV [37,38].

Previous studies have demonstrated the susceptibility of monolayer CrI₃ to ambient oxidation [39,40,41,42], necessitating the systematic investigation of oxidative stability in v_I and 3d-TM co-doped systems. As evidenced by Supplementary Note S3 and Figures S4 and S5, the oxygen-substituted co-doped configurations (TM_i@O_I-CrI₃) exhibit prohibitively high formation energies (~5 eV), confirming their thermodynamic instability. Consequently, our subsequent computational analyses focus exclusively on the magnetic properties of v_I-TM co-doped systems.

2.1. The Formation Energy of Point Defect in CrI₃

The intrinsic point defects and impurity TMs are introduced into the CrI₃ supercells, such as v_I-CrI₃, v_Cr-CrI₃, 3d-TM_i@CrI₃, 3d-TM_i@v_I-CrI₃, and 3d-TM_i@v_Cr-CrI₃. We exclude the v_Cr-CrI₃ and 3d-TM_i@v_Cr-CrI₃ in the following calculations because the formation energy of v_Cr-CrI₃ is significantly higher than that of v_I-CrI₃, as presented in Figure S6. To verify the favorable doping sites of TMs in the v_I-CrI₃, we take the Ti_i@v_I-CrI₃ as an example and consider five hollow doping sites (H_a, H_b, H_c, H_d, and H_e, see Figure 1) for the formation energy calculation (see Figure S7). The results show that the formation energies of Ti_i^••@v_I-CrI₃ systems at the H_b site are lower than those at H_a, H_c, H_d, and H_e. Hence, we only consider the H_b site to be the doping site in the following calculation of TM_i@v_I-CrI₃.

To investigate different possible charge states under Cr-rich and I-rich conditions, we calculate the defect formation energies of v_I-CrI₃, 3d-TM_i@CrI₃, and 3d-TM_i@v_I-CrI₃ for all the possible charge states (−2 to +3) as a function of the Fermi level. Taking v_I-CrI₃ as an example (see Figure S6), the formation energies of v_I-CrI₃ in the −2 and −1 charge states are higher than those in the 0 charge state, while the formation energies of v_I-CrI₃ in the +2 charge state are higher than those in the +1 and 0 charge states. This implies unreasonable charge states have relatively higher formation energies; the charge states we calculated are convergence. Thus, this work only focuses on the charge states that exhibit the lowest formation energy for all investigated systems under Cr-rich and I-rich conditions. The formation energy represents the stability of 3d-TM_i@CrI₃ and 3d-TM_i@v_I-CrI₃, and small formation energy is desired for the stability, making them even more likely to form [43,44]. As shown in Figures S8 and S9, the solid bold red lines indicate the lowest formation energies at specific Fermi levels. Meanwhile, these formation energies are also summarized in Figure 2, which suggests that 47 types of the TM_i@CrI₃ and TM_i@v_I-CrI₃ systems under different charge states are stable. Considering the intrinsic point defects v_I-CrI₃ and (c-d) v_Cr-CrI₃ (Figure S6), there are 49 types of charged systems of v_I-CrI₃, 3d-TM_i@CrI₃, and 3d-TM_i@v_I-CrI₃ that are demonstrated to be stable.

As illustrated in Figure 2, the formation energy in Cr-rich conditions is low compared to I-rich conditions, indicating that 3d-TM_i@CrI₃ and 3d-TM_i@v_I-CrI₃ systems are more favorable to form under Cr-rich conditions. Under Cr-rich conditions, compared to Figure 2a,c, we find that the formation energies of 3d-TM_i@v_I-CrI₃ are higher than those of 3d-TM_i@CrI₃. The same results are found under I-rich conditions, as shown in Figure 2b,d. These results indicate that I vacancies negatively affect the 3d TM doped into the CrI₃ monolayer. However, under Cr-rich conditions, the largest formation energies of 3d-TM_i@v_I-CrI₃ are not larger than 2.5eV. Thus, subsequent studies will focus exclusively on the magnetic properties of 3d-TM_i@CrI₃ and 3d-TM_i@v_I-CrI₃ systems.

As shown in Figure 2a, under Cr-rich conditions, the lowest-formation energy defects are Mn_i^••@CrI₃, Zn_i^••@CrI₃, and V_i^••@CrI₃ (their formation energies are almost the same) as E_F < 0.40 eV; when E_F > 0.40 eV, Cu_i^•@CrI₃ is the lowest-formation energy defect. In Figure 2b, the lowest-formation energy defect is Cu_i^•@CrI₃ for all the values of the Fermi level under I-rich conditions, making them easier to form. However, the formation energy of Ti_i@CrI₃ is the highest for all the values of the Fermi level, which means it is difficult to dope Ti into the CrI₃. For 3d-TM_i@v_I-CrI₃ under Cr-rich conditions (Figure 1c), when E_F < 0.34 eV, the lowest-formation energy defects are Mn_i^••@v_I-CrI₃, Zn_i^••@v_I-CrI₃, Cr_i^••@v_I-CrI₃, and V_i^••@v_I-CrI₃ (their formation energies are almost the same); when E_F > 0.34 eV, Cu_i@v_I-CrI₃ is the lowest-formation energy defect. In Figure 2d, the lowest-formation energy defect is Cu_i@v_I-CrI₃ for all the values of the Fermi level under I-rich conditions.

2.2. Magnetic Exchange Interactions

The exchange energies (E_ex) of CrI₃, 3d-TM_i@CrI₃, and 3d-TM_i@v_I-CrI₃ are shown in Table S5, to explore the effect of point defects on magnetic properties. The E_ex between FM and ferrimagnetic (FiM)/AFM states (E_ex = E_FM − E_FiM/AFM) for all the studied systems are less than 0 eV, indicating that the ferromagnetic state is the ground state. Compared with CrI₃, I vacancies and doping with 3d-TMs enhance ferromagnetism. To illustrate the physical origin of ferromagnetic coupling, Figure 3a–d displays the ferromagnetic superexchange mechanism. In a specific octahedral environment, the d levels of Cr atoms split into a higher energy e_g doublet, and a lower energy t_2g triplet, and three electrons occupy the t_2g triplet. In contrast, the e_g doublet remains empty in CrI₃. Considering that the Cr-I-Cr angle is approximately 90º, a 3d electron in one Cr ion can only transfer to another Cr ion through an e_g-p-t_2g transition, as shown in Figure 3a. The p orbitals of the I atom strongly overlap with the t_2g orbitals of one Cr ion. They are orthogonal to the t_2g orbitals of another Cr atom, and thus the t_2g-p-t_2g transition is forbidden. Similar to t_2g-p-t_2g transition, there is no simultaneous overlap between the p orbitals and the e_g orbitals of both Cr ions in e_g-p-e_g transition, as shown in Figure 3b–c. Given that the t_2g orbitals of the Cr ions are half-occupied while the e_g orbitals are empty, according to the Pauli principle, only FM coupling is allowed.

We now further discuss the transition process of t_2g-p-e_g. Two distinct fourth-order electron transfer mechanisms can be sketched, as shown in Figure 3d. The first involves a spin-up electron transferring from the p orbital of an I atom to the e_g orbital of one Cr atom, while a spin-down electron simultaneously transfers from the p orbital to the t_2g orbital of another Cr atom. Subsequently, both electrons revert back to their original p orbitals (see the hopping processes I and II). The second mechanism is a spin-up electron transferring from the p orbital to the e_g orbital of a Cr atom, and then a spin-up electron from the t_2g orbital of another Cr atom transferring to the p orbital. Following these transitions, the electrons return to their initial locations in the p and t_2g orbitals, respectively (see the hopping processes I and III). As demonstrated in Figure S10, compared to CrI₃, the presence of I vacancies and interstitial doping with TMs causes the formation of an impurity band near the Fermi level in the electronic structure, and the reduction of the energy difference between the unoccupied spin-up e_g/spin-down t_2g orbitals and the p orbitals enhances FM.

Next, we calculate the magnetic exchange parameters J and D between all the nearest-neighbor Cr pairs or Cr-TM pairs, and the Hamiltonian can be written as follows:

$$H=-{\displaystyle \frac{1}{2}}J{\displaystyle \sum _{\langle i,j\rangle }{\overrightarrow{s}}_i\cdot {\overrightarrow{s}}_j}-{{\displaystyle \sum _i{A}_i^z({\overrightarrow{s}}_i)}}^2-{\displaystyle \frac{1}{2}}{\displaystyle \sum _{\langle i,j\rangle }D\cdot ({\overrightarrow{s}}_i\times {\overrightarrow{s}}_j)},$$

(1)

where J is the Heisenberg exchange parameter, D is the antisymmetric magnetic exchange parameter, ${\overrightarrow{s}}_i$ and ${\overrightarrow{s}}_j$ are spin vectors of Cr atoms and TMs, and $A_i^z$ is the anisotropy parameter. The J is calculated as $\overline{J}=-{\displaystyle \frac{E_{\mathrm{FM}}-E_{\mathrm{FiM}/\mathrm{AFM}}}{54\left|{\overrightarrow{s}}_{\mathrm{Cr}}\right|\cdot \left|{\overrightarrow{s}}_{\mathrm{Cr}}\right|+6\left|{\overrightarrow{s}}_{\mathrm{Cr}}\right|\cdot \left|{\overrightarrow{s}}_{\mathrm{TM}}\right|}}$ and the D is obtained by $\overline{D}=-{\displaystyle \frac{E_{\mathrm{CW}}-E_{\mathrm{ACW}}}{6\sqrt{3}|{\overrightarrow{s}}_{\mathrm{Cr}}{|}^2+2\sqrt{3}|{\overrightarrow{s}}_{\mathrm{Cr}}|\cdot |{\overrightarrow{s}}_{\mathrm{TM}}|}}$, where E_FM, E_FiM/AFM, E_CW, and E_ACW are the energy with FM, FiM/AFM, clockwise (CW), and anticlockwise (ACW) orders in the CrI₃, v_I-CrI₃, TM_i@v_I-CrI₃, and TM_i@CrI₃ systems, respectively, as shown in Figure S11. The methodology for calculating the J and D is elaborated in Note S4 [45]. A large D/J ratio tends to form a non-collinear magnetic ordering. When D/J exceeds 20%, the system may form stable chiral spin textures, such as skyrmions [46,47,48,49].

Figure 3e,f displays the values of D and J, and the ratios of D/J for the CrI₃, v_I-CrI₃, 3d-TM_i@CrI₃, and 3d-TM_i@v_I-CrI₃ systems under different charge states, respectively. Compared to CrI₃, the values of J in Zn_i^••@CrI₃, Cu_i^×@v_I-CrI₃, Co_i^•@v_I-CrI₃, Cu_i^•@v_I-CrI₃, and Cr_i^••@v_I-CrI₃ systems significantly decrease. In all the systems, the values of D have a small fluctuation, except those in TM_i@CrI₃ (TM = Cr and Zn) under the +1 charge state, which have a large enhancement. As a result, the |D|/J values in Cu_i^•@v_I-CrI₃, Cr_i^•@CrI₃, and Zn_i^•@CrI₃ are 1.04, 1.07, and 0.69, respectively, which are significantly larger than those in other systems. These values far surpass those of typical materials with skyrmions under appropriate conditions, such as MnSi [50], Cu₂OSeO₃ [51], and Rh/Fe/Ir [52]. To explore the physical origin of the strong DMI, Figure S12a–d gives the differential charge density of Cu_i^•@v_I-CrI₃, Zn_i^•@CrI₃ by $∆\rho (\overrightarrow{r})={\rho }_{\mathrm{system}}(\overrightarrow{r})-{\rho }_{{\mathrm{CrI}}_3}(\overrightarrow{r})$, where ${\rho }_{\mathrm{system}}(\overrightarrow{r})$ and ${\rho }_{{\mathrm{CrI}}_3}(\overrightarrow{r})$ are the charge density of the Cu_i^•@v_I-CrI₃, Zn_i^•@CrI₃, and CrI₃ systems, respectively. One can find that the charge distribution between the I-top and I-bottom layers in the system is antisymmetric, which results in a strong DMI.

To further assess the stability of FM in CrI₃, v_I-CrI₃, 3d-TM_i@v_I-CrI₃, and 3d-TM_i@CrI₃, we calculate the T_C using a 5 × 5 × 1 matrix by performing Monte Carlo simulations with the Metropolis algorithm based on the classic Heisenberg model by the MCSOLVER package [53]. As shown in Table S5 and Figure S13, the T_C of CrI₃ (75 K) is larger than the previously reported value (45 K) [54], which is due to the difference in effective Coulomb interaction correction (U_eff) [55,56]. Compared with CrI₃, the T_C of Zn_i^••@v_I-CrI₃ increases to 88 K, indicating that the FM slightly enhanced.

2.3. Magnetic Anisotropy Energy

Generally, magnetic anisotropy mainly contains the magnetic shape anisotropy energy (E_MSA) resulting from the dipole–dipole interaction and the magnetocrystalline anisotropy energy (E_MCA) induced by the spin-orbit coupling (SOC). Compared to E_MCA, the values of E_MSA are negligible in the CrI₃ system [57]. Therefore, in this work, we only consider the impact of MCA. The MCA is primarily contributed by single-ion anisotropy (SIA) [58,59] and covalent interactions anisotropy (CIA) [57,60]. Figure 4a gives the E_MCA of the monolayer CrI₃, v_I-CrI₃, TM_i@CrI₃, and TM_i@v_I-CrI₃ systems under different charge states, respectively. The E_MCA of CrI₃ is 16.5 meV with an E_SIA value of 14.2 meV and an E_CIA value of 2.3 meV. This indicates that MCA is mainly derived from the SIA, consistent with previous reports [57,60]. Under neutral charge states, the E_MCA values of all the systems are smaller than those of CrI₃, except for the Zn_i^×@v_I-CrI₃ system. Moreover, the E_MCA values for most of the systems in positive charge states are higher than those in neutral states, except for the Cr_i^•@CrI₃ and Zn_i^•@v_I-CrI₃ systems. Compared with TM_i@CrI₃ systems, the E_MCA values of TM_i@v_I-CrI₃ systems are large, except for Mn_i@v_I-CrI₃ under positive charge states. Considering that Cu_i@v_I-CrI₃ system has a large D/J ratio of 1.04 and high uniaxial anisotropy, which can stably enhance the creation of skyrmions [9,10], we subsequently focus on analyzing the origin of E_MCA for the Cu_i@v_I-CrI₃ system.

As shown in Figure 4b, the E_SIA of Cu_i^×@v_I-CrI₃ (11.9 meV) is higher than that of Cu_i^•@v_I-CrI₃ (10.3 meV). To analyze the charge effect on E_SIA, Figure 4c gives the orbital-resolved MAE of Cu_i@v_I-CrI₃ systems under 0 and +1 charge states, respectively. It can be observed that the E_SIA is primarily contributed by the coupling of p_y/p_x and p_y/p_z orbitals for the I atoms under both charge states, while the d orbitals of the Cr and Cu atoms play a minor role. In the neutral charge state, the contribution of p_y/p_x and p_y/p_z is positive and leads to an out-of-plane E_SIA, while the p_y/p_z orbitals contribute to negative E_SIA in the +1 charge state. Compared to the neutral charge state, the contributions of both p_y/p_x and p_y/p_z increase in the +1 charge state, and the variation of p_y/p_z orbitals is higher than that of p_y/p_x orbitals, which leads to the reduction in the total E_SIA.

We now further illuminate the mechanism of the influence of p orbitals of I atoms on the E_SIA under 0 and +1 charge states in the Cu_i@v_I-CrI₃ system based on second-order perturbation theory. The E_SIA can be expressed by the following equation [61]:

$$∆E^{\uparrow \uparrow }=E^{\uparrow \uparrow }(x)-E^{\uparrow \uparrow }(z)={\xi }^2{\displaystyle \sum _{o^{\uparrow },u^{\uparrow }}\frac{{|\langle o^{\uparrow }|L_z|u^{\uparrow }\rangle |}^2-{|\langle o^{\uparrow }|L_x|u^{\uparrow }\rangle |}^2}{E_u^{\uparrow }-E_o^{\uparrow }}},$$

(2)

$$∆E^{\downarrow \uparrow }=E^{\downarrow \uparrow }(x)-E^{\downarrow \uparrow }(z)=-{\xi }^2{\displaystyle \sum _{o^{\downarrow },u^{\uparrow }}\frac{{|\langle o^{\downarrow }|L_z|u^{\uparrow }\rangle |}^2-{|\langle o^{\downarrow }|L_x|u^{\uparrow }\rangle |}^2}{E_u^{\uparrow }-E_o^{\downarrow }}},$$

(3)

where u and o are the energy levels of the unoccupied and occupied states, respectively. ↑ and ↓ denote the spin-up and spin-down states, respectively. L_z and L_x represent the orbital angular momentum components of I atoms. Here, the $∆E^{\downarrow \downarrow }$ and $∆E^{\uparrow \downarrow }$ are neglected due to the low proportions of spin-up unoccupied states near the Fermi level, as shown in Figure 4d. The differences in the spin-orbital matrix element and their corresponding energy separations determine the E_SIA. The relative contributions of the nonzero L_z and L_x matrix elements are $|\langle p_x|L_z|p_y\rangle {|}^2=1$, $|\langle p_y|L_x|p_z\rangle {|}^2=1$. The other matrix elements of $|\langle o^{\pm }|L_z|u^{+}\rangle {|}^2-|\langle o^{\pm }\left|L_x\right|u^{+}\rangle {|}^2$ are shown in Table S6.

According to Equations (2) and (3), the E_SIA contributed by p_y/p_x orbitals originates from the spin-conservation term ($\langle p_{y,u}^{\uparrow }\left|L_z\right|p_{x,o}^{\uparrow }\rangle$) in Cu_i@v_I-CrI₃, which can be regarded as excitations of electrons from the occupied spin-up state to the unoccupied spin-up state. For the p_y/p_z orbitals, their contributed E_SIA comes from the spin-flip term ($\langle p_{z,u}^{\downarrow }\left|L_z\right|p_{x,o}^{\uparrow }\rangle$), which arises from the excitations of electrons from the occupied spin-down state to the unoccupied spin-up state. To better clarify the contribution of p_y/p_x and p_y/p_z orbitals coupling to E_SIA, we show partial DOS (PDOS) of the I atoms in Figure 4d. The spin-up p_y/p_x orbitals contribute to out-of-plane SIA with ΔE_cf of 0.508 eV in the Cu_i^×@v_I-CrI₃ system. Compared to the Cu_i^×@v_I-CrI₃ system, the Fermi level moves to the low energy side in the Cu_i^•@v_I-CrI₃ system, and the crystal field splitting energy (ΔE_cf) decreases. Therefore, the contribution of coupling p_y/p_x orbitals to E_SIA increases in the Cu_i^•@v_I-CrI₃ system. The contribution from p_y/p_z orbitals is small for Cu_i^×@v_I-CrI₃. The exchange splitting energy (ΔE_ex) of p_y/p_z orbitals (0.627 eV) for Cu_i^•@v_I-CrI₃ is smaller than that of Cu_i^×@v_I-CrI₃ (0.863 eV), increasing the in-plane E_SIA. The increase in the positive contribution of p_y/p_x is slightly smaller than that of the negative contribution of p_y/p_z, and thus, the overall E_SIA decreases as the charge state changes from 0 to +1.

The E_MCA of Cu_i^×@v_I-CrI₃ (13.7 meV) is significantly less than that of Cu_i^•@v_I-CrI₃ (33.8 meV), whereas the E_SIA of both charge states is approximately equal. This is due to the E_CIA of Cu_i^•@v_I-CrI₃ (23.5 meV) being substantially greater than that of Cu_i^×@v_I-CrI₃ (1.8 meV). To further elucidate the origin of E_MCA, we explore E_CIA under both charge states. Figure S12a presents the charge density difference of Cu_i@v_I-CrI₃ between the +1 and 0 charge states. One can find that the charge redistribution mainly occurs around the I vacancy (See the region indicated by the red arrows in Figure 5a). Then, we calculate the crystal orbital Hamilton population (COHP) to analyze the changes in covalent interaction in the area. As shown in Figure 5b,c, the $\pi {*}_{d-p}^{\mathrm{up}\text{-}\mathrm{up}}$, ${\sigma }_{d-p}^{\mathrm{up}\text{-}\mathrm{up}}$, and ${\sigma }_{d-s}^{\mathrm{up}\text{-}\mathrm{up}}$ bonds are formed by the overlap between spin-up Cr atoms and I atoms at −0.25 −2.81 and −11.33 eV, respectively. Among them, the $\pi {*}_{d-p}^{\mathrm{up}\text{-}\mathrm{up}}$ is an antibonding state, mainly formed by the hybridization between the d orbitals of the Cr atoms and the p orbitals of the I atoms. The d and p orbitals are composed of $d_{\mathit{xy}}$, $d_{\mathit{yz}}$, $d_{z^2}$, $d_{\mathit{xz}}$, $d_{x^2-y^2}$ and p_y, p_z, p_x with different weights, respectively, as specified in Table S7. Additionally, the ${\sigma }_{d-p}^{\mathrm{up}\text{-}\mathrm{up}}$ and ${\sigma }_{d-s}^{\mathrm{up}\text{-}\mathrm{up}}$ are both bonding states, formed with certain weights by the interaction between the d orbitals of the Cr atoms and the p orbitals and s orbitals of the I atoms, respectively. Similarly, for the spin-down interactions, ${\pi }_{d-p}^{\mathrm{dw}\text{-}\mathrm{dw}}\left(1\right)$, ${\pi }_{d-p}^{\mathrm{dw}\text{-}\mathrm{dw}}\left(2\right)$, ${\sigma }_{s-p}^{\mathrm{dw}\text{-}\mathrm{dw}}$, and ${\sigma }_{d-s}^{\mathrm{dw}\text{-}\mathrm{dw}}$ are formed at −0.70, −1.20, −3.16, and −11.43 eV, respectively. Notably, all four bonds are bonding states. To describe the covalent strength quantitatively, we calculate the integrated COHP (ICOHP) by integrating the energy up to the Fermi level, where more negative values indicate stronger interactions. As shown in Figure 5b, compared to Cu_i^×@v_I-CrI₃, the ICOHP of Cu_i^•@v_I-CrI₃ reduces to −0.513 eV (−0.66 eV) for the spin-up (-down) interactions. This is because the Fermi level moves to the lower energy side, reducing the antibonding state. This indicates that the interaction between spin-up (-down) states from the 0 to the +1 charge state is enhanced, thus leading to the increase in the E_CIA in the charge redistribution area. As can be seen from the above analysis, in the Cu_i@v_I-CrI₃ system, the E_MCA is significantly higher in the +1 charge state than in the 0 charge state, due to the notable increase in the E_CIA.

3. Conclusions

In summary, we systematically investigated the effects of defect charge on MAE and DMI of v_I-CrI₃, 3d-TM_i@CrI_3, and 3d-TM_i@v_I-CrI₃ (TM = Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) using DFT calculations. Compared with I-rich conditions, Cr-rich conditions are more favorable for the formation of these systems. Among these systems, the MAEs of Co_i^••@CrI₃, 3d-TM_i^•@v_I-CrI₃ (TM = V and Cu), and 3d-TM_i^••@v_I-CrI₃ (TM = Ti, V, Mn, Fe, and Zn) all exceed 30 meV. This is derived from the change in the coupling of p_y/p_x orbitals for I atoms near the Fermi level under different charge states. Notably, the D/J ratio of Cu_i^•@v_I-CrI₃ exceeds 1 due to the electrostatic potential differences between the I-top and I-bottom layers, which are conducive to forming stable chiral spin textures. Moreover, I vacancies and TMs enhance the ferromagnetic stability of CrI₃ due to an impurity band near the Fermi level, which reduces the energy difference between the unoccupied spin-up e_g/spin-down t_2g orbitals and the p orbitals. This study provides new possibilities for controlling the MAE and DMI by defect charge states in 2D magnetic materials.

4. Computational Details

All the DFT calculations were performed using the projected augmented wave [62] approach, as implemented in the Vienna ab initio simulation (VASP) [63,64]. Considering the exchange and correlation functional interactions, the Perdew–Burke–Ernzerhof (PBE) within generalized gradient approximation was applied [65]. The plane wave cut-off energy was set to 500 eV. The crystal structure of the monolayer CrI₃ was completely relaxed until a force of less than 0.02 eV/Å per atom and an energy difference of smaller than 10⁻⁷ eV between two convergence steps was observed. The VASPKIT code was applied for postprocessing VASP-calculated data [66]. In addition, the crystal orbital Hamilton population (COHP) analysis was carried out to quantify the strength (the integrated value of COHP, ICOHP) of the Cr-I bond using the LOBSTER code [67]. The figures were visualized using VESTA [68]. Partly occupied d orbitals of the TMs were treated using the GGA + U approach. The methodology for calculating the U_eff of defect systems is elaborated in Supplementary Note S1 [26,36,37,69,70,71,72], and Figures S1 and S2. The U_eff of impurity-related phases used for the PBE functional calculations are summarized in Table S1. The calculating methodology for the formation energy of defects is provided in Note S2 [35,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91] and Figure S3. The lattice constants used in the calculations from experimental data are listed in Table S2 [92,93,94,95,96,97,98,99,100,101]. Table S3 presents the formation enthalpy for these impurity-related phases. The chemical potential obtained in this work is displayed in Table S4. We used the standard Kröger–Vink notation [102] (X_D^q) to represent the defect charge state q (′, ×, • being −1, 0, and +1 charge state, respectively) and lattice site D of a dopant X.

Magnetocrystalline anisotropy (MCA) is calculated by taking the difference between the total energy of the magnetization oriented along the in-plane ($E^{\parallel }$) and out-of-plane ($E^{\perp }$) directions, as follows:

$$E_{\mathrm{MCA}}=E^{\parallel }-E^{\perp }.$$

(4)

A positive value represents uniaxial or perpendicular anisotropy, which means the easy magnetization axis is perpendicular to the x-y plane, i.e., along the z-axis. In contrast, a negative represents planar anisotropy, which suggests the easy magnetization axis is parallel to the x-y plane.