Two-Fold Enhancement of Curie Temperature in Monolayer CrI₃ by High Pressure

Abstract

In recent years, the discovery of the two-dimensional (2D) intrinsically ferromagnetic monolayer CrI₃ has opened up promising avenues for the advancement of spintronic devices. Nevertheless, the relatively low Curie temperature poses a significant challenge for practical applications. Herein, we determine changes in the superexchange interaction of ferromagnetic coupling caused under pressure by using first-principles calculations and Monte Carlo simulations. Based on the superexchange interaction of ferromagnetic coupling, the effect of applying high pressure on the Curie temperature of monolayer CrI₃ is investigated. With a pressure coefficient of 2.0%, the Curie temperature is enhanced to 97.3 K, which is nearly double that of the monolayer CrI₃ without pressure. In addition, the direction of the easy magnetization axis changes from the out-of-plane to the in-plane one when the pressure coefficient is 1.2%. Meanwhile, the band gap of monolayer CrI₃ can be transformed from indirect to direct by applying high pressure. Our work enriches the process of modulating the magnetic and electronic properties of 2D monolayer materials.

1. Introduction

The pursuit of intrinsic magnetism in two-dimensional (2D) monolayer materials has been an ongoing research focus for an extended period [1,2] due to the unique physical properties and the promising applications in spintronics [3,4,5,6,7], magneto-optoelectronics [8,9,10], and data storage [11]. In 2017, intrinsic ferromagnetism was first observed in atomically thin layers of CrI₃ and Cr₂Ge₂Te₆, which opened new avenues for exploring low-dimensional magnetic performance and spintronics devices [8,12]. After that, more and more monolayer 2D materials [13,14,15,16] with intrinsic magnetism were investigated experimentally and theoretically, such as CrBr₃ [17], VBr₃ [18], VI₃ [19], NiBr₃ [20], FePS₃ [21], and VSe₂ [22]. Moreover, many 2D intrinsic magnetic materials have been predicted with the assistance of high-throughput screening and machine learning [23,24,25,26,27,28]. However, most 2D monolayer magnetic materials have a common weakness, which is that the Curie temperature (T_C) is much lower than room temperature, which greatly hinders their applications.

As a representative 2D intrinsically ferromagnetic (FM) materials, the monolayer CrI₃ is a potential candidate for practical application because of its distinct spin-lattice coupling [29] and strong perpendicular magnetic anisotropy [30]. Nevertheless, the T_C of monolayer CrI₃ is only 45 K [8], which makes enhancing its T_C a great challenge. Many attempts have been made to improve the T_C of monolayer CrI₃, such as constructing heterostructures [31,32], adsorbing small molecules [33,34], and doping foreign elements [17,35,36]. For example, the T_C of 2D monolayer can be raised to 99.3 K by constructing a CrI₃/α-In₂Se₃ heterojunction [15]. Constructing a heterojunction with GaN is another effective method, which can increase the T_C of monolayer CrI₃ to 75 K. Meanwhile, the easy magnetization axis transforms from an out-of-plane to an in-plane orientation, with electron doping levels exceeding 0.2 electrons per unit cell [31]. Moreover, the adsorption of gas molecules, including CO, H₂, N₂, and H₂O, can substantially enhance FM coupling in monolayer CrI₃ [33]. Specifically, the adsorption of N₂ can significantly improve the T_C of the pristine monolayer CrI₃ by approximately 2.6 times [33]. It has been reported that doping with RE atoms (RE = Sc, Y, La, Ce, Pr, Nd, Pm, Eu, Gd, Tm, and Lu) enables a significant enhancement of the Curie temperature and the control of total magnetic moments of the CrI₃ monolayer, resulting in an obvious increase in T_C for monolayer CrGdI₆ to 347.58 K, as shown in [37].

To further investigate alternative approaches to enhancing the T_C in monolayer CrI₃, it is essential to elucidate the fundamental mechanisms that are responsible for FM coupling. In monolayer CrI₃, the magnetic properties originate from the partially filled Cr-3d orbitals. The Cr³⁺ ion resides within an octahedral crystal field formed by six surrounding ligands, leading to a splitting of the Cr-3d orbitals into higher-energy, doubly degenerate e_g orbitals and lower-energy, triply degenerate t_2g orbitals. The three 3d electrons of Cr³⁺ occupy the lower-energy t_2g orbitals. Mediated by the intermediate I-5p orbital, the spin direction of the electrons in the Cr-3d orbitals is consistently aligned, resulting in a superexchange interaction between the two Cr ions adjacent to the I ion, which is driven by FM coupling. It is known that the strength of the superexchange interaction is decisively affected by the overlap degree of the wave functions for Cr-d and I-p orbits, which is reflected by the bond length l of Cr-I and the bond angle θ of Cr-I-Cr [38]. Generally, the shorter l is and the nearer θ is to 180°, the stronger the exchange interaction, is and the higher the T_C is [39,40].

According to our recent work [41], introducing I-vacancies and interstitial H-atoms into pristine monolayer CrI₃ can induce internal pressure, leading to obvious changes in the Cr-I bond length (l) and the Cr-I-Cr bond angle (θ), which remarkably improves the value of the T_C. When an interstitial hydrogen atom is introduced into the 2 × 2 × 1 CrI₃ supercell, the T_C can be enhanced to 112.4 K.

Inspired by the aforementioned results, in this paper, we attempt to investigate the effect of applying external pressure on the physical properties and T_C of monolayer CrI₃ by using first-principles calculations and Monte Carlo (MC) simulations. We establish a 2 × 2 × 1 CrI₃ supercell to simulate the magnetic and electronic properties of monolayer CrI₃ under different pressures. One novelty of this study is the use of first-principles calculations and Monte Carlo simulations to determine changes in the superexchange interaction of ferromagnetic coupling caused by pressure, which helps with understanding the relationships between the pressure and the T_C of monolayer CrI₃. Our work illustrates the promising potential of structural changes induced by high pressure in modulating the magnetization orientation and enhancing magnetism in monolayer CrI₃.

2. Materials and Methods

Our first-principles calculations were conducted using the Vienna Ab initio Simulation Package (VASP), employing density functional theory (DFT) [42]. The generalized gradient approximation (GGA) exchange correlation was formulated according to the Perdew–Burke–Ernzerhof (PBE) approach [43]. Taking into account the correlation effects of Cr-3d electrons, the GGA + U method [44] was adopted. The main reason was that there is a strong coulomb interaction between the 3d orbital electrons of Cr in CrI₃, and the general DFT exchange correlation function is not sufficient to describe the above coulomb interaction, resulting in the close proximity or even overlap of the orbitals. The addition of U and J take into account the coulomb repulsion between the local electrons with opposite spin on the same atom, resulting in the splitting of the energy levels, so that the theoretical band gap value is closer to the experimental value. Respectively, the on-site Coulomb interaction U and exchange interaction J parameters were set to 3.0 and 0.9 eV [45]. The core electrons were treated by the projector-augmented wave (PAW) method [46]. A vacuum region of approximately 20 Å was introduced along the z direction to accurately model the 2D monolayer system. The kinetic energy cutoff was set to 500 eV for the plane-wave basis functions. For the structural optimizations, the Brillouin zone was sampled using a 9 × 9 × 1 Monkhorst–Pack k-point mesh.

In the previous study, high pressure was applied on the monolayer MoS₂ to regulate the optoelectronic and band structural properties via a diamond anvil cell and DFT calculations [47]. The fundamental configuration for the DFT calculations involves relaxing all structures within the x-y plane, while the z coordinates are constrained using a conjugate gradient method [47]. As we know, in monolayer CrI₃, the Cr atomic plane is sandwiched by two I atomic planes, which is similar to that of monolayer MoS₂. Thus, in this paper, we used a similar setup to perform the DFT calculation for investigating the effect of pressure on the physical properties of monolayer CrI₃.

To ensure computational efficiency, the convergence criteria for energy and force in the self-consistent field (SCF) calculations were set to 10⁻⁸ eV and 0.001 eV/Å, respectively. Meanwhile, spin–orbit coupling (SOC) interaction was included in the noncollinear calculation when the values of magnetic anisotropy energy (MAE) were calculated. The calculation of structural optimization and the static and electronic state densities were carried out using the GGA + U method. Given that the GGA(PBE) method typically underestimates the energy gap of semiconductor materials, in order to make the calculation more accurate, we employed the Heyd–Scuseria–Ernzerhof (HSE06) [48] screened hybrid functional to calculate their electronic properties and band information. The mixing parameters were AEXX and ALDAX. ALDAX was constrained to be equal to 0.75.

Using a 32 × 32 × 1 2D honeycomb lattice with periodic boundary conditions, the T_C of the samples based on monolayer CrI₃ were calculated using the MC simulations. For each temperature examined, the MC simulations entailed 10⁴ MC steps per site to attain thermal equilibrium. All MC simulations were conducted using the open-source project MCSOLVER [49].

3. Results and Discussion

As shown in Figure 1a,b, the 2D monolayer CrI₃ possesses a rhombohedral BiI₃ structure (space group $R\overline{3}$) at a low temperature. Each Cr³⁺ ion is surrounded by six I⁻ ions, which build an edge-sharing octahedra. Under this environment, the spin direction of the three 3d electrons which occupy the lower energy Cr-t_2g orbits is consistent, resulting in$S={\displaystyle \frac{3}{2}}$. Our spin-polarized calculations show that the magnetic moment on each Cr atom is 3.1 μ_B, which agrees well with the experimental value of ∼3.1 μ_B [50]. Furthermore, high pressure was applied on the monolayer CrI₃ along the z direction. Here, the pressure coefficient ɛ is defined as follows:

$$\varepsilon ={\displaystyle \frac{z-z_0}{z_0}}$$

(1)

where z₀ and z are the coordinates of the I atom along the z direction for the system without and with pressure, respectively. During the calculation process, in order to reflect the influence of pressure, we fixed the z-axis coordinate of the I atom and the z-direction of the lattice constant after setting the z-direction coordinate of the I atom when performing the structure optimization. In accordance with the renowned Mermin–Wagner theorem [51], the MAE serves as a critical parameter in 2D materials, enabling stabilization of the long-range magnetic ordering against thermal fluctuations. Therefore, the values of the MAE of monolayer CrI₃ under different pressures were calculated. Here, the MAE is obtained by calculating the total energy difference (MAE = E_// − E_⊥) of two different magnetization directions (in-plane (E_//) and the perpendicular (E_⊥)). Due to its varying directions in between the x or y direction, while calculating the easy magnetization axis, the spin directions for the in-plane specific configuration were x and y. The results are listed in

Table 1. $J_1$ and J₂, as well as the Curie temperature TC, for monolayer CrI3 under different pressures.

ɛ	l (Å)	θ (deg)	J1 (meV)	J2 (meV)	TC (K)	EMA	MAE (μeV/Cr)
0%	2.75	95.2	−1.89	−0.31	53.5	z	778
0.5%	2.73	96.0	−2.16	−0.27	68.8	z	437
1.0%	2.70	97.0	−2.47	−0.23	82.8	z	124
1.2%	2.69	97.3	−2.60	−0.19	83.1	x	−9.2
1.5%	2.68	97.8	−2.85	−0.16	90.6	x	−234
2.0%	2.66	98.7	−3.23	−0.12	97.3	x	−592

, in which the positive and negative values of MAE show the easy magnetization axis (EMA) along the out-of-plane and in-plane directions, respectively. The MAE of monolayer CrI₃ under zero pressure is 778 μeV/Cr, which is in close agreement with previous reports [52,53]. With the pressure coefficient ɛ increasing to 1.2%, the value of the MAE of monolayer CrI₃ decreases from 778 to −9.2 μeV/Cr. Simultaneously, the EMA of monolayer CrI₃ transforms from the out-of-plane to the in-plane direction. As the pressure coefficient ɛ increases from 1.2% to 2.0%, the MAE notably changes from −9.2 to −592 μeV/Cr, and the direction of EMA is still in-plane. These results suggest that the MAE and EMA of monolayer CrI₃ can be effectively optimized through the application of high pressure.

With the nonzero MAE, the exchange interaction, which is another fundamental parameter for determining the T_C of 2D magnetic systems, was then studied. The magnetic exchange parameter J_i is obtained by the DFT calculations, and the Heisenberg spin Hamiltonian is given by the following [54]:

$$H=E_0+{\sum }_{ij}{J_{1}S}_i{S}_j+{\sum }_{iK}{J_{2}S}_i{S}_K+A{S}_i^z{S}_i^z$$

(2)

where $J_1$ and $J_2$ denote the parameters for the nearest and next-nearest magnetic exchange interaction, respectively; $S_i$ is the spin vector of each magnetic atom; $A$is the anisotropy energy parameter; and $S_i^z$ is the z component of spin vector. During the calculation of $J_1$ and $J_2$, three spin arrangements (FM, Stripy-AFM, and Néel-AFM) were considered. In the FM configuration, all magnetic moments were initialized to align in a uniform direction, which is shown in Figure 1c. The arrangements of the magnetic moments for the Stripy-AFM and Néel-AFM configurations are illustrated in Figure 1d,e, respectively. The magnetic energies of the three spin configurations of monolayer CrI₃ are expressed as follows:

$$E_{\mathrm{F}\mathrm{M}}=E_0+\left(3J_1+6J_2\right){\left|S\right|}^2+A{\left|S\right|}^2$$

(3)

$$E_{\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{y}}=E_0+\left(J_1-2J_2\right){\left|S\right|}^2+A{\left|S\right|}^2$$

(4)

$$E_{\mathrm{N}é\mathrm{e}\mathrm{l}}=E_0+\left(-3J_1+6J_2\right){\left|S\right|}^2+A{\left|S\right|}^2$$

(5)

Correspondingly, the expressions of the magnetic exchange interaction parameters $J_1$ and $J_2$ are

$$J_1={\displaystyle \frac{E_{\mathrm{F}\mathrm{M}}-E_{\mathrm{N}\acute{\mathrm{e}}\mathrm{e}\mathrm{l}}}{{6\left|S\right|}^2}}$$

(6)

$$J_2={\displaystyle \frac{{2E}_{\mathrm{F}\mathrm{M}}-3E_{\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{y}}{+E}_{\mathrm{N}é\mathrm{e}\mathrm{l}}}{{24\left|S\right|}^2}}$$

(7)

Here, $\left|\mathrm{S}\right|={\displaystyle \frac{3}{2}}$.

After comparing the calculated results shown in Table 1, we find that applying high pressure on monolayer CrI₃ can lead to remarkable changes in the $J_1$ and $J_2$. The values of $J_1$ and $J_2$ of monolayer CrI₃ under zero pressure are −1.84 and −0.31 meV, respectively. As the pressure coefficient ɛ increases, the absolute value of $J_1$ changes significantly from 1.89 to 3.23 meV, and that of $J_2$ decreases slightly from 0.31 to 0.12 meV, as shown in Table 1.

In order to quantitatively estimate the T_C value of monolayer CrI₃ under different pressures, MC simulation was conducted based on the Heisenberg model with the values of $J_1$ and $J_2$. The temperature-dependent variations in heat capacity (C_V) and average magnetic moment of the monolayer CrI₃ are shown in Figure 2. As we know, the temperature at which the mean magnetic moment exhibits a sharp decline to nearly zero and the C_V peaks can be identified as the T_C. For monolayer CrI₃, the T_C is estimated to be 53.5 K, as illustrated in Figure 2a, which is consistent with previously reported results [8,52,53]. With the pressure increasing, the T_C value of monolayer CrI₃ enhances dramatically, and the maximum value is 97.3 K when the pressure coefficient ɛ is 2.0%. The T_C values under different pressures are listed in Table 1.

Generally speaking, direct exchange and superexchange interactions are two major kinds of exchange interactions, which are often discussed in research on 2D semiconducting magnetic materials. In the monolayer CrI₃, the FM coupling stems from the superexchange interaction, which can be investigated using the projected density of states (PDOS). As shown in Figure 3, the PDOS peaks of spin-up and spin-down in the valence band and those of spin-down in the conduction band gradually move toward the Fermi energy level as the pressure coefficient ɛ increases. Meanwhile, two more intense resonant states, located at the range of −2 to 2 eV, imply that the hybridization degree between the d-orbital of Cr and p-orbital of I becomes stronger with the increase in pressure. It is well established that the strength of a superexchange interaction is significantly influenced by the degree of overlap between wave functions. Figure 4 illustrates the distribution of wave functions in monolayer CrI₃ under varying pressures. Under the same display standard, when no pressure is applied, the degree of overlap between the spin-up and spin-down electronic state densities of Cr-d and I-p atoms is relatively small. Compared to the monolayer CrI₃ under zero pressure, the application of high pressure markedly enhances the overlap between Cr-d and I-p orbitals under pressure coefficients ɛ of 1.0% and 2.0%. This indicates that the superexchange interaction under ferromagnetic coupling is significantly enhanced under the action of pressure, which accounts for the observed increase in T_C.

As mentioned above, the mechanism of FM coupling in monolayer CrI₃ is a superexchange interaction, whose strength is decisively affected by the bond length l of Cr-I and the bond angle θ of Cr-I-Cr [38]. Moreover, the shorter l is and the nearer θ is to 180°, the stronger the exchange interaction is, and the higher the T_C is [39,40]. Thus, it is essential to investigate the impact of pressure on the Cr-I bond length l and Cr-I-Cr bond angle θ in monolayer CrI₃. Following the structural optimization, the Cr-I bond length l and Cr-I-Cr bond angle θ with zero pressure were determined to be 2.75 Å and 95.2°, which is in agreement with the previously reported values of 2.74 Å and 95.2°. When the pressure coefficients ɛ are 0.5%, 1.0%, 1.2%, 1.5%, and 2.0%, the corresponding values of l and θ are listed in Table 1. Compared with these calculated results, the Cr-I bond length l becomes shorter and the Cr-I-Cr bond angle θ becomes larger with the pressure coefficient ɛ increasing, which can make the e_g levels move closer to the occupied t_2g levels and enhance ferromagnetic exchange. Thus, the T_C is enhanced.

To further investigate the effect of high pressure on the electronic properties of 2D monolayer materials, the band structures of monolayer CrI₃ for the FM phase under different pressures were calculated based on the HSE06 functional. As shown in Figure 5a, the monolayer CrI₃ without applying pressure is an FM insulator with an indirect band gap of 2.03 eV, which is consistent with previously reported results [45]. With the pressure coefficient ɛ increasing to 1.0%, the valence bands of the spin-up are hardly shifted, but the conduction bands gradually move away from the Fermi level, resulting in an increase in the band gap of monolayer CrI₃. When the pressure coefficient ɛ increases to 2.0%, instead of an indirect band gap, a direct electronic band gap of 3.23 eV is observed, which is important for optoelectronic and magneto-optical applications.

4. Conclusions

As shown in the above calculation results, applying high pressure can significantly enhance the T_C of monolayer CrI₃. Moreover, this is also possible to achieve experimentally. According to earlier reports [47], high pressure has been applied to monolayer MoS₂ to regulate the band structure and optoelectronic properties via a diamond anvil cell. Based on these experimental findings, we propose that high pressure can be applied on monolayer CrI₃ via a diamond anvil cell.

In conclusion, the effect of high pressures on the magnetic and electronic properties of monolayer CrI₃ were investigated by using the first-principle calculations and MC simulations. The calculated results indicate that applying high pressure to two-dimensional monolayer CrI₃ along the z direction can transform the direction of EMA from the out-of-plane to the in-plane and remarkably enhance the T_C. In particular, when ɛ is increased to 2.0%, the T_C of monolayer CrI₃ increases from 53.5 to 97.3 K, which is nearly double that of pristine monolayer CrI₃. The changes in the Cr-I bond length and Cr-I-Cr bond angle indicate that the wave function overlap of Cr-d and I-p orbitals increases, which is the reason for the T_C enhancement of monolayer CrI₃. Moreover, as the applied pressure coefficient ɛ reaches 2.0%, an indirect-to-direct change of band gap can be observed in monolayer CrI₃. These findings provide practical methods and strategies for enhancing ferromagnetism in 2D monolayer systems, which is useful for the further potential application of 2D materials in spintronic devices.