Room-temperature ferromagnetism and half-metallicity in monolayer orthorhombic CrS₂

Abstract

Two-dimensional materials with high-temperature ferromagnetism and half-metallicity have the latest applications in spintronic devices. Based on first-principles calculations, we have investigated a novel two-dimensional CrS₂ phase with an orthorhombic lattice. Our results suggest that it is stable in dynamics, thermodynamics, and mechanics. The ground state of monolayer orthorhombic CrS₂ is both ferromagnetic and half-metallic, with a high Curie temperature of 895 K and a large spin-flipping gap on values of 0.804 eV. This room-temperature ferromagnetism and half-metallicity can maintain stability against a strong biaxial strain ranging from −5% to 5%. Meanwhile, increasing strain can significantly maintain the out-of-plane magnetic anisotropy. A density of states analysis, together with the orbital-resolved magnetic anisotropy energy, has revealed that the strain-enhanced MAE is highly related to the 3d-orbital splitting of Cr atoms. Our results suggest the monolayer orthorhombic CrS₂ is an ideal candidate for future spintronics.

Graphical abstract

Keywords

orthorhombic CrS ₂ / Curie temperature / magnetic anisotropy energy / biaxial strain / first-principles calculations

1 Introduction

Since the successful preparation of graphene in 2004 [1], numerous two-dimensional (2D) materials have been discovered, such as h-BN [2], transitional metal dichalcogenides (TMDs) [3, 4], phosphorene [5], and MXenes [6], showing novel properties in electronics and spintronics. The well-known Mermin−Wagner theorem [7, 8] indicates spontaneous magnetization cannot exist in 2D isotropic Heisenberg ferromagnets. Therefore, 2D magnetic materials for spintronics have not appeared until recent years, such as CrI₃ [9, 10], Fe₃GeTe₂ [11, 12], and Cr₂Ge₂Te₆ [13, 14]. However, most 2D ferromagnets have limited their practical applications due to their low Curie temperature (T_C), and lack of ferromagnetic (FM) order [15, 16]. To overcome these limitations, researchers have identified a class of magnetic half-metallic materials that have the advantages of 100% spin polarizability at the Fermi level, additional carriers to stabilize the long-range magnetic order, and high magnetic moments [17-20]. Therefore, in the development of 2D magnetic devices, researchers have focused on materials with these properties.

According to experimental and theoretical reports, chromium-based magnetic materials have the potential to achieve a high T_C and half-metallicity. For example, researchers synthesized a monolayer of 1T (1T'/2H)-CrS₂ using chemical vapor deposition [21-25]. They have found that 2H-CrS₂ is a nonmagnetic (NM) direct band gap semiconductor, while 1T and 1T'-CrS₂ are metallic and half-metallic ferromagnets [21, 23]. Other researchers discovered that adjusting the thickness of 1T-CrTe₂ (8 nm [22], 4 monolayer [24], 7 monolayer [25]) films could result in room temperature ferromagnetism and perpendicular magnetic anisotropy (PMA), and their T_C decreased as thickness lower [24, 25]. Theoretical studies have revealed that 2D CrX₂ (X = O, S, Se, Te) systems exhibit a variety of electronic and magnetic properties, especially in the 1T, 1T', and 2H phases. For instance, CrO₂ in the 2H phase is NM, and the 1T phases are FM and half-metallicity [26, 27]. In the ground state of the CrS₂ [28, 29] and CrSe₂ [30, 31] systems, the 2H phase is in the NM state, 1T phase is antiferromagnetic (AFM) state, and 1T' phase is FM state. Conversely, the 1T phase of CrTe₂ [32, 33] in the ground state exhibits long-range FM order in the monolayer limit. It is important to note that the T_C reaches 1358 K for 1T'-CrS₂ [29] with 6% uniaxial strain and 110 K for 1T-CrSe₂ with 10.8-nm-thick [34]. Chromium tellurium compounds (Cr_1+δTe₂) exhibit a wide range of magnetic order temperatures [35-37] (Cr₃Te₄ = 340 K, Cr₄Te₅ = 318 K, and CrTe = 367 K). These findings raise the question of whether there are other CrX₂ materials with similar structures.

In this work, we have predicted a novel monolayer orthorhombic CrS₂ (ML O-CrS₂) and investigated its electronic structure, magnetic anisotropy energy (MAE), and biaxial strain tunable T_C by using first-principles calculations. The results show that ML O-CrS₂ is FM and a spin-up channel half-metallic with a high T_C (895 K) and MAE (0.034 meV/Cr). Interestingly, the biaxial strain does not change ferromagnetism and half-metallicity and can increase the T_C. The physical mechanism of the magnetic easy axis is explored from the view of atom- and orbit-resolved MAE, which shows that the strong spin−orbit coupling (SOC) introduced by the heavy magnetic Cr atoms plays a crucial role. Our investigations enrich the family of 2D FM materials and may contribute to future experimental studies of novel 2D magnets with high T_C.

2 Calculation method

First-principles calculations were guided by Density Functional Theory (DFT) and implemented in the Vienna Ab initio Simulation Package (VASP) [38, 39]. The exchange-correlation potential was characterized using the Generalized Gradient Approximation (GGA) within the framework of Perdew−Burke−Ernzerh (PBE) formalism. The Projector Augmented Wave (PAW) [40] potential was used to investigate the electron−ion interactions, setting a cutoff energy of 500 eV for ML O-CrS₂. Structural optimizations were performed via the conjugate gradient method, with the convergence criteria set at 10⁻⁵ eV for the total energy and 0.01 eV/Å for force. The Brillouin zones were sampled using a Gramma-centered Monkhorst−Pack [41] k-point mesh with a grid of 7 × 7 × 1 for structural optimizations, then followed by calculations of electronic and magnetic properties using a 9 × 9 × 1 grid. Periodic boundary conditions were applied in all three directions, and a vacuum space of 20 Å was established along the Z-direction to screen interactions between adjacent layers. The Hubbard U correction was applied to compensate for the strong correlation of the 3d electrons in transition metal Cr atoms, with an effective U_eff (U_eff = U − J) set to 3.0 eV, as validated in prior studies [21, 42]. The phonon dispersion spectrum was derived using the finite displacement method on a 4 × 4 × 1 supercell via the PHONOPY package [43]. The thermodynamic stability was verified through Ab Initio Molecular Dynamics (AIMD) simulations conducted in the fixed particle number-volume-temperature (NVT) ensemble, with temperature control achieved through a Nosé‒Hoover thermostat [44].

3 Results and discussion

3.1 Structure and stability

As shown in Fig.1(a), the ML O-CrS₂ is stacking by the S−Cr−S sandwich layers along the Z-direction. Its primitive cell belongs to the P-4m2 (No.115) space group (see red rectangular area). Each Cr⁴⁺ cation is surrounding by six neighboring S²⁻ anions with octahedral coordination. The magnetic properties of the 1T and 1H phases are also calculated, and the results closely align with those observed in previous research [28, 29], thereby confirming the validity of our chosen calculational parameters. Furthermore, Tab.1 presents the optimized lattice and magnetic parameters.

To determine the dynamic stability of the ML O-CrS₂, we calculated its phonon spectrum, as shown in Fig.1(b). The phonon dispersion exhibits a small imaginary frequency around the Γ point (0.2 THz), which can considered as a negligible numerical noise. This indicates ML O-CrS₂ is dynamically stable. We have further performed AIMD simulations, as shown in Fig.1(c). The crystal structure remains stable at 400 K for a duration of 10 ps and shows no distinct structural distortion, suggesting ML O-CrS₂ is thermally stable above room temperature. We then calculated the elastic constant tensor of a symmetric 6 × 6 matrix. The elastic constants are C₁₁ = C₂₂ = 108.7 N/m, C₁₂ = 17.4 N/m, and C₆₆ = 12.8 N/m, and satisfy the Born criterion of stability [45] (C₁₁ > 0, C₆₆ > 0, and C₁₁ C₂₂ > C₁₂ C₁₂), confirming the ML O-CrS₂ is mechanically stable.

3.2 Electronic properties

Fig.2 displays the spin-polarized electronic structure of ML O-CrS₂. The band of the spin-up channel crosses the Fermi level, while the spin-down channel band gap is 2.626 eV (see light blue area). The ML O-CrS₂ shows metallicity in the spin-up channel and semiconducting behavior in the spin-down channel. Thus, it exhibits intrinsic half-metallicity. Using the HSE06 hybrid functional and PBE method (see Supplementary Materials Fig. S1), ML O-CrS₂ still exhibits FM and half-metallicity, which is consistent with the results of PBE+U. This suggests that charge transport is fully controlled by bands in the same spin-up channel, resulting in 100% spin-polarization. Therefore, it may serve as an ideal material for spin injection [18, 20]. The spin-flip band gap refers to the minimum energy value in the bottom of spin-down conduction band with respect to the Fermi level and the absolute value of the top energy of spin-down valence band [46, 47]. Interestingly, there is a significant spin-flipping gap of 0.804 eV. This particular characteristic allows for the transition from a spin-up excited state to a spin-down state at room temperature while still maintaining a stable spin polarization and remaining unaffected by thermal excitation. As a result, it proves valuable as a critical component in various spintronic devices, including sources and sinks of spin-polarized current, spin-polarized electron memory, and spin field-effect transistors [14, 19, 42]. Within the energy range from −3 to 2 eV, the Cr-3d and S-3p orbitals exhibit a significant overlapping peak, indicating considerable orbital hybridization.

3.3 Magnetic properties

The spin Hamiltonian can be expressed as

(1)H=H_0+\left[-{\displaystyle \frac{J}{2}}{\displaystyle \sum _{i\text{j}}{S}_i{S}_{\text{j}}+{\displaystyle \sum _i{A}_e{(S_i^e)}^2}}\right],

(2)J={\displaystyle \frac{E_{\mathrm{A}\mathrm{F}\mathrm{M}}-E_{\mathrm{F}\mathrm{M}}}{16S^2}}.

Here, H and H₀ represent the Hamilton and nonmagnetic energy, respectively. J represents the nearest-neighbor Heisenberg exchange parameter. To simplify the calculation, S_iS_j represents the spin state of Cr atom in i/j cite, S is taken as normalized, and the term with A represents the easy-axis single-ion magnetic anisotropy. E_FM and E_AFM represent the FM and AFM energy, respectively.

To determine the magnetic ground state, we have analyzed potential magnetic configurations, including NM, FM, and AFM order [see Fig.3(a) and (b)]. For the ML O-CrS₂ system, the FM state energy is lower than that of the AFM state, regardless of the adopted function. The FM state remains stable even at six different values of U (0−5 eV, see Supplementary Materials Tables S1 and S2). The spin-charge density further proves that the magnetism mainly originates from the Cr element and partially from the S element (see Supplementary Materials Fig. S2). The spin polarization of S is opposite to that of Cr, which favors an FM coupling between Cr sites, which is consistent with results obtained from Fig.2. The spin exchange parameter (J) was calculated value of 119.519 meV. The PBE, PBE+U, and HSE06 functions have shown a positive J, indicating the presence of FM state in the ML O-CrS₂ (see Supplementary Materials Table S1).

To evaluate the angle-dependent MAE, we conducted noncollinear magnetic calculations considering the influence of SOC. Fig.3(c) illustrates the MAE of ML O-CrS₂ when the magnetization angle is limited to the XZ plane (the X and Y plane being equivalent). The MAE is determined by the energy difference between an arbitrary magnetization direction and a specific one (MAE = E₀₀₁‒E₁₀₀). In ML O-CrS₂, the magnetic easy axis is out-of-plane (Z axis), while the in-plane represents the magnetic hard axis. The SOC effect in 2D systems significantly contributes to the MAE, with Cr, a heavy element, exhibiting a strong SOC effect. This could account for the observed large MAE. The intensity of MAE magnetization is 0.034 meV/Cr, which is comparable to that of 1T-CrO₂ [27] and FeP₄ [20]. The presence of MAE ensures the stable existence of the FM state at higher temperatures by resisting thermal disturbance.

Next, we analyze the microscopic mechanism underpinning the ferromagnetism in the CrS₂ system. Our calculations reveal that the Cr-3d orbitals primarily contribute to the total magnetic moments, and the contribution of S element is negligible. To understand the strong FM coupling in monolayer O-CrS₂, there are three prevalent mechanisms should be considered: direct and super-exchange interactions induced by the coupling of local magnetic moments, and the itinerant electron magnetism originating from the half-metallicity. We first consider the direct exchange coupling mechanism. The nearest-neighbor distance between Cr atoms is 3.66 Å, approaching the sum of radii of two Cr atoms (3.70 Å). As a comparison, monolayer CrCl₃ should very weakly AFM coupling due to the direct exchange coupling [48], which has a Cr‒Cr distance of 3.49 Å slightly smaller than that of monolayer O-CrS₂. A larger distance will further weaken the direct exchange coupling, therefore it should be fractional in monolayer CrS₂. Then we consider the super-exchange mechanism. Typically, the crossover angle from FM to AFM coupling is 127° ± 0.6° for the super-exchange coupling between half-occupied 3d-orbitals [49], the Cr−S−Cr bond angle of 109.66° is inclined towards FM. According to the findings of Soriano et al. [48], the Cr−I−Cr bond angle in CrI₃ is 97.5°, which is characteristic of the FM super-exchange, and the Mo−I−Mo bond angle in MoTeI is 113.43°, which also belongs to FM super-exchange [50]. This suggests that the FM super-exchange in the monolayer O-CrS₂ may be more favorable and is not expected to be particularly strong. It is noteworthy that the CrS₂ system exhibits half-metallicity, a characteristic commonly observed in other metallic systems such as Fe₃GeTe₂ [51], FeTe [52], and 1T-CrTe₂ [23], which are known to exhibit strong ferromagnetism. This observation suggests that itinerant electromagnetism is instrumental in achieving robust FM exchange interactions. By calculating the densities of states for NM, FM, and AFM configurations (see Supplementary Materials Fig. S7), we observe a reduction in their distributions near the Fermi surface. This indicates a tendency for the system’s energy to decrease in the FM state. For the first, second, and third nearest-neighboring Cr−Cr pairs, the exchange parameters can be estimated as J₁ = 119.519 meV, J₂ = 4.471 meV, and J₃ = −9.097 meV, respectively, as FM coupling interactions. Therefore, the FM coupling in the monolayer O-CrS₂ primarily originates from itinerant ferromagnetism.

Based on the classical Heisenberg model, the Monte Carlo (MC) method [53] with the Metropolis algorithm is used to analyze the thermal dynamics of magnetism in the equilibrium state. All the renormalization group MC algorithms described here were executed in the open-source project Spirit package [54]. Fig.3(d) presents the calculated magnetization and specific heat as a function of temperature. The T_C is estimated to be 895 K, and the magnetization curves are well-fitted by using the Curie−Bloch equation [55] in the classical limit (cyan lines). Its T_C is far higher than the experimental results of other 2D FM materials like CrI₃ (45 K) [10] and CrTe₂ (200 K) [25]. In terms of its theoretical value, its T_C is also higher than that of other monolayers like 1T-CrTe₂ (405 K) [32] and lepidocrocite-type CrS₂ (842 K ) [42]. The results suggest the presence of ferromagnetism with T_C above room temperature exists in ML O-CrS₂. To verify the T_C, we employed the same method to simulate a hexagonal lattice ML CrI₃ (see Supplementary Materials Fig. S4), which yielded a T_C value of 55 K, almost consistent with previous experimental works [10].

3.4 Strain engineering

In this work, we explored the relations for the ML O-CrS₂ under the biaxial strain along the XY-plane, which is defined as: {\epsilon }_{xy}= (a-a_0)/a_0\times 100\mathrm{\%}. Here, a and a_0 represent the in-plane lattice constants for strained and unstrained monolayers, respectively. The calculated results show that the FM configuration is the ground state of the ML O-CrS₂ with net magnetic moments per unit cell close to 2μ_B. Further analysis revealed that the FM energy is consistently lower than the AFM energy, indicating that the system maintains good FM properties (see Supplementary Materials Fig. S5). The Cr ions carry most of the magnetic moment, while the adjacent S ions exhibit AFM spin polarization (see Supplementary Materials Table S3). The ML O-CrS₂ has an exchange energy [ΔE = (E_AFM − E_FM) / unit cell] of 1.738 (2.176) eV for −5 (5)%. As a result of increasing distances between adjacent Cr−Cr, ΔE monotonically increases with strain growth, indicating that the system remains FM.

To develop practical spintronic devices with half-metallic ferromagnets, the half-metallic gap should be wide enough to prevent them from spin-flip transitions caused by thermal disturbance and maintain half-metallicity at room temperature. Half-metals can exhibit three energy gaps. The spin-flip gap Δ1(Δ3) is defined as the absolute value of the top(bottom) energy of spin-down valence(conduction) band with relative to the Fermi level. The spin-flip gap Δ2 represents the spin-down channel band gap [46, 47]. In Fig.4(a), when biaxial strain increases, spin-flip gap Δ1 and Δ2 increase significantly, while Δ3 increases slightly. Under 5% tensile strain, the band gap Δ2 is 3.026 eV, which is about 137.8% of that under the −5% compressive strain (2.195 eV), indicating that biaxial strain can significantly increase the spin-flip gap of the system. We analyzed the spin-polarized energy band structure of the ML O-CrS₂ to investigate its electronic property (see Supplementary Materials Fig. S6 and Table S3). The band of the spin-up (spin-down) channel gradually move as the biaxial strain changes. The biaxial strain in the range from ‒5% to 5% does not change energy band structure properties, and the band minimum of the spin-down channel is always localized at the high-symmetry M-point due to the strong localization effect of the 3d orbital of the transition metal Cr atom, maintaining FM in spin-up channel half-metallicity.

Fig.4(b) illustrates the J and MAE function under biaxial strain. A consistent increase in J with strain is observed (see Supplementary Materials Table S4). At a compressive strain of −5%, the value of J is 0.108 eV, which can increase up to 0.136 eV at a tensile strain of 5%. This indicates a 25% increase, suggesting that the T_C of the intrinsic ML O-CrS₂ system [e.g., Fig.4(d)] is also likely to increase. For practical device applications, external tuning of the MAE is crucial. We further discovered that the MAE is sensitive to both biaxial strain and energy, its minimum in the unstrained state. Additionally, the magnetization easy axis consistently remains out-of-plane (Z axis), and biaxial strain does not alter the magnetization easy axis. The MAE significantly decreases with increasing compressive strain, while a similar trend is observed for tensile strain. This suggests that biaxial strain can significantly adjust the MAE. To comprehend why biaxial strain can enhance the PMA for ML O-CrS₂, we initially analyzed the atom-resolved MAEs. As depicted in Fig.4(c), the MAE is primarily contributed by the Cr and S atoms. There is a notable shift in the contribution from S1(S2) atoms towards a more positive MAE, while the contribution from the Cr atoms becomes negative. These results suggest that Cr atoms play a crucial role in enhancing PMA.

To explore the variation of T_C under biaxial strain, we utilized the Heisenberg model to estimate the T_C of the ML O-CrS₂. In Fig.4(d), the T_C linearly increases, with values ranging from 800 K to 1020 K when the biaxial strain ranges from −5% to 5%. This observation aligns with our expectations, as previous research has demonstrated that Cr compounds typically exhibit higher T_C, such as CrTe₂ [32, 33] and CrX₂ (X = S/Se/Te) [42]. Hence, the imposition of biaxial strain results in distortion of the lattice structure, altering the overlap between atomic orbits. This change significantly impacts the exchange interactions among electrons, leading to an increase in the T_C. In Supplementary Materials Table S3, we observed that µ_Cr is significantly larger than µ_S, and both increase monotonically as the biaxial strain intensifies. Therefore, ML O-CrS₂ exhibits considerable potential for application in spin valves, information transmission, and storage between electrical and spin signals.

3.5 Mechanistic analysis of the magnetic anisotropy

According to second-order perturbation theory [52],

(3)MAE=\beta {\xi }^2{\displaystyle \sum _{\mathrm{o},\mathrm{u}}{\displaystyle \frac{{\left|⟨{\mathrm{\Psi }}_{\mathrm{o}}\left|{\hat{L}}_z\right|{\mathrm{\Psi }}_{\mathrm{u}}⟩\right|}^2-{\left|⟨{\mathrm{\Psi }}_{\mathrm{o}}\left|{\hat{L}}_x\right|{\mathrm{\Psi }}_{\mathrm{u}}⟩\right|}^2}{E_{\mathrm{u}}-E_{\mathrm{o}}}}.}

Here, \beta represents the spin−orbit coupling parameter of Cr atoms, and E_{\mathrm{u}}-E_{\mathrm{o}} represents the energy difference between the unoccupied (u) state and the occupied state (o), which is inversely proportional to the MAE. |⟨{\mathrm{\Psi }}_{\mathrm{o}}|{\hat{L}}_z|{\mathrm{\Psi }}_{\mathrm{u}}⟩{|}^2-|⟨{\mathrm{\Psi }}_{\mathrm{o}}|{\hat{L}}_x|{\mathrm{\Psi }}_{\mathrm{u}}⟩{|}^2 represents the spin−orbit angular momentum matrix elements.

To investigate the influence of biaxial strain on the PMA mechanism in ML O-CrS₂, we selected strain values of −5%, 0%, and 5% as representative. The orbital-resolved MAE is depicted in Fig.5(a)‒(c). For Cr atoms under strain values of −5%, it can be observed that the hybridization between d_{z^2} with d_yz [green bars in Fig.5(a)] and d_xz with d_xy (yellow bars) results in PMA contributions. Conversely, the hybridization between {d_{x^2}}_{-y^2} with d_xy (orange bars) and d_yz with d_xz (red bars) forms a robust In-Plane Magnetic Anisotropy (IMA). The relatively small amplitude of Cr’s anisotropy in ML O-CrS₂ is attributed to the competition between two types of hybridizations. The variations in MAE amplitudes when the Cr atoms interact with S atoms are presented in Fig.5(c). Here, PMA continues to increase (green bars), while IMA decreases (orange bars), leading to a weakening of IMA in the Cr atom. When the strain reaches 0% as shown in Fig.5(b), the PMA (green bars) and IMA (orange bars) increase, resulting in an enhancement of PMA in Cr atoms.

Next, we delve into how the electronic states of the Cr-3d orbitals influence the MAE. Fundamentally, the MAE is mainly influence by the SOC effect. In Fig.5(d)−(f), the occupied and unoccupied electronic states of Cr-3d orbitals around the Femi level are predominantly spin-up channels. The peaks of the d_xy, d_xz, and d_{z^2} states near the Femi level decrease, indicating that the d_yz and d_{x^2-y^2} states at deeper energy levels contribute more significantly to the MAE. Consequently, the amplitude of PMA of ML O-CrS₂ from hybridization decreases at a −5% compressive strain. When a tensile strain of 5% is applied, the density of states d_xy and d_xz becomes more delocalized, leading to a continued decrease in the amplitude of PMA from hybridization. However, the d_yz and d_{x^2-y^2} states of Cr move closer to the Femi level and become more localized [Fig.5(e)], resulting in an enhancement of IMA led by the hybridization. In summary, biaxial strain can modulate electronic states of Cr atoms close to ML O-CrS₂, thereby effectively enhancing its PMA.

4 Conclusion

By using the DFT method, we have computed a novel 2D ML O-CrS₂ material. This material exhibits dynamic, thermodynamic, and mechanical stability. The ML O-CrS₂ is an FM with a T_C of 895 K in the unstrained state. The electronic structure reveals that it is a spin-up channel half-metallicity and displays significant out-of-plane magnetic anisotropy of 0.034 meV/Cr in the FM state. Additionally, we explored the magnetic properties of the ML O-CrS₂ under biaxial strain ranging from −5% to 5%. The system consistently exhibits FM and half-metallicity, with a T_C consistently above room temperature. We have scrutinized the orbital- (atom-)resolved MAE and PDOS and ultimately discovered that the microscopically strong PMA originates from the alteration of Cr atoms’ electronic state. The negative contribution of the MAE is suppressed by compressive strain. Therefore, ML O-CrS₂ emerges as a potential candidate for future nanoelectronic applications and warrants further exploration in subsequent experiments.

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