Strain-selected magnetic ordering in 1T′α-CrXY (X, Y = S, Se, Te) monolayers

Abstract

Two-dimensional magnetic materials have garnered considerable attention in recent years owing to their interesting phenomena and potential applications in spintronic devices. In this work, we report that the 1T′α-CrTeSe monolayer, where tellurium atoms are intermittently arranged line by line within the layer, exhibits strain-tuned magnetic properties. We find that the anisotropic nearest-neighbor exchange interactions in this monolayer result in a chiral helimagnetic state. The in-plane biaxial tensile strain suppresses the antiferromagnetic direct exchange interaction and leads to a transition from helimagnetism to ferromagnetism. Its transition temperature is estimated to be about 66 K and can also be tuned to near room temperature by applying strain. Furthermore, we also demonstrate the structural stabilities and strain-tuned magnetic ordering in the CrSeS monolayer. Our study provides not only an ideal platform to explore exotic compounds but also an important opportunity to design controllable magnetic materials in low-dimensional systems.

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I. Introduction

The discoveries of ferromagnetism in the CrI₃ monolayer¹ and the Cr₂Ge₂Te₆ bilayer² have spurred a surge of interest in the theoretical design and experimental preparation of two-dimensional (2D) intrinsic magnetic materials. These materials provide an ideal platform to reveal long-range magnetic ordering in 2D materials and to develop the corresponding applications in next-generation spintronic devices. From the perspective of theoretical design, a large number of 2D magnetic materials have been predicted to be stable and potentially achievable, such as transition-metal dihalides,^3–6 trihalides,^7,8 and dichalcogenides.^9–13 Transition-metal dihalide monolayers have garnered considerable attention in terms of their excellent stabilities and exfoliation,³ such as chromium dichalcogenides.^9,10 Based on these materials, more and more novel structural and physical properties have been revealed by elemental substitution,^14–19 strain engineering^20,21 and so forth.

Element-substituted 2D materials provide an important opportunity to explore interesting polarization^22,23 and intrinsic ferromagnetism with high transition temperature.^19,24 However, there are still some issues that require further investigation. For example, many element-substituted 2D materials with a Janus pattern were studied, while the corresponding experimental characterization efforts are lacking.²⁵ This indicates that some patterns with lower or degenerate energies have not been revealed yet. Furthermore, the phenomenon of non-collinear magnetic ordering arising from geometric frustration is a topic of significant interest.^26–28 Therefore, it is essential to investigate more element-substituted patterns with positive stabilities that may guide experimental synthesis. Moreover, it is necessary and interesting to establish the correlation between their structural and magnetic coupling so as to enhance their performance for promising device applications.

In this paper, based on first-principles calculations and Monte Carlo (MC) simulations, we confirm the structural stabilities and reveal the switchable magnetic coupling of the 1T′α-CrTeSe monolayer. Due to the competition between different intrinsic exchange interactions, the nearest-neighbor ferromagnetic (FM) coupling can be enhanced by applying in-plane biaxial tensile strain, resulting in a chiral helimagnetic (CHM)-to-FM transition. Its transition temperature can also be modified to approach room temperature under biaxial strain. Also, these significant modifications can also be achieved in the CrSeS monolayer. These results offer a promising platform for understanding the tunable magnetic coupling and present an opportunity for manipulating their magnetic ordering in low-dimensional materials.

II. Methods

All first-principles calculations are carried out within the framework of density functional theory as implemented in the Vienna ab initio simulation package.^29,30 The exchange–correlation potential is described by using the Perdew–Burke–Ernzerhof functional within the generalized gradient approximation,³¹ in which the projector-augmented-wave method is adopted, with an energy cutoff of 600 eV.^32,33 The effective U values are tested and the value of U = 3.0 eV is adopted to consider the strong correlation effect among Cr-3d electrons, which is reasonable to describe the critical magnetic properties,^14,34 as shown in Fig. S1 of the SI. The c-axis-fixed structural optimization is performed until the energy convergence criterion of 1 × 10⁻⁶ eV is met and the force on each atom is less than 1 × 10⁻³ eV Å⁻¹. The electronic configurations 3d⁵4s¹, 2s²2p⁴, 3s²3p⁴, 4s²4p⁴, and 5s²5p⁴ are considered as the valence states of chromium, oxygen, sulfur, selenium, and tellurium atoms, respectively. A vacuum region larger than 15 Å is added so that the interactions between the monolayer and its periodic image can be neglected. A k-point mesh of 5 × 9 × 1 is used for structural relaxation, and a 7 × 11 × 1 mesh is used for the calculation of magnetocrystalline anisotropy energy (MAE) and electronic structure. A 2 × 2 × 1 supercell with a 3 × 5 × 1 k-point mesh is used to calculate the exchange coupling parameters. The phonon spectra are calculated by using the finite displacement method as implemented in the PHONOPY code.^35,36 A 3 × 3 × 1 supercell and Γ-point are considered in the ab initio molecular dynamics (AIMD) simulations at 300 K up to 8 ps, in which the NVT ensemble with a Nosé–Hoover thermostat³⁷ is used. A 40 × 50 × 1 supercell including 4000 magnetic atoms is used in MC simulations.

III. Results and discussion

III.A. Structural stabilities

Half-element-substituted trigonal transition metal dihalide monolayers have three common crystal structures, namely the 1T′α, 1T′β,^25,38 and Janus patterns.^15,18 We have recently reported that the 1T′α pattern is the most energetically favored state of the half-halogen-substituted trigonal CrS₂ monolayer, in which both additional electrons and external strains are induced.³⁹ Here, we consider the effects of chalcogen substitution on the structural and magnetic properties in this pattern. Structurally, the selenium and tellurium atoms are alternately arranged in the plane line by line and form the 1T′α-CrTeSe monolayer with the P2₁/m space group, as shown in Fig. 1(a). The calculated lattice parameters, nearest Cr–Cr distances, and Wyckoff sites of the CrTeSe monolayer are listed in Table S1 of the SI. The optimized lattice parameters are a = 6.642 and b = 3.352 Å, and the nearest Cr–Cr distances are d₁ = 3.979, d₂ = 3.352, and d₃ = 3.470 Å, respectively. The Cr–Se–Cr path is 14.7% shorter than the Cr–Te–Cr path as a result of the smaller ionic radius of selenium, indicating a non-negligible local deformation induced by elemental substitution.

Fig. 1 (a) Crystal structure of the 1T′α-CrTeSe monolayer shown in top and side views. Blue, orange, and grey spheres represent the chromium, selenium, and tellurium atoms, respectively. Red, purple, and green arrows indicate the Cr–Se/Se–Cr, Cr–Se/Te–Cr, and Cr–Te/Te–Cr nearest-neighbor exchange paths, respectively, and the atomic distances are labelled as d₁ to d₃. Black dashed lines represent the unit cell. (b) Schematic models of four different magnetic configurations for calculating exchange parameters and further examining the magnetic ground state. The blue and yellow arrows represent the directions of magnetic moments on chromium atoms.

To confirm the dynamical and thermal stabilities, we calculate the phonon spectrum and carry out the AIMD simulations of the 1T′α-CrTeSe monolayer at 300 K up to 8 ps. The lattice dynamics calculations have demonstrated its dynamical stability, as evidenced by the absence of imaginary frequencies throughout the Brillouin zone (see Fig. S2 in the SI). Moreover, the CrTeSe monolayer maintains its structure up to 8 ps during the AIMD simulation, which confirms its thermal stability (see Fig. S3 in the SI). In order to examine its potential for experimental synthesis, we also calculate the formation energy of the monolayer by using the following formula:

E_f = (E_CrTeSe − E_Cr − E_Te − E_Se)/3 (1)

where E_CrTeSe, E_Cr, E_Te, and E_Se are the energies of the monolayer and the stable phases of chromium, tellurium and selenium, respectively. Here, E_f with negative values suggests the synthetic feasibility in terms of energy. The E_f value of the CrTeSe monolayer is estimated to be about −0.407 eV per atom, indicating its energetic stability and the possibilities of experimental synthesis. We also consider the typical Janus pattern to compare their formation energies. The formation energy of the 1T′α pattern is almost degenerate to that of the Janus pattern with an energy difference of 0.061 meV per atom. In addition, it is desirable to compare the energy of the 1T′α pattern with that of disordered substitutions. As shown in Fig. S4 of the SI, the 1T′α pattern is more stable than disordered substitutions with energy differences above 144 meV per atom.

III.B. Magnetic ground state

To calculate the magnetic exchange parameters and reveal the magnetic ground states of the 1T′α-CrTeSe monolayer, we consider a 2 × 2 × 1 supercell with four magnetic configurations, namely, FM, striped antiferromagnetic (sAFM), doubly striped antiferromagnetic (dAFM), and zigzag antiferromagnetic (zAFM), as shown in Fig. 1(b), in which the blue and yellow arrows represent the directions of magnetic moments on chromium atoms. We determine the exchange parameters based on the Heisenberg model as follows:Here, J₁, J₂, and J₃ represent the nearest-neighbor exchange paths in different directions, as illustrated in Fig. 1(a), and S_i and S_j are the spin vectors. Since the majority of Dzyaloshinskii–Moriya interactions are forbidden by the symmetry of this monolayer, it is excluded from our magnetic Hamiltonian. The positive (negative) values indicate FM (AFM) couplings. According to eqn (2), the energies of the four magnetic configurations can be written as

The energies of these four magnetic configurations are listed in Table S2 of the SI, and we derive the parameters as J₁ = 4.065, J₂ = −8.95, and J₃ = 7.035 meV. Direct exchange interactions between cations are highly distance-dependent, as reflected in the Bethe–Slater interaction curve.^40–42 In this monolayer, atomic distances of 3.35–3.98 Å indicate AFM direct exchange interactions along these paths, consistent with previous reports.¹³ They act in competition with the ferromagnetic interactions such as superexchange and Ruderman–Kittel–Kasuya–Yosida (RKKY) interactions. For the Cr–Se/Te–Cr path, the AFM direct exchange interactions are the dominant interactions and cause a negative J₂ value.

We consider the spin–orbit coupling and calculate the MAEs of the 1T′α-CrTeSe monolayer with the spin moments parallelly oriented along different directions. The magnetic-easy axis of the CrTeSe monolayer lies in the xz plane with a rotational angle of 60° with respect to the z axis, as shown in Fig. S5 of the SI. This can be attributed to the competition between the in-plane contribution from selenium and the out-of-plane contribution from tellurium (see Fig. S6 in the SI), which is in line with the previous studies.^43,44 The MAEs along the x′, y′, and z′ directions are listed in Table S3 of the SI, in which x′ is the magnetic-easy axis, y′ and z′ are rotated y and z axes with the same rotation as the x to x′ axis, respectively. Based on the calculated magnetic exchange parameters and MAEs, we then solve the following Heisenberg model by means of MC simulations to reveal the magnetic ground state of the 1T′α-CrTeSe monolayer:

where A_y and A_z are the MAEs along the y and z axes with respect to the x axis. S_i and S_j are the spin vectors. Here we use the MAEs along the x, y, and z directions to simulate the magnetic ground state. Notably, this yields the same result as that obtained from the MAEs along the x′, y′, and z′ directions.

Fig. 2(a) depicts the magnetic ground state of the 1T′α-CrTeSe monolayer. When both J₁ and J₃ are positive, the negative J₂ gives rise to a non-collinear magnetic ground state. For each chromium line along the y direction, magnetic moments rotate in the xz-plane with a step size of π/2. Due to the strong AFM coupling of Cr–Se/Te–Cr paths, magnetic moments invert their rotation direction and continue rotating with a step size of −π/2, leading to a CHM state. In the x direction, two chromium lines with a phase difference of about π/6 form a translation period.

Fig. 2 MC simulation of the ground state of the 1T′α-CrTeSe monolayer. The black lines show the chromium atomic lines along the y direction. Colored ellipses, arrows, and curves represent the xz plane, magnetic moment, and projection of magnetic moment on the z-axis, respectively.

The spin-resolved band structure of the 1T′α-CrTeSe monolayer reveals its metallicity, as evidenced by several bands crossing the Fermi level around the Γ and Y points (see Fig. S7 in the SI). Given its metallic properties, we evaluate the potential impact of long-range exchange interactions on the magnetic ground state, where these interactions are primarily mediated by the RKKY mechanism. We incorporate an additional parameter, J_{lr_eff}, into our spin Hamiltonian to describe the net effect of long-range exchange interactions, as illustrated in Fig. S8 of the SI. Here we scan J_{lr_eff} across a wide range from −1.0 to 1.0 meV. Our analysis demonstrates that the stability of the ground state is maintained for effective long-range coupling strengths up to |J_{lr_eff}| ≤ 0.4 meV.

III.C. Biaxial strain

The nearest exchange interactions in the 1T′α-CrTeSe monolayer depend mainly on the atomic distances, which is sensitive to biaxial strain. Thus, it is interesting to explore the effects of in-plane biaxial tensile strain on the modification of the magnetic coupling in the CrTeSe monolayer. Biaxial strain is defined as ε = (l − l₀)/l₀ × 100%, in which l (l₀) is the lattice parameters under an applied (free from) strain. As shown in Fig. 3, within the considered tensile strain range, the energies of the FM and sAFM states are always lowest and highest, respectively. The energy of the dAFM state is sandwiched between those of the zAFM and sAFM states at strains smaller than 3% and becomes lower than that of the zAFM state at strains larger than 3%. Meanwhile, J₃ slightly increases up to a strain of 2% and then decreases slightly, while J₁ and J₂ monotonically increase over the entire strain range considered, in which J₂ undergoes a transition from negative to positive at a strain of 3%. The strain induced increase of atomic distance suppresses the AFM direct exchange interaction more than the FM interaction and triggers a monotonical increase of J₂. To confirm the magnetic ground state of the CrTeSe monolayer under strain, we calculate its MAEs and the values are shown in Table S4 of the SI. Further MC simulations demonstrate that, at strains below 0.8%, the system resides in a transitional magnetic state characterized by a non-collinear configuration, where all the moments are not fully parallel but maintain a net magnetization along the z-direction. At strains exceeding 0.8%, the moments become fully parallel, indicating the FM state. In particular, although J₂ remains negative when the strain is larger than 0.8% and smaller than 3%, its absolute values are clearly decreased and thus lead to FM states.

Fig. 3 (a) The calculated energies of the 1T′α-CrTeSe monolayer with different magnetic configurations as functions of in-plane tensile biaxial strain (ε). Grey, magenta, blue, and orange represent the FM, sAFM, dAFM, and zAFM states, respectively. (b) The exchange coupling strengths of the CrTeSe monolayer as functions of in-plane tensile biaxial strain. The shaded, striped, and white areas represent the magnetic ground states of the CHM, transitional, and FM configurations, respectively.

Based on the calculated magnetic phase transition and magnetic exchange parameters and MAEs, we carry out MC simulations to estimate the magnetic transition temperature of the 1T′α-CrTeSe monolayer under strain. Here, we consider the strain-free case and typical strains of 0.2%, 0.8% and 6% as examples. As shown in Fig. 4, owing to the tensile strain enhanced magnetic coupling strength, the magnetic transition temperatures for the strain-free case and strains of 0.2%, 0.8%, and 6% are estimated to be about 66, 63, 94, and 279 K, respectively. To demonstrate that the supercells in MC simulations are enough for correct calculations of the transition temperatures, we test several supercells as shown in Fig. S9 of the SI. It is clear that the 40 × 50 supercell is enough for these simulations. We also use the MAEs along the x, y, and z directions that maintain the transition temperature, as shown in Fig. S9 of the SI. The T_C of the CrTeSe monolayer is higher than that of the CrI₃ monolayer (45 K)¹ and the Cr₂Ge₂Te₆ bilayer (28 K),² which indicates its possibilities for further applications.

Fig. 4 MC simulations of the (a) magnetic moments and (b) specific heats of the 1T′α-CrTeSe monolayer with and without in-plane tensile biaxial strain. The lines and symbols with different colors represent different strain environments.

The small strains in the 1T′α-CrTeSe monolayer notably reduce the energy consumption and mechanical complexity required for potential strain-controlled devices. In experiments, strain can be applied to monolayers using various methods, such as lattice mismatch between substrates and epitaxial thin films.⁴⁵ Based on this method, flexible graphene-based transistors with a large range of strains have been successfully achieved,^46–49 indicating the potential spintronic applications of the CrTeSe monolayer.

III.D. CrSO and CrSeS monolayers

To systematically explore the structural and magnetic properties of chalcogen-substitution 1T′α patterns, we further consider the structural stabilities of CrSO and CrSeS monolayers. We calculate the phonon spectra and find that the CrSO monolayer exhibits obvious imaginary frequency, while the CrSeS monolayer is dynamically stable (see Fig. S10 in the SI). We also carry out the AIMD simulation and confirm the thermal stability of the CrSeS monolayer (see Fig. S11 in the SI). The formation energy of −0.496 eV per atom indicates its possibilities of experimental synthesis. By comparing the energies of the four magnetic configurations of the CrSeS monolayer, we derive its exchange parameters as J₁ = 5.94, J₂ = −4.14, and J₃ = 4.00 meV. Meanwhile, we calculate the MAEs and the obtained values of E_y and E_z are 1348 and 161 µeV, respectively. MC simulations suggest that these comparable exchange parameters lead to a 120°-AFM ground state in the CrSeS monolayer.

In order to understand the effect of biaxial strain on the magnetic properties of the 1T′α-CrSeS monolayer, we calculate its exchange parameters and MAEs under a biaxial strain (ε) of 6%. Under biaxial strain, its exchange parameters are increased to J₁ = 10.23, J₂ = 6.28, and J₃ = 10.30 meV, and the values of E_y and E_z are changed to 2026 and 588 µeV, respectively. All positive exchange parameters indicate the FM ground state of the tensile CrSeS monolayer. By solving the Heisenberg model, we calculate the transition temperatures of strain-free and tensile CrSeS monolayers and the values are shown in Fig. S12 of the SI. The transition temperature of the strain-free monolayer is estimated to be about 45 K, while that of the tensile monolayer increases above room temperature and reached 315 K. The tensile strain induced change in the magnetic properties of this monolayer confirms the Cr–Cr distance-dominated modification of magnetic coupling in the CrTeSe monolayer and provides another switchable 2D magnetic material.

IV. Conclusions

In conclusion, we performed systematic first-principles investigations on the half-element-substituted CrTeSe monolayer with a line-by-line 1T′α pattern. In this case, the 1T′α and Janus structures have almost degenerate energies and both their formation energies are negative, in which the 1T′α structure is also dynamically and thermally stable. Under in-plane tensile biaxial strain, the antiferromagnetic nearest-neighbor coupling can be tuned to ferromagnetic coupling, resulting in a transition from chiral helimagnetism to ferromagnetism. The strain-tuned magnetic properties of the CrTeSe monolayer are attributed to the competition between different intrinsic exchange interactions, in which the competition depends mainly on the nearest Cr–Cr distances. By applying strain, the transition temperature of this monolayer can be tuned to approach room temperature. Furthermore, based on the same mechanism, the transition from the AFM to the FM coupling can be achieved in the CrSeS monolayer. Our calculations unveiled that the intermittently element-substituted 1T′α monolayers have interesting nearest-neighbor exchange interactions with remarkable controllability, making them promising candidates for use in next-generation low-dimensional spintronic applications.

Conflicts of interest

There are no conflicts to declare.

Data availability

All data generated or analyzed during this study are included in this article and its supplementary information (SI). See supplementary information for the tests of computational parameters, analyses of MAEs and structural stabilities and long-range exchange interactions, calculated energies of different configurations and band structure for the 1T′α-CrTeSe monolayer, and the discussions of structural stabilities and transition temperatures for the CrSO and CrSeS monolayers. See DOI: https://doi.org/10.1039/d5cp03532e.

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