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. 2022 Nov 16;7(47):43316–43320. doi: 10.1021/acsomega.2c06754

Interlayer Coupling Induced Sharp Increase of the Curie Temperature in a Two-Dimensional MnSn Multilayer

Panjun Feng †, Xiaohui Zhang ‡, Shuo Zhang †, Dapeng Liu †, Miao Gao §, Fengjie Ma ∥, Xun-Wang Yan †,*, Zhi-Yuan Xie ‡,*
PMCID: PMC9713886  PMID: 36467953

Abstract

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The MnSn monolayer synthesized recently is a novel two-dimensional ferromagnetic material with a hexagonal lattice, in which three Mn atoms come together to form a trimer, making it remarkably different from other magnetic two-dimensional materials. Most impressively, there occurs a sharp increase of the Curie temperature from 54 to 225 K when the number of layers increases from 1 to 3. However, no quantitative explanation has been reported in previous studies. Herein, by means of the first-principles calculation method and the Monte Carlo method, we demonstrate that strong interlayer ferromagnetic coupling plays an essential role in enhancing its critical temperature, which acts as a magnetic field to stabilize the ferromagnetism in MnSn multilayers. Our work not only explains the sharp increase of the Curie temperature of the MnSn film in experiments but also reveals that the interlayer coupling is a new routine to achieve high-temperature ferromagnetism in two-dimensional materials.

1. Introduction

The magnetic properties of two-dimensional materials have always been a hot topic in condensed matter physics and materials science fields. Graphene is the first two-dimensional material and possesses excellent mechanical properties, thermal conductivity, and electrical conductivity. However, nonmagnetism prevents graphene from application in the spintronics area. Thus, various means are used to realize the magnetism in graphene-related systems such as doping, chemical modification, defect, and boundary engineering. Unfortunately, such magnetic properties are disordered, nonintrinsic, and difficult to control. In 2016, Je-Geun Park proposed the concept of two-dimensional magnetic van der Waals materials.1 In the same year, two-dimensional antiferromagnetic van der Waals materials NiPS3 and FePS3 were synthesized.2,3 In 2017, two-dimensional ferromagnetic materials Cr2Ge2Te6 and CrI3 were discovered for the first time.4,5 In 2018, room-temperature or near-room-temperature ferromagnetism of transition-metal dihalides VSe2 and MnSe2 was discovered.6,7 Since then, the experimental and theoretical study of two-dimensional materials with intrinsic ferromagnetism has become a booming research field.8−13

For ferromagnetic two-dimensional materials synthesized experimentally, there are three main kinds of structures. The first one is the XY3 structure represented by CrI3, including CrBr3, CrCl3, VI3, VBr3, FePS3, MnPS3, NiPS3, Cr2Ge2Te6, Cr2Si2Te6, and CrCuP2S6.8,14 In these two-dimensional magnets, the core structure is an XY3 octahedron, in which Y and X are at the apex and center of the octahedron, respectively, and each octahedron shares edges with the three surrounding octahedrons. The second typical structure is the XY2 structure represented by VSe2, including VS2, MnSe2, and NiI2.15 Its core structural unit is still an octahedron with a vertex of Y, but each octahedron shares edges with six surrounding octahedrons. The third structure is the XY structure, including FeTe and MnSe,16 in which Y atoms form a tetrahedron and X is in the center of the tetrahedron. Thus, from the perspective of structure, the two-dimensional magnets synthesized in the experiment are mainly composed of octahedrons or tetrahedrons. Recently, the ferromagnetic MnSn monolayer and multilayer,9 with the distinct structural features from the above materials, were fabricated by the molecular beam epitaxy growth method.

The atomic structure of the MnSn layer is from the experimental measurement.9 The MnSn monolayer has a hexagonal lattice with the symmetry of the P6̅2m space group (No.189), and the lattice parameters are a = b = 6.57 Å. Each unit cell is composed of three Mn atoms and three Sn atoms. More interestingly, three Mn atoms are gathered together to form a trimer with an equilateral triangular shape, and Sn atoms make up a distorted kagome lattice. The top view of the MnSn monolayer is shown in Figure 1a. According to the experimental samples reported in ref (9), the MnSn layer is synthesized on the Sn/Si buffer layer. Thus, we include a Sn buffer layer in our calculations to consider the influence of the buffer layer. The side views of the MnSn monolayer, bilayer, and trilayer on a Sn buffer layer are shown in Figure 1b–d, respectively. The Mn trimer and the distorted kagome lattice of Sn atoms are so unique that they do not appear in the reported crystal structures. These endow the MnSn material with particular electronic and magnetic properties. The most impressive feature is the sharp increase of the Curie temperature from 54, 225 to 235 K when the number of layers increases from 1, 3 to 4, and to date, no quantitative explanation has been reported.

Figure 1.

Figure 1

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2 × 2 unit cell of the MnSn monolayer, bilayer, and trilayer. (a) Top view; the unit cell is marked with a solid line rhombus. (b) Side view of the monolayer. (c) Side view of the bilayer. (d) Side view of the trilayer.

In this study, by means of the first-principles calculation and Monte Carlo simulation method, we demonstrate that interlayer ferromagnetic coupling plays a significant role in realizing high-temperature ferromagnetism in the MnSn multilayer. In addition, we consider interlayer magnetic coupling as a magnetic field normal to the MnSn layer and enclose it into a two-dimensional Heisenberg model to evaluate the critical temperature, which provides a practical approach to estimate the critical temperature for multiple layers of magnetic materials.

2. Computational Method

The calculations are carried out based on the plane-wave pseudopotential method enclosed in the VASP package, and the projector augmented-wave (PAW) pseudopotential with the Perdew–Burke–Ernzerhof (PBE) functional17−20 is used. The GGA + U method is employed in our calculations because MnSn is a 3d transition-metal compound. The Hubbard U value of 5.66 eV is determined by the linear response approach developed by Matteo Cococcioni.21 The plane-wave basis cutoff is 600 eV, and the convergence thresholds of total energy and force are 10–5 eV and 0.01 eV/Å, respectively. The vacuum layer is set to 20 Å to avoid interlayer interaction, and a mesh of 24 × 24 × 1 k-points is used to calculate the Brillouin zone integration for the primitive unit cell and a 12 × 12 × 1 k-point mesh is used for 2 × 2 supercells. The temperature of phase transition is evaluated by the Monte Carlo method, and the 100 × 100 lattice is used to represent the magnetic triangle lattice of the MnSn layer.22

3. Results and Discussion

Figure 1 displays the 2 × 2 unit cell of the MnSn structure, including the top view (a) and side views for the monolayer (b), bilayer (c), and trilayer (d). In the MnSn layer, each Mn trimer acts as one lattice site and all sites make up a triangle lattice. We first investigate the magnetism of the MnSn monolayer. To find out the magnetic ground state, we calculate the energies of the MnSn monolayer with three magnetic orderings revealed by Figure 2. The energies are −39.026, −39.009, and −39.016 eV per unit cell for FM, AFM1, and AFM2 orderings, respectively, indicating that the MnSn monolayer on the Sn buffer layer exhibits a ferromagnetic behavior. The band structure of the MnSn monolayer in ferromagnetic ordering is presented in Figure 3a, in which multiple bands cross the Fermi level, indicating its metallic behavior. Figure 3b exhibits the projected density of states on Mn 3d orbitals and the Sn atom. The electronic states near the Fermi level are dominated by Sn atoms, while the spin polarization of Mn electrons makes a decisive contribution to the magnetization of the MnSn layer. As shown in Figure 3b, the spin-up states of five d orbitals are almost fully occupied and its spin-down states are empty, which results in the large local moment 4.7 μB.

Figure 2.

Figure 2

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(a) Top view of the MnSn monolayer. J1 and J2 are the nearest and next-nearest neighboring magnetic interactions in the triangle lattice. (b) FM ordering, (c) AFM1 ordering, and (d) AFM2 ordering. The solid line rhombus represents the magnetic unit cell. To exhibit the magnetic orderings more clearly, the atomic structure is displayed with the wire frame.

Figure 3.

Figure 3

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(a) Spin-up and spin-down band structures of the MnSn monolayer. (b) Projected density of states on Mn 3d orbitals and Sn atoms.

To judge whether the ferromagnetic coupling of two adjacent MnSn layers is favorable or not, we calculate the energies of the MnSn bilayer and trilayer with different magnetic orders to estimate the interlayer magnetic interactions. For the MnSn bilayer, we chose a 1 × 2 × 2 supercell and display four magnetic orders from its side view, which are shown in Figure 4a–d, corresponding to FM′, AFM1′, AFM2′, and AFM3′, respectively. For AFM1′ and AFM2′ orderings with the antiparallel moments in one layer, the energy of AFM2′ is lower than that of AFM1′, indicating that ferromagnetic interlayer coupling is preferred in the case of intralayer antiferromagnetism. For FM′ and AFM3′ orderings with parallel moments in one layer, the energy of FM′ is lower than that of AFM3′, indicating that ferromagnetic interlayer coupling is also preferred in the case of intralayer ferromagnetism. Thus, for the MnSn bilayer, the interlayer coupling is ferromagnetic. What about three layers? We build a 1 × 1 × 3 supercell composed of three MnSn layers to exhibit different interlayer magnetic couplings, and the related FM″, AFM1″, and AFM2″ are presented in Figure 4e–g, respectively. From AFM1″, AFM2″ to FM″, the energy goes down because of more ferromagnetic couplings. Therefore, this further demonstrates that the interlayer magnetic couplings are ferromagnetic for MnSn multilayers. In brief, the interlayer ferromagnetic coupling is confirmed by the calculations of the MnSn bilayer and the MnSn trilayer, which is in good agreement with the results in the experiments.9

Figure 4.

Figure 4

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Side views of the 2 × 1 × 2 supercell and the 1 × 1 × 3 supercell. Red and green arrows indicate the directions of the magnetic moments on the Mn trimer. Four magnetic orders for the MnSn bilayer and three magnetic orders for the MnSn trilayer are shown in (a) FM′, (b) AFM1′, (c) AFM2′, (d) AFM3′, (e) FM″, (f) AFM1″, and (g) AFM2″, respectively. J⊥, J⊥1, and J⊥ are the interlayer magnetic couplings between two neighboring layers.

Curie temperature is a vital parameter for ferromagnetic materials. At present, Curie temperature lower than room temperature is the main obstacle limiting the application of two-dimensional ferromagnetic materials in microscopic electronic devices. In our previous work,11 we suggested that the dimerization of magnetic ions is an effective approach to realize high-temperature ferromagnetism. In the MnSn layer, three Mn ions form a trimer, and this leads to a large moment on one lattice site. Compared with a single Mn ion, the Mn trimer can obviously increase the magnetic interaction energy between two lattice sites. The critical temperature of magnetic phase transition is usually evaluated through solving the Heisenberg model with the Monte Carlo method.23 For the two-dimensional Heisenberg model, the Hamiltonian is defined as

3. 1

in which the symbols j and j′ represent the nearest and next-nearest neighboring sites of the i site in the triangle lattice, and α is the coordinate component x, y, or z. A is the single-site magnetic anisotropic energy. We perform the noncollinear magnetism calculations with the spin along the (1 0 0), (0 1 0), and (0 0 1) axis for different magnetic orderings, and the magnetic exchange interactions can be derived from these energies. There is no significant difference in the exchange interactions along different directions. For the MnSn monolayer, the exchange couplings are J1x = J1y = J1z = −5.46 meV/S2 and J2x = J2y = J2z = 1.34 meV/S2, and the single-site magnetic anisotropic energy is −0.636 meV/S2 with the easy axis normal to the MnSn layer. The Curie temperature is 43 K, as reflected by the variation of the magnetic moment (M) and magnetic susceptibility (Inline graphic) with temperature in Figure 5a.

Figure 5.

Figure 5

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Average magnetic moment M and magnetic susceptibility χ as functions of temperature from the solution of the Heisenberg model on the triangle lattice for the MnSn monolayer (a) and bilayer (b).

As for the MnSn bilayer and multilayer, how to evaluate its Curie temperature on the basis of the results from electronic structure simulations is an interesting question. We chose one MnSn layer as our research object, in which the exchange couplings in the layer are computed, similar to the calculations of J1α and J2α in Expression 1. For this selected layer, the ferromagnetic interactions from its adjacent layer are considered effective magnetic fields normal to the MnSn layer, which can be derived from the energies of different magnetic orders. For the bilayer or multilayer system, the following Hamiltonian is used.

3. 2

where the last term is the interlayer coupling. Apart from the last term, the Hamiltonian in Expression 2 is the same as in Expression 1. For the MnSn bilayer, the interlayer coupling J⊥ is -84.16 meV/S2. Compared to the exchange couplings of J1x = J1y = J1z = −5.46 meV/S2, the interlayer coupling J⊥ of −84.16 meV/S2 is a large value. It acts as a strong magnetic field on the MnSn layer, which forces the moments in a certain direction. The last term ∑iJ⊥(S⊥zSiz) is regarded as ∑iHeff · Siz. Thus, the effective magnetic field is Heff = J⊥S⊥z. By solving the Hamiltonian of the Heisenberg model in Expression 2 with the Monte Carlo method, the curve of magnetic phase transitions for the MnSn bilayer is obtained and is shown in Figure 5b. The Curie temperature of the MnSn bilayer is 142 K, much higher than the value of 43 K for the monolayer, which is because strong interlayer ferromagnetic interaction stabilizes its ferromagnetism. For the MnSn trilayer, the interlayer couplings J⊥1 and J⊥ are −102.6 meV/S2 and −4.7 meV/S2, and the sum of J⊥1 and J⊥ is J⊥ = −107.3 meV/S2, higher than −84.16 meV/S2 in the case of the bilayer. From the bilayer to the trilayer, the increase of interlayer coupling J⊥ is due to the coupling from the third layer, which results in stronger ferromagnetism. From another point of view, the interlayer magnetic coupling is a new knob to adjust the ferromagnetism in two-dimensional materials, which provides another approach to enhance the Curie temperature of two-dimensional materials.

4. Conclusions

In summary, we systematically investigate the magnetic properties of the MnSn monolayer and the MnSn multilayers by means of the first-principles calculation and the Monte Carlo method. From the MnSn monolayer to the bilayer, the calculated Curie temperature varies from 43 to 142 K owing to the inclusion of strong ferromagnetic interactions between layers. Thus, we demonstrate that strong interlayer ferromagnetic coupling plays an essential role in enhancing its critical temperature, which acts as a magnetic field to stabilize the ferromagnetism in the MnSn multilayers. Our work not only explains the sharp increase of Curie temperature of the MnSn film from 54 to 225 K in experiments but also reveals that enhancing the interlayer coupling is a new routine to realize high-temperature ferromagnetism in two-dimensional materials.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12274255, 11974207, 11974194, 11774420, and 12074040), the National R&D Program of China (Grant No. 2017YFA0302900), and the Major Basic Program of Natural Science Foundation of Shandong Province (Grant No. ZR2021ZD01).

Author Contributions

⊥ P.F. and X.Z. contributed equally to this work.

The authors declare no competing financial interest.

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