The static and dynamic magnetic properties of monolayer iron dioxide and iron dichalcogenides

Abstract

The electronic structures, and static and dynamic magnetic properties of monolayer iron dioxide and iron dichalcogenides (FeX₂ (X = O, S, Se, Te)) are investigated using first-principle calculations in conjunction with Monte Carlo (MC) simulation and atomic spin dynamics (ASD) simulation. In this rarely studied family of monolayer binary compounds a variety of possible phases are discovered, including narrow bandgap a semiconductor (FeO₂), half-metal (FeS₂) and metal (FeSe₂ and FeTe₂), and all the ground states are ferromagnetic (FM). Based on the magnetic exchange interactions, the temperature dependence of the average magnetic moment per unit cell and magnetic susceptibility of monolayer FeX₂ are predicted. The Curie temperatures (T_Cs) are estimated and magnon dispersions as a function of temperature are demonstrated, revealing a new family of pristine monolayers with transition temperatures (96–168 K) above the liquid-nitrogen temperature.

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1. Introduction

The discovery of monolayer graphene in 2004 (ref. 1) opened the gate of the two-dimensional (2D) materials' world. Even though great success has been made, the lack of a band gap becomes the Achilles' heel of graphene, its IV analogs including silicene,² germanene³ and stanene,⁴ and other 2D metals, such as monolayer iron,⁵ rhodium⁶ and hafnium.⁷ On the other hand, the graphene-like monolayer hexagonal boron nitride has a band gap of 6.07 eV.⁸ The cleft between the 2D metal and insulator has not been filled until the discovery of the semiconducting monolayer molybdenum disulfide. Inspired by this, more 2D materials with different structures are discovered in recent years: for instance, phosphorene with high hole mobility,⁹ binary compounds such as SnSe with a narrow band gap,¹⁰ and ternary compounds such as FePS₃ (ref. 11) and Nb₃SiTe₆.¹² Following the pace of the MoS₂, the family of transition metal dichalcogenides (TMDs) steps into the sight of scientists. The study of bulk TMDs have last for more than half a century and abundant properties been excavated. The monolayer TMDs inherit their bulk parents' properties: semiconductors such as MoS₂ with bandgap of about 1.8 eV;¹³ semimetals such as WTe₂ with band overlap of 0.21 eV;¹⁴ true metals such as VS₂;¹⁵ superconductors with charge density wave behavior such as NbSe₂.¹⁶ Also, due to the dimensional reducing, some additional characteristics are raised, for instance, the valleytronics and valley-based optoelectronics application of MoS₂.^17–19 Considering the notable achievements already made, magnetic behaviors and application potentials in spintronics of monolayer TMDs are very intriguing. Some theoretical attempts have been done on pristine and doped monolayer TMDs: the monolayer VX₂ (X = S, Se) display FM states with a small magnetic moment of 0.43 and 0.58 μB per unit cell for VS₂ and VSe₂, respectively;²⁰ the monolayer semiconductors MnX₂ (X = O, S, Se) can still hold FM states at room temperature under 5% strains;^21,22 Mn-doped monolayer MoS₂ was predicted to be 2D dilute magnetic semiconductor.^23,24

As one of the most important traditional magnetic elements, iron has riveted little attention in TMDs' research tide. No iron dioxide has been found in nature and minerals of iron dichalcogenides often exist as pyrite and marcasite [Fig. 1(a) and (b)], which neither of them is van der Waals layered compound. Probably based on the unfortunate fact, the nonexistence of the monolayer FeX₂ was taken for granted. But according to Ataca et al., the phonon dispersion of monolayer FeX₂ show imaginary frequency in no region or a tiny region near Γ, which means the monolayer FeX₂ should be structurally stable, at least in a small size.²⁵ Owing to the magnetic nature of Fe element and the miniature trend of information storage and processing on low dimensional structures, the monolayer FeX₂ is a new family of promising materials for the study of magnetism and spintronics. In this work, by using first-principle calculation in conjunction with Monte Carlo (MC) and atomic spin dynamics (ASD) simulations, the electronic and magnetic properties of monolayer FeX₂ will be systematically explored and discussed.

Fig. 1 (a) and (b) Crystal structures of pyrite and marcasite; (c) and (d) top and side views of monolayer FeX₂. The dark blue balls represent the Fe atoms and the yellow balls represent the chalcogenide atoms. The light blue rhombus stands for the primitive unit cell of the two dimensional lattice.

2. Calculation methods

The spin-polarized DFT with the Vienna ab initio simulation package (VASP) code²⁶ is applied to investigate the monolayer FeX₂. The projector augmented wave (PAW) potentials²⁷ are employed to describe the core electrons and the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzernhof (PBE)²⁸ is adopted for exchange and correlation. The kinetic energy cutoff of 520 eV is used and the tolerance for electron convergence is set as 1 × 10⁻⁷ eV while the force is converged within 0.001 eV Å⁻¹ for all calculation. The Brillouin zone (BZ) sampling is done using a 21 × 21 × 1 and 7 × 7 × 1 grid for unit cell and 2 × 2 supercell calculations, respectively. All atomic positions are optimized by minimization of the total energy and atomic forces. The vacuum separation is taken to be more than 25 Å to screen the interactions between adjacent unit cells. The Metropolis MC calculation²⁹ is used to simulate the magnetic order of the system and the ASD simulation is to get the magnon dispersion with the Uppsala Atomistic Spin Dynamics (UppASD) project.^30,31

3. Results and discussion

At first, the geometric structures of monolayer FeX₂ should be relaxed. All FeX₂ own same structure with monolayer MoS₂ in hexagonal structure as shown in Fig. 1(c) and (d). The Fe atom covalently bonded to six chalcogen atoms, forming a sandwich structure with Fe sub-layer placed in between the chalcogens. The lattice constants are 2.678, 3.100, 3.250 and 3.500 Å with the Fe–X bond length of 1.902, 2.251, 2.391 and 2.581 Å for FeO₂, FeS₂, FeSe₂ and FeTe₂, respectively. These results are slightly larger than previous report with the LDA exchange correlation potential.²⁵ All the Fe–X bond lengths are very close to their bulk isomers, such as the pyrite and marcasite. This might imply the Fe–X interaction strengths have no dramatically change with the dimensional reduction.

The spin-polarized electronic band structures of the monolayer FeX₂ are illustrated in Fig. 2. The non-degeneration of the spin-up and spin-down bands suggest that the ground states are magnetic. The metallic behaviors are obvious since several bands on both spin channels cross the Fermi level. According to the symmetry of the lattice, the d-orbitals of Fe are split into three sets: one single degenerated d_z² and two two-hold degenerated d_xz,yz and d_x²−y²,xy orbitals. Detailed analysis reveals that the d-orbitals make remarkable impact on the conductivity of the monolayer FeX₂: at the near-K and near-M region in the BZ, part of the d_z² and d_x²−y²,xy orbitals distribute on the one spin-down band which crosses the Fermi level while part of the d_xz,yz orbitals pervade one the several spin-up ones. On the other hand, the near-Γ region of the conducting band is mainly contributed by the chalcogens. Ascribed to the limitation of the GGA-PBE exchange, the band gap of the FeX₂ might be underestimated. To obtain more accurate description of the electrical properties, the state-of-art hybrid functional HSE06 is utilized for the band structures of the FeX₂. As a result, the FeO₂ opens a gap of about 0.6 eV on spin-up channel at Γ point and larger gap on other channel, and the spin-down part of FeS₂ emerges a gap of nearly 2 eV. Namely, FeO₂ is more likely to be a semiconductor and FeS₂ is a half metal, while FeSe₂ and FeTe₂ still remain metallic.

Fig. 2 Spin-polarized electronic band structures of monolayer FeO₂, FeS₂, FeSe₂ and FeTe₂, respectively, from left to right. The blue solid and red dash lines stand for the spin-up and spin-down channels, respectively. The green dash dot lines stand for the Fermi level.

The magnetic moments of the Fe in monolayer FeX₂ are listed in Table 1. Too little spin charge accumulates near the chalcogen atoms to be important in all four compounds. Thus, we will focus the magnetic behaviors on the Fe atoms. In a 2 × 2 supercell, the energies of two different magnetic configurations are compared: one with the all four Fe atoms' magnetic moments set parallel, refers as FM state, and the other one with two spin-up and two spin-down moments, refers as antiferromagnetic (AFM) state, as depicted in Fig. 3a. Then the nearest-neighbor (NN) exchange interaction between two Fe atoms can be calculated with J = (E_AFM − E_FM)/(8 × 2S²), which the E_AFM and E_FM represent the energies of AFM state and FM state, respectively, and the S stands for the magnetic moment of Fe atom. The exchange interactions in four compounds are also shown in Table 1. In the four FeX₂ compounds, the FM situations are the ground states, as the values of Js are all positive. From the electronic density of states, the double exchange interaction dominates the FM stability in all FeX₂ compounds, and the p–d exchange interaction gradually increases from FeO₂ to FeTe₂ as the chalcogen atom gets larger and the p electrons become more delocalized.

Table 1 Magnetic moments of the Fe, the exchange interactions and the T_Cs with the MC simulation of the monolayer FeX₂

	S (μB)	J (meV)	T C (K)
FeO2	1.83	3.78	96
FeS2	1.42	5.87	144
FeSe2	1.77	6.72	168
FeTe2	1.85	6.64	166

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Fig. 3 (a) Magnetic charge densities of the FM and AFM states of monolayer FeX₂, the red and green clouds respectively represent the spin-up and spin-down charge density; (b) average magnetic moment per unit cell and the magnetic susceptibility of the FeTe₂ from the MC simulation with respect to the temperature. The lines for view guide.

Then, with the assumption of only the NN exchange interactions taken into account, the T_Cs of the monolayer FeX₂ are estimated with the MC simulation, which provides a numerically exact solution,³² also shown in Table 1. Here, we ignore the temperature influence on the exchange interaction. The MC simulations are taken on the 40 × 40 supercell at certain temperature for enough steps. In different temperatures, the magnetic susceptibility can be obtained with, where S_i is the magnetic moment, N is the number of the ensembles and T is the temperature.³³ The peak of magnetic susceptibility indicates a second-order phase transition, namely the ferromagnetic–paramagnetic phase transition, which corresponds to the T_C of the system. Since the exclusive of the exchange interactions of longer distance, which might be important for magnetic order stability,³⁴ could depress the T_C of the system, the results of our MC simulations probably give the lower bound. This indicates that the iron dichalcogenides might be good candidates as two dimensional materials for spintronic and electronic application.

Using the exchange interaction we have obtained, the dynamics properties of magnetic monolayer FeX₂ can be further studied. Based on the Landau–Lifshitz–Gilbert equations, the ASD approach is exploited to get the magnon dispersions. In Fig. 4, the magnon dispersions at different temperatures of monolayer FeTe₂ are drawn. First, at the low temperature of 0.1 K, the Gilbert damping factor α is set to 0.0003, a value small enough to ignore the coupling to the temperature bath.³⁵ As only one layer involved, one acoustic branch of magnon in the dispersion is found, while no optical branch observed in Fig. 4a. A parabolic shape of magnon band near Γ point infers the FM ground state of the monolayer FeTe₂. The magnon energy at the BZ boundary is relatively large with two local maximum: about 100 meV at K point and about 108 meV at M point. The magnon spectra of FeO₂, FeS₂ and FeSe₂ are very similar to the FeTe₂'s, but with maximal magnon energies at M of 60, 94 and 109 meV, respectively.

Fig. 4 Magnon dispersions of the monolayer FeTe₂ with (a) temperature of 0.1 K and damping of 0.0003; (b) temperature of 130 K and damping of 0.03.

In the other situation, α is assumed to be 0.03 because of the temperature raised to 130 K, which is close to the MC estimated T_C of FeTe₂. We can tell from the Fig. 4b that the branch is broadened which means the decrease of magnon's life-time, and as a result the magnon is softened as the energies at the K and M is reduced to about 80 and 85 meV, respectively. The significant changes may be ascribed to the temperature effect. As the temperature rises, the stability of the magnetic order would be become weaker and weaker. The magnon will disappear and the magnon dispersion will totally collapse if the temperature is above T_C.

4. Conclusion

In contrast to the monolayers without a bandgap such as graphene, silicene and monolayer Fe, monolayer FeX₂ may be a semiconductor (FeO₂) with a bandgap, a half metal (FeS₂) or a metal. Also different from monolayers with a bandgap such as phosphorence, SnSe, FePS₃ and Nb₃SiTe₆, monolayer FeX₂ show magnetism, more specifically, various static and dynamic magnetic properties. In fact, monolayer FeX₂ is a rare family of materials that can have a bandgap and show magnetism. The rich behaviors of the FeX₂ systems may be useful for various applications based on the systems.

To summarize, we systematically investigate the electronic structures and magnetic properties of the monolayer iron dioxide and iron dichalcogenides using the first-principle calculation in combination with MC simulation and ASD simulations. The various electronic properties of monolayer FeX₂ family range from the semiconductor and half-metal, to the true metal would hold potentials in future two-dimensional electronics. The FM ground states also make monolayer FeX₂ as the test bed for the spintronics. Although the limitation of the T_Cs is the barrier, things could be better with involving long range magnetic exchange interactions which we ignored in this paper, and the T_Cs could be increased by applying an external strain, such as epitaxial growth on certain crystal surface. The spin dynamic behaviors of monolayer FeX₂ are also explored. The acoustic branch of magnon which manifests the stability of the monolayer FeX₂ is found at the low temperature, and the softening of the magnon at the high temperature reveals that the temperature rise weakens the magnetic order. Undoubtedly, as reduced dimensional magnets, the monolayer iron dioxide and iron dichalcogenides could stand a chance to promote the development of the nanotechnology and make themselves a room in the two dimensional world.

Acknowledgements

This work was supported by the Natural Science Foundation of China (NSFC) through Grant No. 11174212.

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