Magnetic anisotropy and phononic properties of two-dimensional ferromagnetic Fe3GeS2 monolayer

Summary In 2023, Fe3GeS2 monolayer with Curie temperature of 630 K is predicted, which is promising to be used in next-generation spintronic devices. However, its magnetic anisotropy and phononic properties are still unclear. In this paper, we implemented the first-principles calculations on Fe3GeS2 monolayer, and found its ferromagnetic ground state with robustness to the −1.5%–1.3% biaxial strain. Meanwhile, the out-of-plane magnetic anisotropy dominated by dipolar interaction is found in Fe3GeS2 monolayer. Finally, we studied the phononic properties to identify the dynamical stability of Fe3GeS2 monolayer and highlight the contribution from the anharmonic interaction of optical phonons to the thermal expansion coefficient. We also find two single-phonon modes can be used to design quantum mechanical resonators with a wide cool-temperature range. These results can provide a comprehensive understanding of the magnetism and phonon properties of two-dimensional (2D) Fe3GeS2, beneficial for the application of 2D Fe3GeS2 in spintronics.


INTRODUCTION
The successful exfoliation of two-dimensional (2D) graphene has opened up substantial research topics on the fascinating properties and potential applications of 2D materials.In the initial stage of studying 2D materials, researchers focused on black phosphorene, transition metal dichalcogenides (TMDs), hexagonal boron nitride, etc., all of which are nonmagnetic (NM).In 2017, the 2D ferromagnetic (FM) CrGeTe 3 and CrI 3 were mechanically exfoliated from the bulk crystals by Gong et al. and Huang et al., 1,2 which shed light on the applications of 2D materials in spintronics.Subsequently, a large of 2D magnets are predicted and prepared, including CrCl 3 , 3 Fe 3 GeTe 2 , 4 CrTe 2 , 5 MnBi 2 Te 4 6 with nontrivial topology, Weyl half-semimetal PtCl 3 , 7 and antiferromagnetic (AFM) MnPS 3 8 and MXenes. 9,10In recent years, these 2D magnets attracted extensive research attention in the field of materials science and condensed matter physics, due to their interesting properties, such as large spin Seebeck coefficient, 11 controllable magnetoresistance, 12 and quantized anomalous Hall effect. 13These intriguing properties render 2D magnets promising candidates in fabricating spintronic devices.Currently, spintronics devices have been widely applied in the fields of signal transfer, 14 data storage, 15 biomedicine, 16 and energy conversion. 17owever, a fatal flaw of 2D magnets is their weak magnetic stability and low magnetic phase-transition temperature (Curie temperature for FM; Ne ´el temperature for AFM), which hinders strongly their application in spintronics.For instance, the Curie/Ne ´el temperatures are $60 K for CrGeTe 3 1 , $45 K for CrI 3 2 , and 89 K for FePS 3 , 18 which is much lower than the room temperature.Therefore, in recent years, researchers have devoted themselves to exploring methods to improve the magnetic phase-transition temperature of 2D magnetic materials or design novel 2D magnetic materials with high Curie/Ne ´el temperature.At present, it is recognized that the out-of-plane magnetic anisotropy is key for 2D magnets to break the Mermin-Wagner theorem and to resist thermal fluctuation.The relationship between Curie/Ne ´el temperature and the out-of-plane magnetic anisotropy can be written as: (Equation 1) where J lf , K lf , and A zz are the isotropic magnetic exchange coupling coefficient, out-of-plane magnetic anisotropic parameter, and out-ofplane single-point anisotropic parameter, respectively.K B is the Boltzmann constant, while S 0 is the altitude of spin vector on each magnetic lattice.The details of Equation 1 can be referred to a study by Wang et al. 19 Hence, many strategies have been employed to enhance the out-of-plane magnetic anisotropy and elevate the Curie/Ne ´el temperature of 2D magnets, such as strain engineering, 20,21 charge doping, 22,23 surface functionalization, 24,25 atomic doping, 26,27 intercalation, [28][29][30] and external manipulation. 31,324][35] For example, Wen et al. 36 prepared 2D FM CuCr 2 Te 4 flakes with thickness-dependent Curie temperature (260 K-320 K) by the heteroepitaxial growth.Based on Fe 3 GeTe 2 monolayer with the Curie temperature of 150 K, 37 Yang et al. 38 predicted 60 types of easy exfoliable and highly stable magnetic A 3 BX 2 monolayers by machine

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learning and high-throughput computation.Among these 60 types of A 3 BX 2 monolayer, Fe 3 GeS 2 monolayer with excellent dynamical and thermal stability owns the highest Curie temperature of 630 K, much higher than room temperature 300 K. Therefore, if it can be successfully prepared in time, it will bring revolutionary progress to spintronic devices based on 2D magnets, making 2D spintronic devices have great application potential in quantum communication, data storage, biomedicine, and other fields.However, in their study, the Curie temperature was estimated using Monte Carlo simulation based on the magnetic exchange coupling, without exploring the magnetic anisotropy in Fe 3 GeS 2 monolayer.
In this paper, we would explore the magnetic anisotropy in Fe 3 GeS 2 monolayer by the first-principles calculations.Besides, in the spintronic devices fabricated by the 2D magnets, the performance and reliability of the devices heavily depend on the thermal conduct and thermal expansion of 2D magnets.Phononic properties of 2D magnets are the basis for analyzing and exploring the thermal conduct and thermal expansion.Therefore, we would also delve into the phononic properties and thermal expansion of Fe 3 GeS 2 monolayer, laying a theoretical foundation for its application in 2D spintronic devices.We found the out-of-plane magnetic anisotropy dominated by dipolar interaction in Fe 3 GeS 2 monolayer, robust to the À1.5%-1.3%biaxial strain.Meantime, we also find two single-phonon modes with frequencies of 9.08 THz and 11.12 THz, which can be used to design quantum mechanical resonators with wide cool-temperature range.Besides, we also calculate the Gru ¨neisen constants of all phonons and thermal expansion coefficient (TEC) of FM Fe 3 GeS 2 monolayer, highlighting the contribution from the anharmonic interaction of optical phonons to the TEC.

Computational details
In our simulations, the calculations of structural relaxation, electronic density of states, and phonon dispersion were obtained by Device Studio program, 39 which provides several functions for crystal visualization, modeling and simulation.In this paper, all these calculations were implemented by DS-PAW software integrated in Device Studio program. 40In DS-PAW software, the projected augmented wave (PAW) method was employed to describe the coupling between atomic nuclei and extra-nuclear electrons, while we selected the Perdew-Burke-Ernzerhof (PBE) method of general gradient approximation (GGA) as exchange-correlation functional. 41,42To suppress the non-physical interaction between adjacent layers, a 15 A ˚vacuum space was imposed along the out-of-plane direction.The convergence limits of energy and force were set as 10 À7 eV and 0.001 eV/A ˚in the relaxation of geometrical structure with cutoff energy of 500 eV and 53531 Monkhorst-Pack (MP) grid.The phonon dispersions were calculated using an MP grid of 93931 for a 23231 supercell, based on the density function perturbation theory (DFPT). 43Besides, it has been reported in previous studies [44][45][46] that including the Hubbard ''U'' into the PBE functional will overestimate the lattice parameters and magnetic moment of based-Fe 3 GeTe 2 materials seriously.Meantime, we calculated the band structures of Fe 3 GeS 2 monolayer with different ''U'' values, as shown in Figure 1.It can be observed that the metallic state of Fe 3 GeS 2 monolayer is robust to the value of Hubbard ''U''.Therefore, the ''U'' parameter was not included in our calculations.

RESULTS AND DISCUSSION
In this paper, the model of crystal Fe 3 GeS 2 is established by using the bulk Fe 3 GeTe 2 as a template.The bulk Fe 3 GeS 2 is hexagonal, with the space group of P6 3 /mmc (No. 194).The Fe 3 GeS 2 monolayer is exfoliated from the bulk Fe 3 GeS 2 , and its space group is P-6m2 (No. 187).The top and side views of Fe 3 GeS 2 monolayer are shown in Figures 2A and 2B, where the Fe, Ge, and S atoms are represented by the brown, grayish purple and gold balls, respectively.In Fe 3 GeS 2 monolayer, there are five sub-layers, where Fe 3 Ge substructure is sandwiched by two S layers.The red dashed line marks the unit cell of Fe 3 GeS 2 monolayer in Figure 2A.In this unit cell, there are six atoms, including three Fe atoms, one Ge atom, and two S atoms.In the unit cell of Fe 3 GeS 2 monolayer, these three Fe atoms can be divided in two inequivalent types Fe 1 (including the top and bottom Fe atoms) and Fe 2 , based on their positions and the crystal symmetry.Hence, there are four possible magnetic phases in the unit cell of Fe 3 GeS 2 monolayer, including FM, AFM-1, AFM-2, and NM magnetic configurations, as shown in Figure 3.When both the spins of two Fe 1 atoms are parallel to that of Fe 2 atom, it results in an FM phase, as shown in Figure 3A.In AFM-1 phase, the spin of top Fe 1 is antiparallel to the bottom Fe 1 , but is parallel to that of Fe 2 .If the coupling between Fe 2 and two Fe 1 atoms is AFM, while the coupling between two Fe 1 atoms is FM, it leads to AFM-2 phase.To determine the magnetic ground state, we calculated the energies of Fe 3 GeS 2 monolayer at these four possible magnetic phases, as shown in Figure 4. We can find the Fe 3 GeS 2 monolayer with FM phase owns the lowest energy at lattice constant of 3.95 A ˚.In Figures 3C and 3D, it can be found that the energies of AFM-1, AFM-2 and NM phases are $1.322 eV, $1.323 eV, and $1.346 eV higher than the FM phase as lattice constant is of 3.95 A ˚. Meantime, with the lattice constant increasing, these energy differences also elevate significantly.When the lattice constant is smaller than 3.95 A ˚, all of the energy differences between AFM-1, AFM-2, NM, and FM phases are still positive, indicating the energy of FM phase is lowest among these four possible magnetic phases.According to the least-energy principle, it can be drawn that the magnetic ground state of Fe 3 GeS 2 monolayer is FM, and the optimized lattice constant is a = b = 3.95 A ˚. Furthermore, this FM magnetic ground state of Fe 3 GeS 2 monolayer is robust to the biaxial strain within the range of À1.5%-1.3%.
For 2D magnets, magnetic anisotropy is essential to resist thermal fluctuation and keep the long-range magnetic order.Magnetic anisotropy, introduced by the spin-orbit coupling, is composed of magnetocrystalline anisotropic energy (C-MAE) and dipolar magnetic anisotropic energy (D-MAE).In this paper, C-MAE is calculated by the XXZ model 47 : (Equation 2) where the E XX (E ZZ ) is the energy of 2D magnet with the spin along x-(z-) direction.In the XXZ model, the energy with spin along the x-direction is considered equal to the spin along the y-direction, resulting in-plane magnetic isotropy.Thus, the C-MAE only considers the energy difference of 2D magnets as the spin is along x-and z-directions.The dipolar interaction energy in magnet can be calculated by 48 : where g of 2 is the gyromagnetic ratio, m B and m 0 are the Bohr magnon and the vacuum permeability, respectively.R ij is the position vector between the i-and j-magnetic lattices, while the spin angular momentum of the i-(j-) magnetic lattice is S i (S j ).According to Equation 3, it can be found that the dipolar interaction energy relies on the relative position of spin pair and their spin orientations.Early in 2002, Politi et al. 49 reported the dipolar coupling between single-domain FM particles could induce long-range FM order.Based on the strong and tunable dipole-dipole interaction, Young et al. 50realized fast two-qubit entangling gates, providing significant speedups for quantum algorithms.Meantime, Utesov 51 proposed that dipolar forces originating from the dipole-dipole interaction can lead to biaxial anisotropy in the reciprocal space of antiferromagnets with skyrmion, and the dipolar forces is considered as the critical ingredient to stabilize the nanometer-sized skyrmions in antiferromagnets. 52Hence, it is recognized that dipolar forces can result in the complicated sequences of magnetic phase transition in magnets under an external magnetic field. 53,54To obtain the D-MAE, we make a difference in the dipole interaction energy of spin .89 to 4.00 A ˚.These results suggest the Fe 3 GeS 2 monolayer owns out-of-plane magnetic anisotropy dominated by the dipolar interaction, and this out-of-plane magnetic anisotropy is robust to the biaxial strain within the range of À1.5%-1.3%.It is worth emphasizing that this out-of-plane magnetic anisotropy dominated by the dipolar interaction is rare in previous synthesized 2D ferromagnets, such CrGeTe 3 , CrI 3 , Fe 3 GeTe 2 , which is promising to realize dipolar-induced magnon chirality.
To identify the dynamic stability of optimized Fe 3 GeS 2 monolayer, we calculated its phonon dispersion based on the DFPT, shown in Figure 6.There is little imaginary frequency in the phonon dispersion and phonon density of states (PDOS), suggesting the dynamic stability of our optimized Fe 3 GeS 2 monolayer.In phonon dispersion, there are 18 phonon branches including 3 acoustic branches and 15 optical branches, because there are six atoms in the unit cell of Fe 3 GeS 2 monolayer.These three acoustic branches are named as longitudinal acoustic (LA), transverse acoustic (TA), out-of-plane acoustic (ZA) phonons, and their eigenvectors describe the translation of the Fe 3 GeS 2 monolayer along x-, y-, and z-directions, respectively.It has been reported that the thermal conductivity usually is dominated by the anharmonic interaction of acoustic phonons, 55,56 which is fundament for the performance and reliability of the 2D devices.We mark the LA, TA, and ZA branches by the red, green, and purple solid lines in phonon dispersion, as shown in Figure 6A.To estimate the anharmonicity of acoustic phonons in Fe 3 GeS 2 monolayer, we calculated the Gru ¨neisen constant (g q,s ) by 57 : g q;s = À a 2u q;s $ du q;s da ; (Equation 4) where u q,s is the phonon frequency of the s-branch at the wave vector of q, and a is the lattice constant.In this paper, the Gru ¨neisen constant was calculated by applying a 2% biaxial strain to lattice at 0 K, and the obtained Gru ¨neisen constants of acoustic phonons are presented in Figure 7A.As known, a larger absolute value of Gru ¨neisen constant suggests a stronger anharmonicity.In Figure 6A, the absolute Gru ¨neisen constant of ZA phonon is larger than LA and TA phonons, indicating the strong anharmonic interaction with other phonon modes.Meantime, the Gru ¨neisen constant of ZA remains positive with the wave vector changing, suggesting a softening phonon frequency of ZA mode with lattice expansion.At the long-wavelength limit (near G point), the Gru ¨neisen constants of LA and TA phonons are 0.18 and 4.42, respectively, which is much smaller than ZA phonon (37.24).These results are different from other 2D materials where ZA phonon owns remarkable negative Gru ¨neisen constant, such as graphene, 58 biphenylene, 59 and black phosphorene. 60As the wave vector increases, the Gru ¨neisen constant of LA phonon also remains positive, but there is a negative Gru ¨neisen constant (À0.51) of TA phonon.Compared with acoustic phonons, optical phonon modes can be used to measure thermal conductivity, 8 evaluate the thickness of 2D materials, 61 and design quantum mechanical resonator. 62Here, we also investigate and analyze the optical phonons in Fe 3 GeS 2 monolayer.In the phonon dispersion of Fe 3 GeS 2 monolayer, there are fifteen optical phonons named from O01 to O15, as shown in Figure 6A.We can find two sing-phonon optical modes O12 and O15 with frequencies of 9.08 THz and 11.12 THz, respectively, which is rare in other 2D materials, because PDOS always is composed of the mixed contribution from coupled phonon modes.In sing-phonon optical mode, there is no degeneracy, rendering it promising to design quantum mechanical resonators.In designing quantum mechanical resonators, how to cool the resonator to its ground state is a critical problem.Generally, the cooling temperature T cool should be smaller than hf =k B , 63 and h, f, k B are Planck's constant, phonon frequency, and Boltzmann's constant, respectively.For O12 and O15 modes, the corresponding maximum cooling temperatures T cool are 432.63K and 529.83K, higher than room temperature and can be realized by standard cryogenic methods.Meantime, the Gru ¨neisen constants of fifteen optical phonons at the center of Brillouin zone (Gpoint) were also calculated by Equation 4, as shown in Figure 7B.Obviously, the O15 mode owns a negative Gru ¨neisen constant of À0.4, suggesting a frequency softening with lattice expansion.The largest Gru ¨neisen constant of 5.85 occurs at O04 mode, but is still smaller than the ZA phonon at the long-wavelength limit.
In 2D devices, the difference in TEC between the substrates and 2D material can induce strain inevitably, and this strain increases with the temperature.At high temperature, this induced strain could destroy the geometrical structure of 2D devices, leading to the performance degradation of 2D devices.Based on these Gru ¨neisen constants, the TEC a can be calculated by 64 : (Equation 5) where V 0 and B 2D are the volume of Fe 3 GeS 2 primitive cell and the bulk modulus of Fe 3 GeS 2 monolayer.The bulk modulus of Fe 3 GeS 2 monolayer is calculated as 2,165.67N/m by using the Yang's modulus and Poisson ratio in a study by Long and Yang. 38The calculated TEC of Fe 3 GeS 2 monolayer is plotted in Figure 8.At 20 K, the TEC contributed by acoustic phonons is 1.39 3 10 À5 1/K, which is 97.89% of that considering the contribution from both acoustic and optical phonons.As temperature increases to 300 K, the TEC by only considering the contribution from acoustic phonons is 2.80 3 10 À5 1/K.Meantime, the TEC by considering the contribution from both acoustic and optical phonons is up to 11.84 3 10 À5 1/K whose 76.35% comes from the optical phonons.Obviously, the contribution from optical phonons grows remarkably with temperature, because the Gru ¨neisen constants of optical phonons are comparable to acoustic phonons, as shown in Figure 7C.With temperature increasing, more and more optical phonons are excited, and then the anharmonic interaction between optical phonons becomes strengthening.Therefore, the contribution from optical phonons to TEC of Fe 3 GeS 2 monolayer is enhanced significantly, which has also been observed in SiP 2 monolayer. 65

Conclusions
According to first-principles calculations implemented in Device Studio program, we find the magnetic ground state of Fe 3 GeS 2 monolayer is the FM, and this FM state is robust to the À1.5%-1.3%biaxial strain.Meantime, we also find the out-of-plane magnetic anisotropy is dominated by dipolar interaction in Fe 3 GeS 2 monolayer.This out-of-plane magnetic anisotropy is beneficial for Fe 3 GeS 2 monolayer to resist thermal fluctuation and sustain long-range FM order.In phonon dispersion, there are two single-phonon modes with frequencies of 9.08 THz and 11.12 THz, which can be used to design quantum mechanical resonator with a wide cool temperature range.Besides, the Gru ¨neisen constants of all phonons and TEC of FM Fe 3 GeS 2 monolayer were calculated, which highlights the contribution from the anharmonic interaction of optical phonons to the TEC.Our study provides theoretical support for the application of FM Fe 3 GeS 2 monolayer in future spintronic devices.

Limitations of the study
This study implemented first-principles calculations by Device Studio program to investigate the magnetic anisotropy and phononic properties of ideal 2D FM Fe 3 GeS 2 monolayer with high Curie temperature of 630 K, and found the out-of-plane magnetic anisotropy dominated by dipolar interaction, two single-phonon modes with frequencies of 9.08 THz and 11.12 THz, and the large contribution from optical phonons to thermal expansion.Nonetheless, this study is not without its limitations that warrant further attention.Generally, the first-principles calculation and Monte Carlo simulation often overestimate the magnetic phase-transition temperature of 2D magnets, underscoring the necessity for additional experimental corroboration.The largest limitation of the study is that 2D FM Fe 3 GeS 2 has not been successfully prepared, and experimental corroboration of its interesting properties cannot be carried out.This limitation presents a pivotal direction for future research endeavors on 2D FM Fe 3 GeS 2 .

Lead contact
Further information and requests for resources and reagents should be directed to and will be fulfilled by the lead contact, Ke Wang (kewang@xupt.edu.cn).

Materials availability
This study did not generate new unique materials.

Figure 2 .
Figure 2. Geometrical structure and high-symmetry path of Fe 3 GeS 2 monolayer (A and B) are the top and side views of geometrical structure, and (C) is the high-symmetry path in the irreducible Brillouin zone.In (A) and (B), the Fe, Ge, and S atoms are represented by the brown, grayish purple and gold balls, respectively.

Figure 4 .
Figure 4.The curve of energy at possible magnetic phases versus strain (A) is the energy of Fe 3 GeS 2 monolayer at FM phase, while (B-D) are the energy differences between FM and, AFM-1, AMF-2 and NM phases obtained by DS-PAW, respectively.

Figure 6 .
Figure 6.The phonon dispersion and phonon density of states of Fe 3 GeS 2 monolayer obtained by DS-PAW (A) is phonon dispersion, while (B) shows the phonon density of states.

Figure 7 .
Figure 7. Gru ¨neisen constants of phonons (A) is Gru ¨neisen constants of acoustic phonons along the high-symmetry path, while (B and C) are Gru ¨neisen constants for fifteen optical phonons at the center of Brillouin zone and all phonons, respectively.