Xi Chen(陈熙), Zheng-Zhe Lin(林正喆), and Li-Rong Cheng(程丽蓉)

School of Physics and Optoelectronic Engineering,Xidian University,Xi’an 710071,China

Keywords: two-dimensional(2D)ferromagnetism,spin wave,magnetic anisotropy

1. Introduction

Atomically thin two-dimensional (2D) crystals can present exceptional electronic structures as a result of reduced dimensionality. The success in 2D material research brings vast opportunities to pursue emerging physical properties.Various 2D crystals are discovered,ranging from semiconductors,highly correlated materials to superconductors. Recently,magnetic crystals are added to the 2D material family.[1–3]In particular, intrinsic ferromagnetism in 2D materials is a new form of condensed matter. Unlike the ferromagnetism in conventional ultrathin metals, ferromagnetic (FM) 2D materials hold magnetic order in reduced dimensionality and may exhibit new physics for spintronics.

With the rapid development, 2D materials have permeated into many research areas[4–9]and also bring opportunities for finding ferromagnetism in reduced dimensionality.Although the Mermin–Wagner theorem[10]asserts that 2D longrange FM order cannot exist in isotropic magnetic systems,a recent discovery has found 2D layered systems possessing intrinsic magnetocrystalline anisotropy against thermal fluctuations (e.g., CrI3,[11–15]Cr2Ge2Te6,[16]and CrSiTe3[17]).However, the Curie temperatures of CrI3, Cr2Ge2Te6, and CrSiTe3are only dozens of Kelvin. In recent years,Fe3GeTe2has been found to have a Curie temperature close to room temperature[18–25](150–220 K depending on Fe occupancy).By the doping of the ionic gate, the Curie temperature of atomically thin Fe3GeTe2is dramatically elevated to room temperature.[18,26]The discovery of Fe3GeTe2provides a new chance for ultra-thin spintronics. Magnetically ordered 2D crystals open vast possibilities for novel physical phenomena and new device concepts.

In this paper, a theoretical model is proposed to understand the magnetic interactions in 2D Fe3GeTe2,and the density functional theory (DFT) calculations are employed to evaluate the key parameters. We uncover the mechanism of magnetic anisotropy in maintaining the magnetic order of 2D systems. The model reveals a new form of long-range ferromagnetism and suggests a physical picture beyond the Stoner model. The Curie temperature of Fe3GeTe2is then predicted by the model. Our theoretical model can also be applied to other 2D itinerant ferromagnetism systems.

2. Theory

In traditional theory,the Mermin–Wigner theorem[10]asserts that gapless excitation can occur in systems with continuous symmetry. In isotropic 2D systems,the gapless quasiparticle spectrum leads to a divergence of thermal distribution,which indicates the absence of magnetic order at finite temperature. However,anisotropic 2D systems hold low-energy gaps which protect the long-range magnetic order. To understand the magnetic order in Fe3GeTe2,let us start with the spin wave theory. The possible Heisenberg Hamiltonian in anisotropic 2D systems reads

In the absence of magnetic anisotropy(i.e., D=0 and λ =0 which leads to Eg=0),the integral goes

This divergence is consistent with the Mermin–Wagner theorem, indicating that 2D ferromagnetism cannot exist without magnetic anisotropy.

3. Model and computation

The 2D Fe3GeTe2lattice(Fig.1(a))has hexagonal symmetry with the primitive cell belonging to the space group P¯6m2.The Fe atoms in one primitive cell are located in two inequivalent Wyckoff sites. In each primitive cell,two Fe atoms are located at the same position in the 2D plane at different heights(atoms 1 and 2 in Fig.1(b)).The last Fe atom(atoms 3 in Fig.1(b)) is sandwiched between the two Te atoms. The neighboring Fe atoms 1 and 2 constitute a magnetic group. J1and J2denote its Heisenberg coupling with 1stNNs and 2ndNNs. The magnetization of Fe3GeTe2lattice is mainly decided by these groups. The magnetic moments of atom 3 are then induced by the three adjacent groups(which is explained in supplementary section 3). In the following text, we treat each Fe atom group (composed of 1 and 2) as a whole magnetic object.

Fig.1. Structure and magnetic coupling of Fe3GeTe2. (a)The structure of Fe3GeTe2 with the primitive cell shown by dashed lines. (b)The atomic group composed of atoms 1 and 2,and the coupling J1 with 1st NNs and J2 with 2nd NNs. (c)6×1 supercell and four different configurations for the magnetization of the atomic groups. The arrows denote the directions of magnetic moments on the Fe atomic groups.

It is worth discussing the choice of DFT functional here. To verify the reliability of LDA, we also perform comparison calculations using the Perdew–Burke–Ernzerhof(PBE)functional[41]and the hybrid Heyd–Scuseria–Ernzerhof(HSE06) functional.[42,43]The LDA functional gives a magnetic moment m = 4.55µBper primitive cell, which is in the range of experimental values (m=3.60–4.89 µB).[20,21]However, the obtained magnetic moments by the PBE (m=6.32µB)and HSE06(m=7.11µB)functionals are drastically overestimated. The band structures (supplementary Fig.S1)show a feature of band shift in FM Fe3GeTe2monolayer.The PBE or HSE06 functional present a larger band shift than LDA. The disagreement between theoretical (PBE or HSE06) and experimental magnetic moments also appears in the case of Fe-based superconductors.[44]This suggests that LDA, which is used throughout this work, is suitable to describe the electronic structure of Fe3GeTe2.

Within the Heisenberg model, the interatomic couplings are evaluated via the energy-mapping method.[45]A 6×1 supercell(Fig.1(c))is employed as a model. The configurations α, β, γ, and δ exhibit different magnetic moments on the Fe atomic groups. DFT calculations without SOC count the energy of Heisenberg coupling without magnetic anisotropy.According to the Heisenberg model,their energies read

Based on the DFT total energies without SOC,the Heisenberg coupling parameters J1and J2are obtained by the ordinary least squares method(see supplementary section 2).

4. Results and discussion

4.1. DFT calculations

The mission of DFT calculations is to obtain the spin S of the Fe atomic group and the parameters in Eq.(1)(D,λ,J1and J2). LDA provides a total magnetic moment m=4.55µBper primitive cell,and the magnetic moment of the Fe atomic group is 3.58 µB. In DFT, the calculations of magnetic moments are based on atomic Wigner–Seitz radii, which are somewhat vague. Here we take an integral on the magnetic moment and can convince that S=2 is the spin owing by the Fe atomic group.

Next, D and λ can be fitted from the MAE calculation. DFT calculations reveal that 2D Fe3GeTe2monolayer is uniaxial with the easy axis along the z-direction perpendicular to the 2D surface. Based on the hexagonal symmetry of Fe3GeTe2, the MAE only depends on θ, which is the angle between lattice magnetization and the z axis.[46]According to Eq. (1), the MAE can be written as MAE =(D+n1λ)S2sin2θ. On the other hand, the MAE of a hexagonal crystal can be fit into[46]MAE ≈K sin2θ. DFT calculation results (Fig.3(c)) fit K = 2.46 meV/cell. So we get D+n1λ ≈K/S2=0.62 meV.

4.2. Magnon spectrum

The magnon spectrum can be obtained from the spin wave Hamiltonian Eq.(5).The hexagonal Fe3GeTe2has 1stNN and 2ndNN number n1=6 and n2=6. The Bloch Hamiltonian for the lattice reads

Fig.2. Magnon spectrum and the Curie temperature. (a) The energy spectrum of spin wave magnon. (b) Three-dimensional plot of the magnon spectrum. (c)The magnetic moment of one primitive cell versus temperature T.

4.3. Curie temperature

We now provide a rough estimate of the Curie temperature based on nonlinear spin wave theory. We use the expressions Eqs.(2)and(3)for spin operators, and expand them to fourth order in the bosonic operators

At intermediate temperatures,there is a finite number of spin waves that are accounted for the high-order terms in bosonic operators when substituting the previous expansion in the spin Hamiltonian. The spin Hamiltonian contains four field operators and therefore is not rigorously solvable. Here, the effect of the spin wave population in the Hamiltonian is described by the substitution

Here Ciis the perturbation terms. The last step removes the constant term (do not affect the physics of Hamiltonian) and ignores the high-order perturbation. Using a mean-field approximation,we have

By Eqs.(12)and(13)we have

The above formulas result in a substitution DS →DM,λS →λM, J1S →J1M, and J2S →J2M in the Hamiltonian HSWA.So, the magnetization is expressed by a self-consistent equation

At a given temperature T, the magnetization M(T) can then be derived by numerically solving this equation. This equation implies that the temperature T reduces quasiparticle energy and decreases the low-energy gap of magnon. The effect destroys the magnetic order in the system. The numerical results are shown in Fig.2(c). M(T)drastically decreases with increasing temperature when T >100 K. According to the results,we get a Curie temperature Tc=154 K.In a recent experiment,[18]the magnetic hysteresis of monolayer and bilayer Fe3GeTe2is detected to vanish at T =100 K.People also used polar refractive magnetic circular dichroism microscopy to measure layer-dependent Tcof Fe3GeTe2and detected a Curie temperature of about 70 K with a size of 2µm.[18]These experiment results are generally consistent with our calculation.

4.4. Stoner ferromagnetism

Fe3GeTe2monolayer is metallic with a non-integer magnetic moment. This inspires us to apply the Stoner model to understand the mechanism of yielding the magnetic order.The DFT bands of Fe3GeTe2monolayer(Fig.3(a))show a feature of band shift. The shapes of corresponding spin-up and spindown bands are similar. The spin-up band has lower energy,while the spin-down band has higher energy. From the density of states (DOS) (Fig.3(b)), we can see that the electron states near the Fermi level are mainly contributed by Fe 3d orbitals.To calculate the average band splitting ∆near the Fermi level, five corresponding bands (bold lines in Fig.3(a)) are chosen. Their average energy gap is ∆=1.17 eV.According to the Stoner model,[47]we have ∆=Um0,where m0=m/3=1.52 µBis the average magnetic moment per Fe atom. Then the Hubbard U=0.77 eV is determined.It is worth noting that this U is the result of electron interactions within LDA,which does not have to add it into LDA (if U is added to LDA, the exchange-correlation is stronger and that results in a band profile more close to PBE (see supplementary Fig.S1(d)) and a larger magnetic moment m). By spin-non-polarized DFT calculations,we obtain a DOS DNM(EF)=1.76 states/eV/atom at the Fermi level of non-magnetic (NM) Fe3GeTe2monolayer.Finally, the Stoner criterion, i.e., UDNM(EF)=1.36>1, is examined to establish. This criterion reveals that the ferromagnetism of Fe3GeTe2monolayer is spontaneous. To further consider the influence of magnetic anisotropy, SOC is added to the calculation of Hubbard U. For the magnetization angle θ =0◦–90◦,U =0.78–0.77 eV, which changes only by 1%.Thus,the SOC has little effect on the electron correlation. The magnetic anisotropy should arise from the energy difference of the crystal field acting on the itinerant electrons with different magnetization directions.

To understand the formation of ferromagnetism in Fe3GeTe2, we further analyze the mechanism by the Stoner model. When a rigid band splitting ∆ happens in NM Fe3GeTe2(Fig.3(d)), the spin-up electron number n↑is then larger than the spin-down electron number n↓,for which their sum n↑+n↓=n should be a constant and equal to the total electron number. By the DOS of NM Fe3GeTe2, the electron numbers per Fe atom can be counted as

Then we express the magnetic moment per atom as m0(∆)=n↑−n↓, by which an inverse function ∆=∆(m0)can be obtained numerically. On the other hand,the band splitting ∆is decided by the Hubbard model. On every Fe atom,the repulsion between spin-up and spin-down electrons in 3d orbitals leads to an energy difference of single quasiparticle,i.e.,

Then the band splitting reads ∆=U(n↑−n↓)=Um0.The above two equations,∆=∆(m0)and ∆=U(n↑−n↓)=Um0,are plotted in Fig.3(d) with increasing m0. It can be seen that they have a trivial intersection m0=0 and a non-trivial intersection m0=1.9 which is close to the previous DFT calculation (m0=1.52). The above analysis provides an understanding of the spontaneous magnetization in Fe3GeTe2.

Fig.3. (a) Spin-resolved band structure of Fe3GeTe2 monolayer without SOC. Red (blue) lines denote spin-up (spin-down) bands. (b)DOS of Fe3GeTe2 monolayer. (c)MAE of Fe3GeTe2 monolayer. (d)Band shift and the Stone criterion.

5. Conclusion

In this work, we attempt to unravel the origin of 2D ferromagnetism in Fe3GeTe2monolayer. By combining the theoretical model and DFT simulations,a physical picture is built to describe the magnetic interactions in 2D Fe3GeTe2. DFT calculations are employed to evaluate the Heisenberg coupling and magnetic anisotropy. The model reveals long-range ferromagnetic order in 2D systems should be maintained by magnetic anisotropy. The predicted Curie temperature agrees with a recent experiment. The Stoner model provides an insight into the spontaneous magnetization in Fe3GeTe2and results in a prediction close to DFT calculations. Our model is successful in understanding the magnetization mechanism in 2D Fe3GeTe2. In future studies, the model can be extended to other 2D systems.