Two-dimensional Heisenberg models with materials-dependent superexchange interactions

Jia-Wen Li Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijng 100049, China Zhen Zhang Key Laboratory of Multifunctional Nanomaterials and Smart Systems, Division of Advanced Materials, Suzhou Institute of Nano-Tech and Nano-Bionics, Chinese Academy of Sciences, Suzhou, 215123 China Jing-Yang You Department of Physics, National University of Singapore, Science Drive, Singapore 117551 Bo Gu [email protected] Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijng 100049, China CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijng 100190, China Physical Science Laboratory, Huairou National Comprehensive Science Center, Beijing 101400, China Gang Su [email protected] Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijng 100049, China CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijng 100190, China Physical Science Laboratory, Huairou National Comprehensive Science Center, Beijing 101400, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijng 100049, China

Abstract

The two-dimensional (2D) van der Waals ferromagnetic semiconductors, such as CrI₃ and Cr₂Ge₂Te₆, and the 2D ferromagnetic metals, such as Fe₃GeTe₂ and MnSe₂, have been obtained in recent experiments and attracted a lot of attentions. The superexchange interaction has been suggested to dominate the magnetic interactions in these 2D magnetic systems. In the usual theoretical studies, the expression of the 2D Heisenberg models were fixed by hand due to experiences. Here, we propose a method to determine the expression of the 2D Heisenberg models by counting the possible superexchange paths with the density functional theory (DFT) and Wannier function calculations. With this method, we obtain a 2D Heisenberg model with six different nearest-neighbor exchange coupling constants for the 2D ferromagnetic metal Cr₃Te₆, which is very different for the crystal structure of Cr atoms in Cr₃Te₆. The calculated Curie temperature Tc = 328 K is close to the Tc = 344 K of 2D Cr₃Te₆ reported in recent experiment. In addition, we predict two stable 2D ferromagnetic semiconductors Cr₃O₆ and Mn₃O₆ sharing the same crystal structure of Cr₃Te₆. The similar Heisenberg models are obtained for 2D Cr₃O₆ and Mn₃O₆, where the calculated Tc is 218 K and 208 K, respectively. Our method offers a general approach to determine the expression of Heisenberg models for these 2D magnetic semiconductors and metals, and builds up a solid basis for further studies.

I Introduction

Recently, the successful synthesis of two-dimensional (2D) van der Waals ferromagnetic semiconductors in experiments, such as CrI₃ [1] and Cr₂Ge₂Te₆ [2] has attracted extensive attentions to 2D ferromagnetic materials. According to Mermin-Wagner theorem [3], the magnetic anisotropy is essential to produce the long-range magnetic order in 2D systems. For the 2D magnetic semiconductors obtained in experiments, the Curie temperature Tc is still much lower than room temperature. For example, Tc = 45 K in CrI₃ [1], 30 K in Cr₂Ge₂Te₆ [2], 34 K in CrBr₃ [4], 17 K in CrCl₃ [5], 75 K in Cr₂S₃ [6, 7], etc. For applications, the ferromagnetic semiconductors with Tc higher than room temperature are highly required [8, 9, 10, 11]. On the other hand, the 2D van der Waals ferromagnetic metals with high Tc have been obtained in recent experiments. For example, Tc = 140 K in CrTe [12], 300 K in CrTe₂ [13, 14], 344 K in Cr₃Te₆ [15], 160 K in Cr₃Te₄ [16], 280 K in CrSe [17], 300 K in Fe₃GeTe₂ [18, 19], 270 K in Fe₄GeTe₂ [20], 229 K in Fe₅GeTe₂ [21, 22], 300 K in MnSe₂ [23], etc.

In these 2D van der Waals ferromagnetic materials, the superexchange interaction has been suggested to dominate the magnetic interactions. The superexchange interaction describes the indirect magnetic interaction between two magnetic cations mediated by the neighboring non-magnetic anions [24, 25, 26]. The superexchange interaction has been discussed in the 2D magnetic semiconductors. Based on the superexchange interaction, the strain-enhanced Tc in 2D ferromagnetic semiconductor Cr₂Ge₂Se₆ can be understood by the decreased energy difference between the d electrons of cation Cr atoms and the p electrons of anion Se atoms [27]. The similar superexchange picture was obtained in several 2D ferromagnetic semiconductors, including the great enhancement of Tc in bilayer heterostructures Cr₂Ge₂Te₆/PtSe₂ [28], the high Tc in technetium-based semiconductors TcSiTe₃, TcGeSe₃ and TcGeTe₃ [29], and the electric field enhanced Tc in the monolayer MnBi₂Te₄ [30]. The superexchange interaction has also been discussed in the semiconductor heterostructure CrI₃/MoTe₂ [31], and 2D semiconductor Cr₂Ge₂Te₆ with molecular adsorption [32].

In addition, the superexchange interaction has also been obtained in the 2D van der Waals ferromagnetic metals. By adding vacancies, the angles of the superexchange interaction paths of 2D metals VSe₂ and MnSe₂ will change, thereby tuning the superexchange coupling strength [33]. It is found that biaxial strain changes the angle of superexchange paths in 2D metal Fe₃GeTe₂, and affects Tc [34]. Under tensile strain, the ferromagnetism of the 2D magnetic metal CoB₆ is enhanced, due to the competition between superexchange and direct exchange interactions [35].

It is important to determine the spin Hamiltonian for the magnetic materials, in order to theoretically study the magnetic properties, such as Tc. In the usual theoretical studies, the expression of the spin Hamiltonian needs to be fixed by hand according to the experiences. By the four-state method and density functional theory (DFT) calculations [36, 37, 38], the exchange coupling parameters of the spin Hamiltonian, such as the nearest neighbor, the next nearest neighbor, inter-layer, etc, can be obtained. Then the Tc can be estimated through Monte Carlo simulations [38]. With different spin Hamiltonians chosen by hand, sometimes different results are obtained in calculations. Is it possible to determine the spin Hamiltonian by the help of calculations rather than by the experiences ?

In this paper, we propose a method to establish the 2D Heisenberg models for the 2D van der Waals magnetic materials, when the superexchange interactions dominate. Through the DFT and Wannier function calculations, we can calculate the exchange coupling between any two magnetic cations, by counting the possible superexchange paths. By this method, we obtain a 2D Heisenberg model with six different nearest-neighbor exchange coupling constants for the 2D van der Waals ferromagnetic metal Cr₃Te₆ [15], where the calculated Tc = 328 K is close to the Tc = 344 K reported in the experiment. In addition, based on the crystal structure of 2D Cr₃Te₆, we predict two 2D magnetic semiconductors Cr₃O₆ and Mn₃O₆ with Tc of 218 K and 208 K, and energy gap of 0.99 eV and 0.75 eV, respectively.

II Computational methods

Our calculations were based on the DFT as implemented in the Vienna ab initio simulation package (VASP) [39]. The exchange-correlation potential is described with the Perdew-Burke-Ernzerhof (PBE) form of the generalized gradient approximation (GGA) [40]. The electron-ion potential is described by the projector-augmented wave (PAW) method [41]. We carried out the calculation of GGA + U with U = 3.2 eV, a reasonable U value for the 3d electrons of Cr in Cr₃Te₆ [15]. The band structures for 2D Cr₃O₆ and Mn₃O₆ were calculated in HSE06 hybrid functional [42]. The plane-wave cutoff energy is set to be 500 eV. Spin polarization is taken into account in structure optimization. To prevent interlayer interaction in the supercell of 2D systems, the vacuum layer of 16 Å is included. The 5 $\times$ 9 $\times$ 1, 5 $\times$ 9 $\times$ 1 and 7 $\times$ 11 $\times$ 1 Monkhorst Pack k-point meshed were used for the Brillouin zone (BZ) sampling for 2D Cr₃O₆, Cr₃Te₆ and Mn₃O₆, respectively [43]. The structures of 2D Cr₃O₆ and Mn₃O₆ were fully relaxed, where the convergence precision of energy and force were $10^{-6}$ and $10^{-3}$ eV/Å, respectively. The phonon spectra were obtained in a 3×3×1 supercell with the PHONOPY package [44]. The Wannier90 code was used to construct a tight-binding Hamiltonian [45, 46] to calculate the magnetic coupling constant. In the calculation of molecular dynamics, a 3 $\times$ 4 $\times$ 1 supercell (108 atoms) was built, and we took the NVT ensemble (constant-temperature, constant-volume ensemble) and maintained a temperature of 250 K with a step size of 3 fs and a total duration of 6 ps.

III Method to determine the 2D Heisenberg model: an example of 2D Cr₃Te₆

Refer to caption — Figure 1: Crystal structure of Cr₃Te₆ . (a) Top view (b) Side view.

III.1 Calculate exchange coupling J from superexchange paths

The crystal structure of 2D Cr₃Te₆ is shown in Fig. 1, where the space goup is Pm (No.6). In experiment, it is a ferromagnetic metal with high Tc = 344 K [15]. To theoretically study its magnetic properties, we considered seven different magnetic configurations, including a ferromagnetic (FM) , a ferrimagnetic (FIM), and five antiferromagnetic (AFM) configurations, as discussed in Supplemental Materials [47]. The calculation results show that the magnetic ground state is ferromagnetic, consistent with the experimental results. Since the superexchange interaction has been suggested to dominate the magnetic interactions in these 2D van der Waals ferromagnetic semiconductors and metals, we study the superexchange interactions in 2D Cr₃Te₆.

The superexchange interaction can be reasonably descried by a simple Cr-Te-Cr model [48], as shown in Fig. 2. There are two Cr atoms at sites i and j, and one Te atom at site k between the two Cr atoms. By the perturbation calculation, the superexchange coupling J_ij between the two Cr atoms can be obtained as [48],

	$\displaystyle J_{ij}=$	$\displaystyle(\frac{1}{E_{\uparrow\downarrow}^{2}}-\frac{1}{E_{\uparrow\uparrow}^{2}})\sum\limits_{k,p,d}\|V_{ik}\|^{2}J_{kj}^{pd}$		(1)
	$\displaystyle=$	$\displaystyle\frac{1}{A}\sum\limits_{k,p,d}\|V_{ik}\|^{2}J^{pd}_{kj}.$		(1)

The indirect exchange coupling J_ij is consisting of two processes. One is the direct exchange process between the d electron of Cr at site j and the p electrons of Te at site k, presented by J ${}_{kj}^{pd}$ . The other is the electron hopping process between p electrons of Te atom at site k and d electrons of Cr atom at site i, presented by —V ${}_{ik}|^{2}/A$ . V_ik is the hopping parameter between d electrons of Cr atom at site i and p electrons of Te atom at site k. Here, A = 1/(1/E ${}_{\uparrow\downarrow}^{2}$ -1/E ${}_{\uparrow\uparrow}^{2}$ ), and is taken as a pending parameter. E_↑↑ and E_↑↓ are energies of two d electrons at Cr atom at site i with parallel and antiparallel spins, respectively. The direct exchange coupling J ${}_{kj}^{pd}$ can be expressed as [27, 28, 29, 30]:

\displaystyle J_{kj}^{pd}=\frac{2|V_{kj}|^{2}}{|E_{k}^{p}-E_{j}^{d}|}.

(2)

V_kj is the hopping parameter between p electrons of Te atom at site k and d electrons of Cr atom at site j. E ${}_{k}^{p}$ is the energy of p electrons of Te atom at site k, and E ${}_{j}^{d}$ is the energy of d electrons of Cr atom at site j.

By the DFT and Wannier function calculations, the parameters V_ik, V_kj, E ${}_{k}^{p}$ , and E ${}_{j}^{d}$ in Eqs. (1) and (2) can be calculated. The J_ijA can be obtained by counting all the possible k sites of Te atoms, p orbitals of Te atoms, and d orbitals of Cr atoms.

Table 1: For 2D Cr₃Te₆, the calculated exchange coupling parameters J_ijA in Eqs.(1) and (2), by the density functional theory and Wannier functional calculations. A is a pending parameter. The unit of J_ijA is meV³.

J₁₁A	J₂₂A	J₃₃A	J₁₂A	J₁₃A	J₂₃A
40	26	53	29	44	83

Table 2: For 2D magnetic metal Cr₃Te₆ and semiconductors Cr₃O₆ and Mn₃O₆, the parameter A (in unit of meV^-2) in Eq. (1), the exchange couping parameters J_ijS² and the magnetic anisotropy parameter DS² (in unit of meV) in the Hamiltonian in Eq. (3), and the estimated Curie temperature Tc. See text for details.

Materials	A	J₁₁S²	J₂₂S²	J₃₃S²	J₁₂S²	J₁₃S²	J₂₃S²	DS²	Tc (K)
Cr₃Te₆	-27	-17.1	-11.5	-24.4	-12.6	-19.6	-37.4	-0.14	328
Cr₃O₆	-36	-18.9	-14.6	-10.1	-18.7	-1.8	-3.1	0.04	218
Mn₃O₆	-465	-11.9	-7.6	-50.4	-15.9	-5.2	-10.7	-0.09	208

From the calculated results in Table 1, it is suggested that there are six different nearest-neighbor couplings, denoted as J₁₁, J₂₂, J₃₃, J₁₂, J₁₃, and J₂₃, as shown in Fig. 3(b). Accordingly, there are three kinds of Cr atoms, noted as Cr₁, Cr₂, and Cr₃. Based on the results in Table 1, the effective spin Hamiltonian can be written as

$\displaystyle H=$	$\displaystyle J_{11}\sum\limits_{n}\vec{S}_{1n}\cdot\vec{S}_{1n}+J_{22}\sum\limits_{n}\vec{S}_{2n}\cdot\vec{S}_{2n}+J_{33}\sum\limits_{n}\vec{S}_{3n}\cdot\vec{S}_{3n}$	(3)
$\displaystyle+$	$\displaystyle J_{12}\sum\limits_{n}\vec{S}_{1n}\cdot\vec{S}_{2n}+J_{13}\sum\limits_{n}\vec{S}_{1n}\cdot\vec{S}_{3n}+J_{23}\sum\limits_{n}\vec{S}_{2n}\cdot\vec{S}_{3n}$
$\displaystyle+$	$\displaystyle D\sum\limits_{n}(S_{1nz}^{2}+S_{2nz}^{2}+S_{3nz}^{2}),$

where J_ij means magnetic coupling between Cr_i and Cr_j, as indicated in Fig. 3(b). D represents the magnetic anisotropy energy (MAE) of Cr₃Te₆.

III.2 Determine the parameters D and A

The single-ion magnetic anisotropy parameter DS² can be obtained by: DS²=(E_⟂-E_∥)/6, where E_⟂ and E_∥ are energies of Cr₃Te₆ with out-of-plane and in-plane polarizations in FM state, respectively. It has DS² = -0.14 meV/Cr for 2D Cr₃Te₆, which is in agreement with the value of -0.13 meV/Cr reported in the previous study of Cr₃Te₆ [15].

The parameter A can be calculated in the following way. Considering a FM and an AFM configurations, the total energy of Eq. (3) without MAE term can be respectively expressed as [47]:

$\displaystyle E_{FM}$	$\displaystyle=$	$\displaystyle 2J_{11}S_{1}^{2}+2J_{22}S_{2}^{2}+2J_{33}S_{3}^{2}+8J_{12}S_{1}S_{2}$
		$\displaystyle+2J_{23}S_{2}S_{3}+8J_{13}S_{1}S_{3}+E_{0}$
	$\displaystyle=$	$\displaystyle 11838/A+E_{0},$
$\displaystyle E_{AFM1}$	$\displaystyle=$	$\displaystyle 2J_{11}S_{1}^{2}+2J_{22}S_{2}^{2}-2J_{33}S_{3}^{2}-8J_{12}S_{1}S_{2}+E_{0}$
	$\displaystyle=$	$\displaystyle-2502/A+E_{0}.$

The results in Table 1 are used to obtain the final expressions in Eq. (III.2). Since two parameters A and E₀ are kept, two spin configurations FM and AFM1 are considered here. Discussion on the choice of spin configurations is given in Supplemental Materials [47]. For the FM spin configuration, the ground state of Cr₃Te₆, the total energy is taken as E_FM = 0 for the energy reference. The total energy of AFM1, E_AFM1 = 535 meV is obtained by the DFT calculation. The parameters A and E₀ are obtained by solving Eq. (III.2), and the six exchange coupling parameters J_ij can be obtained by Table 1. The results are given in Table 2.

III.3 Estimate Tc by Monte Carlo simulation

To calculate the Curie temperature, we used the Monte Carlo program for the Heisenberg-type Hamiltonian in Eq. (3) with parameters in Table 2. The Monte Carlo simulation was performed on a 30 $\sqrt{3}$ $\times$ 30 $\sqrt{3}$ lattice with more than 1 $\times$ 10⁶ steps for each temperature. The first two-third steps were discarded, and the last one-thirds steps were used to calculate the temperature-dependent physical quantities. As shown in Table 2 and Fig. 4 (d), the calculated Tc = 328 K for 2D Cr₃Te₆, close to the Tc = 344 K of 2D Cr₃Te₆ in the experiment [15]. Discussion on the choice of spin configurations and the estimation of exchange couplings J_ij and Tc is given in Supplemental Materials [47].

IV Prediction of Two High Curie Temperature Magnetic Semiconductors Cr₃O₆ and Mn₃O₆

Inspired by the high Tc in the 2D magnetic metal Cr₃Te₆, we explore the possible high Tc magnetic semiconductors with the same crystal structure of Cr₃Te₆ by the DFT calculations. We obtain two stable ferromagnetic semiconductors Cr₃O₆ and Mn₃O₆. In order to study the stability of the 2D Cr₃O₆ and Mn₃O₆, we calculate the phonon spectrum. As shown in Supplemental Materials [47], there is no imaginary frequency, indicating the dynamical stability. In addition, we performed molecular dynamics simulations of Cr₃O₆ and Mn₃O₆ at 250 K, taking the NVT ensemble (constant temperature and volume) and run for 6 ps. The results show that 2D Cr₃O₆ and Mn₃O₆ are thermodynamically stable [47]. These calculation results suggest that 2D Cr₃O₆ and Mn₃O₆ may be feasible in experiment.

The band structure of 2D Cr₃O₆ and Mn₃O₆ is shown in Figs. 4(a) and 4(b), respectively, where the band gap is 0.99 eV for Cr₃O₆ and 0.75 eV for Mn₃O₆. As shown in Figs. 4(a) and (b), the band gap for 2D Cr₃O₆ and Mn₃O₆ is 0.99 eV and 0.75 eV, respectively. When applying an out-of-plane electric field with a range of $\pm$ 0.3 V/Å, which is possible in experiment [49], the band gap of Cr₃O₆ (Mn₃O₆) increases (decreases) with increasing electric field, as shown in Fig. 4(c). By the same calculation method above, the parameter A, the similar Heisenberg models in Eq. 3 with six nearest-neighbor exchange coupling J_ij are obtained for the 2D Cr₃O₆ and Mn₃O₆. The parameters A, J_ij and D are calculated and shown in Table 2. The spin polarization of Cr₃O₆ and Mn₃O₆ is in-plane (DS² = 0.04 meV) and out-of-plane (DS² = -0.09 meV), respectively. Fig. 4(d) shows the magnetization as a function of temperature for 2D Cr₃Te₆, Cr₃O₆ and Mn₃O₆. The calculated Curie temperature is Tc = 218 K for 2D Cr₃O₆ and Tc = 208 K for 2D Mn₃O₆, respectively.

V Conclusion

Based on the DFT and Wannier function calculations, we propose a method for constructing the 2D Heisenberg model with the superexchange interactions. By this method, we obtain a 2D Heisenberg model with six different nearest-neighbor exchange couplings for the 2D ferromagnetic metal Cr₃Te₆. The calculated Curie temperature Tc = 328 K is close to the Tc = 344 K of Cr₃Te₆ in the experiment. In addition, we predicted two 2D magnetic semiconductors: Cr₃O₆ with band gap of 0.99 eV and Tc = 218 K, and Mn₃O₆ with band gap of 0.75 eV and Tc = 208 K, where the similar 2D Heisenberg models are obtained. The complex Heisenberg model developed from the simple crystal structure shows the power of our method to study the magnetic properties in these 2D magnetic metals and semiconductors.

Acknowledgements

This work is supported in part by the National Natural Science Foundation of China (Grants No. 12074378 and No. 11834014), the Beijing Natural Science Foundation (Grant No. Z190011), the National Key R&D Program of China (Grant No. 2018YFA0305800), the Beijing Municipal Science and Technology Commission (Grant No. Z191100007219013), the Chinese Academy of Sciences (Grants No. YSBR-030 and No. Y929013EA2), and the Strategic Priority Research Program of Chinese Academy of Sciences (Grants No. XDB28000000 and No. XDB33000000).

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