Prediction of two-dimensional antiferromagnetic ferroelasticity in an AgF₂ monolayer

Abstract

Two-dimensional multiferroics, simultaneously harboring antiferromagneticity and ferroelasticity, are essential and highly sought for miniaturized device applications, such as high-density data storage, but thus far they have rarely been explored. Herein, using first principles calculations, we identified two-dimensional antiferromagnetic ferroelasticity in an AgF₂ monolayer that is dynamically and thermally stable, and can be easily fabricated from its bulk. The AgF₂ monolayer is an antiferromagnetic semiconductor with large spin polarization, and with great structural distortion due to its intrinsic Jahn–Teller effect when thinning the AgF₂ down to a monolayer. Additionally, it features excellent ferroelasticity with high transition signal and a low switching barrier, rendering the room-temperature nonvolatile memory accessible. Such coexistence of antiferromagneticity and ferroelasticity is of great significance to the study of two-dimensional multiferroics and also renders the AgF₂ monolayer a promising platform for future multifunctional device applications.

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New concepts

In this work, we propose a novel two-dimensional multiferroic material consisting of single-layer AgF₂ that can be used for next-generation information storage applications. We suggest that single-layer AgF₂ is an antiferromagnetic ferroelastic semiconductor with large spin polarization. It experiences a structural distortion when AgF₂ is thinned down to a monolayer, which is attributed to the Jahn–Teller effect. The AgF₂ monolayer features excellent ferroelasticity with high transition signal and low switching barrier, enabling potential applications in room-temperature nonvolatile memory. Such coexistence of antiferromagneticity and ferroelasticity is of great significance to the study of two-dimensional multiferroics and also renders the AgF₂ monolayer a promising platform for future multifunctional device applications.

Introduction

Multiferroics are singular materials¹ that possess two or three of the so-called ferroic orders: ferromagnetism, ferroelectricity, and ferroelasticity. Their roles are crucial in a great deal of device applications, such as information storage, spintronic devices, non-volatile memory, and magnetic sensors.^2–10 Moreover, exploiting several functionalities in a single material provides an unprecedented opportunity to develop new applications, such as high-density multiscale data storage. For example, Gajek et al.¹¹ demonstrated four-state memory based on multiferroics in La_0.1Bi_0.9MnO₃, which in principle permits data to be electrically written and magnetically read. This not only takes advantage of the best aspects of magnetic data storage and ferroelectric random access memory, but also avoids the shortcomings related to producing the giant local electric fields required to write and read ferroelectric random memory. Therefore, the existence of multiferroics in two-dimensional (2D) materials would be more attractive for miniaturized device applications. Thus far, research on 2D multiferroics has mainly focused on ferromagnetic ferroelectricity,^12–16 and many systems have been identified that belong to this class, such as (CrBr₃)₂Li,¹⁷ CrI₃,¹⁸ Hf₂VC₂F₂,¹⁹ and CuCrP₂S₆.²⁰

There has recently been great interest in 2D ferroelastic materials due to their novel properties that are suitable for a significant number of promising applications such as shape memory, templating electronic nanostructures, superelasticity, and mechanical switches.^21–26 The unique fingerprint of ferroelasticity is the existence of two or more equally stable orientation variants that can be transformed to each other without diffusion by applying external stress.^27,28 Typical examples include phosphorene, phosphorene analogues,²⁹ t-TiN,³⁰ t-YN,³¹ BP₅,³² AgCl,³³ borophane,³⁴ and SnO.³⁵ Furthermore, some of them have demonstrated ferroelectric ferroelasticity or ferromagnetic ferroelasticity.^{29,32,33,35,36} Also, antiferromagnetism has recently been achieved in 2D materials, such as MnPSe₃³⁷ and MnBi₂Te₄.³⁸ Compared with conventional antiferromagnets, 2D antiferromagnetic materials exhibit many emergent behaviors, such as magnon condensates, exotic magnetism, quantum phase transitions, and criticality.³⁹ Considering the merits of 2D ferroelasticity and antiferromagnetism, as well as the fact that multiferroics have been extended to include antiferroic orders in three-dimensional materials,¹ it would be highly interesting to explore whether antiferromagnetic ferroelasticity can exist in 2D materials.⁴⁰

In the present work, on the basis of first principles calculations, we report the discovery of such 2D multiferroics in an AgF₂ monolayer that was found to be dynamically and thermally stable, and required only a small energetic cost to be derived from its bulk, suggesting its high possibility of experimentally achieving exfoliation. An AgF₂ monolayer is an intrinsic magnetic semiconductor with large spin polarization, and its ground state is antiferromagnetic. Moreover, due to its unique structure stemming from the Jahn–Teller distortion, strain-driven 120° variant switching was achieved in an AgF₂ monolayer with a high transition signal, suggesting intrinsic ferroelasticity. The low ferroelastic switching barrier enables 2D ferroelasticity in the AgF₂ monolayer to be eminently accessible in experiments. Therefore, the AgF₂ monolayer is expected to be a promising 2D antiferromagnetic ferroelastic multiferroic. Because of these combined properties, the AgF₂ monolayer is an excellent candidate for future multifunctional applications in a wide range of technologies, in particular, for high-density multiscale data storage.

Method

All calculations based on density functional theory (DFT) were performed using the Vienna ab initio Simulation Package (VASP).⁴¹ The exchange–correlation interaction was treated by generalized gradient approximation (GGA) parametrized by Perdew, Burke, and Ernzerhof (PBE). The ion-electron interaction was treated by the projector augmented wave method (PAW).^42,43 The Monkhorst–Pack k-mesh was set to 5 × 9 × 1, and the kinetic cutoff energy was set to 550 eV. All structures were fully relaxed until the force on each atom was less than 0.01 eV Å⁻¹. The electronic iteration convergence criterion was set to 1 × 10⁻⁶ eV. To avoid spurious interactions between periodic images, a vacuum space of at least 15 Å was introduced. The energy barriers were calculated using the nudged elastic band (NEB) method. The phonon band was calculated using the PHONOPY code.⁴⁴Ab initio molecular dynamics (AIMD) simulations were performed at 500 K and for 3 ps with a time step of 3 fs using a NVT ensemble.

Results

Bulk AgF₂ has been synthesized since 1971 and has also been previously examined.⁴⁵ Fig. S1a (ESI†) shows the crystal structure of bulk AgF₂. The lattice constants were optimized to be a = 5.327 Å, b = 5.733 Å, and c = 5.968 Å, which are consistent with the experimental results (a = 5.073 Å, b = 5.529 Å, and c = 5.813 Å).⁴⁵ The band structure and phonon bands of bulk AgF₂ are shown in Fig. S1b and S3 (ESI†). As shown in Fig. S1a (ESI†), bulk AgF₂ exhibits a layered structure in which the layers are stacked along the c axis by van der Waals interaction. Such a particular structure strongly implies that single-layer AgF₂ might be peeled off from the bulk in a manner similar to that of graphene from graphite.

To quantitatively estimate the experimental feasibility of exfoliating single-layer AgF₂ from its bulk, we calculated the cleavage energy, which was found to be 28 meV Å⁻². This value is quite close to that of graphene (23 meV Å⁻²) and MoS₂ (26 meV Å⁻²),⁴⁶ indicating that the exfoliation of single-layer AgF₂ is highly possible in experiments. Fig. 1a shows the optimized crystal structure of the AgF₂ monolayer. Clearly, its single-layer structure taken from the bulk experiences a significant distortion, leading to the structure shown in Fig. 1a. The AgF₂ monolayer exhibits a tetragonal lattice with the space group of P2_1/c. The lattice constants were found to be a = 6.02 Å and b = 4.15 Å.

Fig. 1 (a) Crystal structure of the AgF₂ monolayer from top and side views. (b) 2D Brillouin zone for the AgF₂ monolayer. (c) Phonon bands of the AgF₂ monolayer. (d) The variation of total free energy during AIMD (500 K, 3 ps) simulations. The inset in (d) shows the crystal structures of the AgF₂ monolayer taken from the end of the AIMD simulations.

It is interesting to stress that the cleavage energy for single-layer AgF₂ is small, although there is significant distortion in the single-layer structure. To understand this feature, we also calculated the cleavage energy of single-layer AgF₂ without the distortion, which was estimated to be only 36.2 meV Å⁻². This value is also small and comparable to that of graphene and MoS₂. The structural distortion further decreases the cleavage energy to 28 meV Å⁻². Such structural distortion arises from the particular structure of bulk AgF₂. As shown in Fig. S2 (ESI†), in bulk AgF₂, each Ag atom is surrounded by six F atoms, wherein four F atoms at 2.12 Å are from the same sheet, and two apical F atoms at 2.75 Å are from two neighboring sheets. When thinning down to a single layer, the absence of two apical F atoms from the neighboring sheets results in the structural distortion.

To verify the stability of the AgF₂ monolayer, we performed phonon spectra calculations and AIMD simulations. As shown in Fig. 1c, the monolayer phonon spectrum shows no obvious imaginary frequency in the entire Brillouin region except for the tiny imaginary frequency (approximately 0.16 THz) near the Γ point, which is negligible. During the AIMD simulation (3 ps at 500 K), neither broken bond nor geometry reconstruction was observed; see Fig. 1d. The fluctuation of total free energy with time is also shown in the inset in Fig. 1d. Based on these results, we confirmed that the AgF₂ monolayer is dynamically and thermally stable. To reflect the bonding characteristic in the AgF₂ monolayer, we calculated its electron localization function (ELF) map, which is plotted in Fig. S5 (ESI†). We found that the electron localizations were mainly distributed around the Ag and F atoms as well as the centers between them, indicating the covalent bonding characteristic for the Ag–F bond.

Due to the spontaneous structural distortion, the crystal structure of the AgF₂ monolayer is significantly different from that of the single layer in bulk AgF₂, which is expected to achieve distinct properties in the AgF₂ monolayer. To this end, we first investigated the magnetic properties of the AgF₂ monolayer. We found that in contrast to the case of bulk AgF₂, interestingly, the AgF₂ monolayer is spin polarized. The magnetic moments are mainly contributed by Ag atoms, while a tiny part originates from F atoms. The local magnetic moment of an Ag atom was found to be 0.39 μ_B. It should be noted that the existence of a magnetic moment does not guarantee the formation of magnetic coupling. Then, we studied the magnetic coupling in the lattice of the AgF₂ monolayer. Because each Ag atom has four nearest-neighboring and two next-neighboring Ag atoms, we considered the following three different magnetic configurations: (1) ferromagnetic coupling (FM) – all magnetic moments are spin-parallel; (2) antiferromagnetic coupling (AFM-1) – the magnetic moments of Ag atom and its nearest-neighboring Ag atoms are spin-antiparallel; (3) antiferromagnetic coupling (AFM-2) – the magnetic moments of Ag atoms and its two next-neighboring Ag atoms are spin-antiparallel.

The spin density distributions for these three magnetic configurations are calculated following the equation: Δρ = ρ↑ − ρ↓, wherein ρ↑ and ρ↓ represent the spin-up and spin-down charge density, respectively. The corresponding results are shown in Fig. 2a. Among these three considered magnetic configurations for the AgF₂ monolayer, the ground state with the lowest energy is AFM-1. It is energetically more stable than the FM (AFM-2) state by 39.3 (36.7) meV per f.u. The underlying mechanism for the AFM-1 coupling in the AgF₂ monolayer is sought in the competition between the through-bond (super-exchange) and through-space (direct-exchange) interactions,^47,48 which have been widely adopted in many previous works on 2D magnetic systems.^49–52 This mechanism can qualitatively describe the interactions between the magnetic moments. For the through-bond (super-exchange) interaction, one Ag atom with down-spin (up-spin) density would induce up-spin (down-spin) density on the adjacent F atom, leading to a ferromagnetic coupling between the adjacent Ag atoms. For the through-space (direct-exchange) interaction, one Ag atom with down-spin (up-spin) density would directly induce up-spin (down-spin) density on the adjacent Ag atom, without the mediation of F atoms, which gives rise to an antiferromagnetic coupling between the adjacent Ag atoms. Both through-bond (super-exchange) and through-space (direct-exchange) interactions sensitively depend on the distance between the adjacent Ag atoms. As shown in Fig. 2a, due to the shorter distance of Ag1 (Ag3)–Ag2 (3.65 Å), the through-space (direct-exchange) interaction plays a dominant role in the magnetic coupling between Ag1 (Ag3) and Ag2. For Ag1–Ag3, the through-bond (super-exchange) interaction plays a dominant role in the magnetic coupling between Ag1 and Ag3, which results from their longer distance (4.14 Å). Based on these results, we can easily understand why the magnetic ground state for AgF₂ monolayer is AFM-1.

Fig. 2 (a) Spin density distributions for the AgF₂ monolayer under different magnetic configurations. The isosurface value is set to 0.2 electrons per ų. Blue and green isosurfaces in (a) correspond to positive and negative spin densities, respectively. (b) Band structures of the AgF₂ monolayer. (c) Projected density of states (PDOS) of the AgF₂ monolayer. The Fermi level was set to 0 eV.

Fig. 2b shows the band structures of the AgF₂ monolayer with the AFM-1 state. Clearly, the AgF₂ monolayer exhibits semiconducting behavior with an indirect bandgap of 0.31 eV, wherein the conduction band minimum (CBM) is located at the X point and the valence band maximum (VBM) lies at the M point. It should be noted that this value is based on the PBE functional, which might be smaller than the experimental value. We further employed the HSE06 functional to obtain a more precise band gap. As listed in Table S1 (ESI†), the band gap based on the HSE06 functional is estimated to be 2.87 eV. Based on the projected density of states shown in Fig. 2c, we found that the VBM and CBM of AgF₂ monolayer are mainly contributed by d orbitals of Ag atoms and p orbitals of F atoms. Accordingly, we demonstrated that the AgF₂ monolayer is a 2D antiferromagnetic semiconductor, which is highly sought for spintronic device applications.

To obtain further insight into this behavior, we also investigated the corresponding properties of single-layer AgF₂ isolated from the bulk without optimization, which is referred to as h-AgF₂. We found that the ground state for the h-AgF₂ monolayer is nonmagnetic. The band structures in Fig. S4 (ESI†) show that it is a conductor. Therefore, the h-AgF₂ monolayer is a nonmagnetic conductor, which is similar to that of its bulk phase; see Fig. S1 (ESI†). The discrepancy between the AgF₂ and h-AgF₂ monolayers can thus be attributed to the spontaneous structural distortion. Upon considering the distortion, it was determined that the coordinated environment of the Ag atoms as well as the ground state were modified.

Considering the particular structure, we next studied the mechanical properties of the AgF₂ monolayer. The AgF₂ monolayer has four independent elastic constants, which are found to be C₁₁ = 36.2 N m⁻¹, C₁₂ = 5.3 N m⁻¹, C₂₂ = 6.65 N m⁻¹, and C₆₆ = 7.61 N m⁻¹. They meet the Born criteria: C₁₁C₂₂ − C₁₂² > 0 and C₆₆ > 0, indicating that the AgF₂ monolayer is mechanically stable. On the basis of the elastic constants, the Young's modulus and Poisson ratio of the AgF₂ monolayer along the in-plane θ can be obtained by the following expression:⁵³

Here,

A = (C₁₁C₂₂ − C₁₂²)/C₆₆ − 2C₁₂,

B = C₁₁ + C₂₂ − (C₁₁C₂₂ − C₁₂²)/C₆₆.

The corresponding angle-dependent Young's modulus and Poisson's ratio of the AgF₂ monolayer are presented in Fig. 3b and d. It can be seen that Y(θ) for the AgF₂ monolayer varies from 6 N m⁻¹ to 32 N m⁻¹, suggesting the significant mechanical anisotropy. Moreover, these values are comparable to that of phosphorene (24–102 N m⁻¹), but remarkably smaller than that of MoS₂ (120–240 N m⁻¹) and graphene (340 N m⁻¹), which ensures their mechanical flexibility and is beneficial for the ferroelastic transition induced by external strain. The Poisson's ratio ν(θ) is used to characterize the transverse synthetic strain of materials under the corresponding axial strain. As shown in Fig. 3d, Poisson's ratio of the AgF₂ monolayer also shows strong anisotropy, and interestingly, a negative Poisson's ratio of −0.075 is achieved.

Fig. 3 (a) Illustration of the Jahn–Teller distortion (elongation) for an octahedral complex and (c) the corresponding evaluations of the splitting of Ag-d orbitals. (b) Young's modulus and (d) Poisson's ratio of AgF₂ monolayer varies with the in-plane angle θ. θ = 0° in (b and d) corresponds to the a axis.

When taking into account its crystal symmetry, 2D ferroelasticity is also expected in the AgF₂ monolayer. For convenience in our discussion, we consider the basic building block, namely the deformed hexagon of the AgF₂ monolayer, as the structural unit. We define three diagonal lines of the deformed hexagon as a₁, a₂, and a₃, respectively, as shown in Fig. 4a. According to the relationship of a₁, a₂, and a₃, three energetically equivalent orientation states can be achieved for the AgF₂ monolayer, i.e., F₁ (a₁ = 8.30 Å and a₂ = a₃ = 7.31 Å), F₂ (a₂ = 8.30 Å and a₁ = a₃ = 7.31 Å), and F₃ (a₃ = 8.30 Å and a₁ = a₂ = 7.31 Å) (Fig. 4a), which can be considered as the three ferroelastic ground states for the AgF₂ monolayer. These three ground states can be transformed into each other. If we take F₁ as the initial ferroelastic ground state, when applying an external stress along the a₁ direction, the longer lattice switches to the a₂ or a₃ axis, that is, the AgF₂ monolayer transforms into the ground state F₂ or F₃. Such behavior does not result in any bond breaking and only imposes a variant transformation, which is equivalent to asymmetry operations, namely a 120° rotation. The T phase of the AgF₂ monolayer was determined as the paraelastic state, which is referred to as the P state (Fig. 4a), with lattice parameters of a₁ = a₂ = a₃ = 7.20 Å. The phonon spectra of the P state are shown in Fig. 4c, from which we can see that it is unstable. This suggests that it would experience spontaneous lattice relaxation and transform into the ground state F₁, F₂, or F₃.

Fig. 4 (a) Schematic diagram of ferroelastic switching among the three different ferroelastic variants of the AgF₂ monolayer. (b) Energy profiles of the ferroelastic transition for the AgF₂ monolayer as a function of step number in NEB calculations. (c) Phonon bands of paraelastic state P.

The underlying mechanism for the spontaneous lattice relaxation can be attributed to the Jahn–Teller distortion. As shown in Fig. 3a, the Ag atom in T-AgF₂ conforms to the octahedral coordination, and the five degenerate d orbitals split into two subsets (i.e., the e_g levels with higher energy and t_2g levels with lower energy) because of the disruption of symmetry under the octahedral crystal field. For one Ag²⁺, there are nine d electrons: six electrons fully occupy the three t_2g orbitals, two electrons fully occupy one e_g orbital, and one electron half-occupies one e_g orbital. This results in one local magnetic moment for one Ag atom. Moreover, the unequivalent distribution of the electrons on the two e_g orbitals would result in the Jahn–Teller distortions, forming a system with lower symmetry and thus further splitting the energy levels; see Fig. 3c. Therefore, the structural phase transition from T-AgF₂ to AgF₂ monolayers spontaneously occurs, exhibiting barrier-free behavior. Due to its three equivalent axes, there are three equivalent distorted ground states for the AgF₂ monolayers.

One important factor that determines the ferroelastic performance of the AgF₂ monolayer is the reversible strain, which relates to the signal intensity of ferroelastic switching. Here, ferroelastic reversible strain is defined as (a_i/a_j − 1) × 100, where a_i ≠ a_j and i/j = 1, 2, and 3. The ferroelastic reversible strain for the AgF₂ monolayer is calculated to be 13.5%, compared with that of t-YN (14.4%)³¹ and GeS (17.8%),²⁹ but significantly smaller than that of BP₅ (41.4%),³² borophane (42%),³⁴ and phosphorene (37.9%).²⁹ It should be mentioned that typical 3D ferroelastic materials normally exhibit a reversible strain ranging from 0.5% to 3%. Such a large value of the AgF₂ monolayer indicates that the 2D ferroelasticity would exhibit a strong ferroelastic switching signal, which is beneficial for practical ferroelastic applications.

To further understand the ferroelastic behaviors of the AgF₂ monolayer, we investigated the ferroelastic transformation pathway using the nudged elastic band (NEB) method. Because the transformation pathways between any two of these three ground states are identical for the AgF₂ monolayer, we will only consider the process from initial state F₁ to final state F₂ as an example. All intermediate states with different lattice sizes between the ferroelastic ground and paraelastic states are automatically created by NEB, and the obtained result is shown in Fig. 4b. Clearly, the paraelastic state P bridges the F₁ and F₂ states. In the interconversion, there is no other metastable state. The overall activation barrier for ferroelastic switching in the AgF₂ monolayer was found to be 51 meV per atom. This value is larger than those of phosphorene analogues (1.3–4.2 meV per atom)²⁹ and t-YN (33 meV per atom),³¹ but smaller than those of phosphorene (200 meV per atom)²⁹ and borophane (100 meV per atom).³⁴ Such a moderate activation barrier suggests that the AgF₂ monolayer would exhibit fast ferroelastic switching when external strain is introduced. This feature, combined with the strong switching signal, enables its excellent 2D ferroelasticity. We thus can conclude that the AgF₂ monolayer is a long-sought 2D antiferromagnetic ferroelastic multiferroic. Concerning the ferroelastic transition process shown in Fig. 4b, we can see another interesting point; that is, the magnetic ground state of the AgF₂ monolayer will be changed during this process. Around the initial state F₁ to final state F₂, the magnetic ground state is AFM-1, while it is FM for the states around the paraelastic state P. It is interesting that the paraelastic structure might be established when including a square substrate with appropriate lattice parameters.

At last, we discuss the ferroelastic domain boundaries of the AgF₂ monolayer. Similar to ferroelectricity and ferromagnetism, the ferroelastic transition is generally subjected to the domain-wall motion.⁵⁴ Additionally, the formation of domain boundaries between adjacent variants might result in intriguing properties. Because there are three different variants for the AgF₂ monolayer, three possible types of boundaries will be produced: F₁–F₂, F₁–F₃, and F₂–F₃. The atomic structures of these three domain boundaries are shown in Fig. 5, in which different ferroelastic domains are highlighted in different colors. These three types of ferroelastic domain boundaries are equivalent and can be transformed to each other by rotating 120 degrees. Unlike two-dimensional domain boundaries in three-dimensional materials, these domain boundaries are quasi-one-dimensional. The formation energy of the domain boundaries was found to be 56 meV Å⁻¹, which is comparable with that of T′-TMDs (27–52 meV Å⁻¹),⁵⁵ indicating that the formation of such a type of domain boundary is possible and stable.

Fig. 5 Crystal structures of the ferroelastic domain boundaries between (a) F₁–F₂, (b) F₁–F₃, and (c) F₂–F₃. Different colors highlight the different ferroelastic domains.

Conclusion

In summary, we systematically investigated the multiferronic properties of an AgF₂ monolayer on the basis of first principles calculations. The AgF₂ monolayer was found to be dynamically and thermally stable, and required only a small energetic cost to be obtained from its bulk. We found that the AgF₂ monolayer is an intrinsic magnetic semiconductor with large spin polarization, and its ground state is antiferromagnetic. The AFM-1 coupling was attributed to the combined effects of through-bond and through-space interaction. Additionally, arising from the structural distortion, the AgF₂ monolayer exhibited a strain-driven 120° variant switching, with a high transition signal and low ferroelastic switching barrier. This enables the 2D ferroelasticity in the AgF₂ monolayer that is desirable for practical applications. We thus demonstrated a 2D antiferromagnetic ferroelastic multiferroic in an AgF₂ monolayer, which holds great potential for multifunctional applications in a wide range of technologies.

Conflicts of interest

The authors declare no competing financial interests.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 11804190), Shandong Provincial Natural Science Foundation of China (No. ZR2019QA011 and ZR2019MEM013), Shandong Provincial Key Research and Development Program (Major Scientific and Technological Innovation Project) (No. 2019JZZY010302), Shandong Provincial Key Research and Development Program (No. 2019RKE27004), Qilu Young Scholar Program of Shandong University, and the Taishan Scholar Program of Shandong Province.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0nh00362j

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