Two-dimensional vanadium tetrafluoride with antiferromagnetic ferroelasticity and bidirectional negative Poisson's ratio

Abstract

Two-dimensional (2D) antiferromagnetic ferroelasticity and bidirectional auxeticity are highly sought for next-generation nanoelectronics, spintronics, or mechanical devices, but are rarely reported. Herein, using first-principles calculations, we report the coexistence of these intriguing properties in the vanadium tetrafluoride (VF₄) monolayer. 2D VF₄ is a direct wide-gap semiconductor with antiferromagnetic order stemming from the synergy between the direct exchange and superexchange of the d_xy orbitals in adjacent V atoms. Notably, 2D VF₄ features ferroelasticity with a moderate reversible strain and low transition barrier, enabling manipulation of the in-plane magnetic easy axis via external strain. Additionally, 2D VF₄ simultaneously harbors an ultra-large negative Poisson's ratio (NPR) in both the in-plane and out-of-plane directions due to the hinged connection between the neighboring VF₆ octahedra. Our work highlights a new material for the exploration of 2D multiferroic physics and the integration of antiferromagnetic ferroelasticity and large bidirectional auxeticity renders 2D VF₄ a versatile candidate for the development of novel multifunctional devices.

---

Introduction

2D multiferroics, referring to the integration of two or more ferroic orders in a singular 2D material, offer the unique opportunity to control one ferroicity via using another and hold great promise for applications in next-generation electronic/spintronic devices.^1–10 Among various ferroic properties, antiferromagnetism (AFM) has aroused great research interest due to the robustness against perturbation of magnetic fields, the absence of parasitic stray fields, ultrahigh resonance frequency, and large magnetotransport effects.^11,12 However, AFM based multiferroics are rarely reported, partially due to the mutual exclusion between ferroelectricity and magnetism, known as the d⁰ rule.^13,14 Ferroelasticity (FA), in contrast, shows better compatibility with other ferroic properties because neither breaking centrosymmetry nor partially filled d orbitals are required.^15,16 The integration of AFM and FA in one material would be ideal for the exploration of AFM based multiferroics. So far, only the weakly-magnetic AgF₂ monolayer is reported with the combination of AFM and FA,¹⁷ but there is no magnetoelastic (AFM–FA) coupling. Therefore, searching for new materials with strong AFM–FA coupling remains a great challenge in the 2D multiferroic world.

Apart from multiferroic properties, materials with NPR, also known as auxetic materials, are another research focus in the 2D regime. When stretched/compressed in the longitudinal direction, auxetic materials will expand/shrink in the transverse direction, providing many novel mechanical properties including enhanced indentation/shear/fracture resistance,^18–20 and acoustic/vibration absorption,^21,22 and promising novel applications in biomedicine, sports equipment, and defense.^21,23 Currently, most reported 2D auxetic materials only demonstrate unidirectional auxeticity along either the in-plane or out-of-plane direction.²⁴ Only 2D X₃M₂ (X = S, Se; M = N, P, As) and Ag₂S were reported with unusual bidirectional auxeticity,^25,26 but none of them is layered in the bulk phase, suggesting a low possibility of experimental fabrication. Especially, the in-plane auxetic performance of X₃M₂ monolayers is poor due to the small NPR value (−0.026). Hence, it is highly desired to explore new 2D auxetic materials with both experimental accessibility and large bidirectional NPRs.

In this work, we predict a novel multifunctional 2D material, namely the VF₄ monolayer. 2D VF₄ with high dynamical, thermal, and mechanical stabilities could be easily exfoliated from the layered bulk counterpart, which is experimentally accessible. The synergy between the direct exchange in neighboring V atoms and superexchange mediated by the F atoms leads to a robust antiferromagnetic order. Notably, the two equally stable lattice variants with different orientations can transfer to each other with a low energy barrier, suggesting intrinsic ferroelasticity. More interestingly, 2D VF₄ exhibits ultra-large bidirectional auxeticity due to the hinged connection between the neighboring VF₆ octahedra.

Computational details

Our density functional theory (DFT) calculations were carried out using the Vienna Ab-initio Simulation Package (VASP).^27–29 The generalized gradient approximation (GGA) of the Perdew–Burke–Ernzerhof (PBE) type functional^30,31 was applied as it can provide an accurate description of the lattice parameters of bulk VF₄ (see Table S1, ESI†). The projector augmented wave (PAW) method was applied to treat the electron–core interactions.³² The van der Waals (vdW) interaction was included by the DFT-D3 method.³³ The cutoff energy for plane waves was set to 520 eV and the Brillouin zone was sampled by a 6 × 6 × 1 mesh. A vacuum space of 15 Å was introduced to avoid the interaction between neighboring images. The structure was fully relaxed until the residual forces and the energy were converged to 0.005 eV Å⁻¹ and 1 × 10⁻⁷ eV, respectively. The electronic structure was calculated using the Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional³⁴ and the finite displacement method was used to calculate the phonon spectrum as implemented in the PHONOPY code.³⁵ The NEB method was applied to study the ferroelastic phase transition³⁶ and 15 images were applied to give a refined transition path and an accurate transition barrier. The convergence of the transition barrier with the number of images is shown in Fig. S1 (ESI†). An ab initio molecular dynamics (AIMD) simulation was carried out for a total of 5 ps with a time step of 1.0 fs at 300 K and the Andersen thermostat was used to control the temperature.³⁷

Results and discussion

Bulk VF₄ crystallizes in a monoclinic structure with the space group P2₁/c and was experimentally synthesized three decades ago.³⁸ The lattice parameters are calculated to be a = 5.43 Å, b = 5.18 Å, c = 5.38 Å, and β = 59.4° at the PBE-D3 level, which are in good agreement with the experimental values (a = 5.38 Å, b = 5.17 Å, c = 5.34 Å, and β = 59.7°).³⁸ As shown in Fig. 1a, bulk VF₄ is a layered material with the VF₄ layers stacked along the [101] direction. The weak van der Waals interlayer bonding suggests that monolayer VF₄ could be exfoliated from the layered bulk counterpart.

Fig. 1 (a) and (c) present the atomic structure of bulk and monolayer VF₄, respectively. (b) and (d) present the cleavage energy and phonon spectrum of the VF₄ monolayer, respectively.

To quantitively explore the possibility of exfoliating monolayer VF₄ from the layered bulk phase, the cleavage energy (E_cl) is calculated, i.e. the energy that is required to overcome the weak interlayer van der Waals bonding.³⁹ As shown in Fig. 1b, E_cl of monolayer VF₄ is only 15 meV Å⁻², which is smaller than that of graphene (21 meV Å⁻²), MoS₂ (26 meV Å⁻²), and phosphorene (23 meV Å⁻²),³⁹ indicating the high feasibility of obtaining a VF₄ monolayer through mechanical exfoliation methods in experiments. As shown in Fig. 1c, the atomic structure of 2D VF₄ consists of VF₆ octahedra with their vertices joined and displays a buckled geometry along the b-direction. The lattice parameters are calculated to be a = 5.39 Å and b = 5.05 Å. To evaluate the stability of 2D VF₄, the phonon spectrum is first calculated as shown in Fig. 1d. Only a tiny imaginary frequency (−0.14 THz) is found around the Gamma point for one of the acoustic modes, indicating the dynamical stability of 2D VF₄.^25,40,41 Furthermore, the total energy and structure of 2D VF₄ only fluctuate slightly during the AIMD simulations for 5 ps as shown in Fig. S2 (ESI†), suggesting its thermal stability. Additionally, the calculated elastic constants for 2D VF₄ are C₁₁ = 34.81 N m⁻¹, C₂₂ = 18.90 N m⁻¹, C₁₂ = 14.84 N m⁻¹, and C₆₆ = 30.58 N m⁻¹, respectively. As C₁₁, C₆₆ > 0 and C₁₁ × C₂₂ > C₁₂², the Born mechanical stability criteria for 2D materials are fully fulfilled.⁴²

2D VF₄ may possess magnetism due to the unpaired electron in the V⁴⁺ ions. To explore this effect, four magnetic configurations (denoted as FM, AFM-1, AFM-2, and AFM-3)^13,43 in a larger supercell (2 × 2 × 1) are considered for 2D VF₄ and the corresponding spin density distributions are presented in Fig. 2. AFM-2 ordering with the nearest magnetic moments antiferromagnetically aligned is the energetically most favorable magnetic configuration (see Table S2, ESI†). The spin density is localized around the V cations with a local magnetic moment of 1 μ_B per atom. The magnetic anisotropy energy (MAE) is further evaluated in the presence of spin–orbit coupling (SOC). Three in-plane directions (the [100], [010], and [110] directions) and four out-of-plane directions (the [111], [101], [011], and [001] directions) are considered and the calculated energies along these directions are summarized in Table S3 (ESI†). Clearly, the easy magnetization-axis of the VF₄ monolayer lies along the longer lattice vector direction, i.e. the [100] direction, showing an in-plane magnetization feature. The largest MAE reaches up to 40 μeV per V atom, which is much larger than those of Fe (1 μeV per Fe atom) and Ni (3 μeV per Ni atom)⁴⁴ and comparable to other 2D antiferromagnets like Mn₂C (69 μeV per Mn atom)⁴⁵ and VOCl₂ (17 μeV per V atom).⁴³

Fig. 2 Spin density distribution of the VF₄ monolayer under different magnetic configurations. The isosurface value is set to 0.02 electrons per ų. The blue and green isosurfaces represent the spin-up and spin-down densities, respectively.

To understand the origin of the antiferromagnetic ordering, we first need to know the electronic structure of 2D VF₄. Fig. 3 presents the spin-polarized and orbital-resolved band structures calculated using the HSE06 functional. 2D VF₄ is a direct semiconductor with a wide bandgap of 3.96 eV. Crystal field theory indicates that the d orbitals in a perfect octahedral environment shall split into double-degenerate e_g and triple-degenerate t_2g orbitals.⁴⁶ However, as shown in Fig. 3e, the V–F bonds within the xy-plane are significantly longer than those along the z-direction, suggesting the existence of Jahn–Teller distortion.⁴⁷ Such a distortion further splits the d orbitals into four subgroups as revealed by the orbital-resolved band structures (see Fig. 3c and d): the d_xy, d_x²−y², and d_z² singlets, and the d_xz and d_yz doublet, with the lowest d_xy orbital occupied.

Fig. 3 (a) and (b) Spin-up and spin-down band structures of monolayer VF₄, respectively. (c) and (d) Orbital projected band structures for V atoms. (e) Local structure of the VF₆ octahedron and the d orbital splitting under the octahedral crystal field and Jahn–Teller distortion. (f) Schematic illustration of the direct d–d exchange and the d–p–d superexchange interaction between the adjacent V atoms. The arrows represent the orientation of spin.

Based on the above orbital analysis, we can establish the magnetic exchange paths along the diagonal direction of V–F–V chains, as shown in Fig. 3f. One typical exchange interaction is antiferromagnetic direct exchange between adjacent d_xy orbitals of V atoms, obeying the Pauli exclusion principle.⁴⁸ Another one is the F-mediated superexchange interaction following the d_xy–p_x/y–d_xy linear paths, which also favors antiparallel spin alignments in terms of the Goodenough–Kanamori–Anderson rule.^49–51 The superexchange interaction can be further revealed by the squared real-space wavefunction of the highest valence band at the gamma point (see Fig. S3, ESI†), which clearly shows the characteristics of V: d_xy and F: p_x/y orbitals. The coexistence of the above two exchange interactions results in the AFM-2 ordering in the ground state.

The unique atomic geometries in the VF₄ monolayer also enable the ferroelastic transition between the two energy-equivalent states: the a > b state (S) and the a < b state (S′). The nudged elastic band (NEB) method is used to investigate the ferroelastic transformation pathway, where both the lattice and atomic positions are spontaneously optimized to obtain an accurate energy barrier. As shown in Fig. 4, the S and S′ states are connected by an intermediate paraelastic state (a = b = 5.28 Å). Using this state as a reference, the transformation strain matrix calculated within Green–Lagrange strain tensor theory⁵² indicates that there is a 2.1% tensile strain along the x/y-direction and a 4.3% compressive strain along the y/x-direction for the S/S′ state. The overall transition barrier is only 43 meV per unit cell and the reversible strain defined as (|b|/|a| − 1) × 100% is calculated to be 6.7%, which are comparable to those of phosphorene analogues such as SnS and SnSe.⁵ The low switching barrier suggests high feasibility of experimental realization and the moderate reversible strain can avoid structural failure before switching is achieved.⁵³ Therefore, 2D VF₄ is a multiferroic material with both antiferromagnetic order and intrinsic ferroelasticity. Moreover, the in-plane magnetic easy axis is strongly coupled to the lattice direction (along the longer lattice vector), making it possible to tune the spin orientation via strain. As shown in Fig. 4, the easy axis will rotate 90° when S is transferred to S′, and thus the spin orientation will also experience a 90° rotation during the ferroelastic transition. Such a feature is expected to offer new opportunity for the design of controllable spintronic devices.

Fig. 4 The energy profiles of the ferroelastic transition for single layer VF₄ in NEB calculations. The insets show the initial state (a > b), the intermediate state (a = b), and the final state (a < b) during the ferroelastic transition. The blue arrows indicate the spin orientations in the S and S′ state.

Then the in-plane Poisson's ratio v_in as a function of θ is calculated as shown in Fig. 5a using the following equation:⁵⁴

where A = (C₁₁C₂₂ − C₁₂²)/C₆₆ − 2C₁₂ and B = C₁₁ + C₂₂ − (C₁₁C₂₂ − C₁₂²)/C₆₆. Clearly, v_in exhibits anisotropy with two local maxima along the x and y-directions, respectively. Notably, an NPR as high as −0.26 can be found when strain is applied along the diagonal directions. The out-of-plane Poisson's ratio (v_out) is also explored by checking the resultant strain (ε_rs) under applied uniaxial strains (ε_x/y). As shown in Fig. 5b, the thickness of the VF₄ monolayer increases linearly with an increase in uniaxial strain along the y-direction, indicating an out-of-plane NPR. v_out is then obtained by fitting ε_rs = −v_out × ε_y and can reach up to −0.62. Hence, the VF₄ monolayer possesses large bidirectional auxeticity, which is extremely scarce in the 2D regime. Compared to previously reported 2D auxetic materials (see Fig. S4, ESI†), both v_in and v_out in 2D VF₄ are among the highest NPR values ever reported, making it a promising candidate for nanomechanical devices.

Fig. 5 (a) The in-plane Poisson's ratio as a function of θ for 2D VF₄. θ = 0° represents the x-direction. The NPR region is highlighted in red. (b) Resultant out-of-plane strain as a function of applied strain along the y-direction. The response of ε_z to ε_x is plotted in Fig. S5a (ESI†), showing a positive Poisson's ratio. (c) and (d) The schematic diagram showing the atomic mechanism for in-plane and out-of-plane auxeticity, respectively.

Our further analysis indicates that the bidirectional NPR in 2D VF₄ is attributed to the hinged connection between adjacent VF₆ octahedra. As shown in Fig. 5c, when stretched along one diagonal direction, the buckled –V–F–V– chain along that direction tends to align linearly, leading to the counterclockwise/clockwise rotation of the V_IF₆/V_IIF₆ octahedron. Such rotation pushes away the octahedra along the other diagonal direction, resulting in the in-plane auxeticity. Similarly, tensile strain along the y-direction would give rise to the counterclockwise/clockwise rotation of the V_IF₆/V_IIF₆ octahedron within the yz-plane (see Fig. 5d), leading to a decrease in the angle between F_I–F_II and the z-axis (θ). Meanwhile, the distance between F_I and F_II (l) barely changes (see Fig. S5b, ESI†) because the rotation of VF₆ octahedra counteracts its structural variation in the presence of strain. Therefore, the thickness of 2D VF₄ (h), defined as h = l × cos θ, would increase when stretching along the y-direction, resulting in the out-of-plane NPR.

Conclusions

In conclusion, we theoretically predicted the coexistence of antiferromagnetism, ferroelasticity, and bidirectional auxeticity in a stable VF₄ monolayer. 2D VF₄ could be easily exfoliated from the layered bulk phase due to the very low cleavage energy (15 meV Å⁻²). The Jahn–Teller distortion leads to a single-electron occupied d_xy orbital, and the coexistence of direct exchange and superexchange between adjacent d_xy orbitals results in long-range antiferromagnetic order. The magnetic easy axis lies along the x-direction, which can be switched to the y-direction through the ferroelastic phase transition, indicating robust magnetoelastic coupling. Besides, the hinged joints between adjacent VF₆ octahedra enable bidirectional ultra-large NPRs (−0.26 and −0.62 along the in-plane and out-of-plane directions, respectively). The predicted VF₄ monolayer can be an ideal platform to explore novel 2D multiferroics and holds great potential for the development of nanosized electronic, spintronic, and mechanic devices.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

We acknowledge generous grants of high-performance computing resources provided by NCI National Facility and the Pawsey Supercomputing Centre through the National Computational Merit Allocation Scheme supported by the Australian Government and the Government of Western Australia. A. D. also greatly appreciates the financial support of the Australian Research Council under Discovery Project (DP170103598).

References

1. H. Wang, X. Li, J. Sun, Z. Liu and J. Yang, 2D Mater., 2017, 4, 045020.
2. H. Wang and X. Qian, 2D Mater., 2017, 4, 015042.
3. Q. Yang, W. Xiong, L. Zhu, G. Gao and M. Wu, J. Am. Chem. Soc., 2017, 139, 11506–11512.
4. J.-J. Zhang, L. Lin, Y. Zhang, M. Wu, B. I. Yakobson and S. Dong, J. Am. Chem. Soc., 2018, 140, 9768–9773.
5. M. Wu and X. C. Zeng, Nano Lett., 2016, 16, 3236–3241.
6. C. Gong, E. M. Kim, Y. Wang, G. Lee and X. Zhang, Nat. Commun., 2019, 10, 2657.
7. Y. Zhao, L. Lin, Q. Zhou, Y. Li, S. Yuan, Q. Chen, S. Dong and J. Wang, Nano Lett., 2018, 18, 2943–2949.
8. Y. Gao, M. Wu and X. C. Zeng, Nanoscale Horiz., 2019, 4, 1106–1112.
9. T. Hu and E. Kan, Wiley Interdiscip. Rev.: Comput. Mol. Sci., 2019, 9, e1409.
10. C. Huang, Y. Du, H. Wu, H. Xiang, K. Deng and E. Kan, Phys. Rev. Lett., 2018, 120, 147601.
11. M. B. Jungfleisch, W. Zhang and A. Hoffmann, Phys. Lett. A, 2018, 382, 865–871.
12. V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono and Y. Tserkovnyak, Rev. Mod. Phys., 2018, 90, 015005.
13. H. Tan, M. Li, H. Liu, Z. Liu, Y. Li and W. Duan, Phys. Rev. B, 2019, 99, 195434.
14. L. Weston, X. Y. Cui, S. P. Ringer and C. Stampfl, Phys. Rev. B, 2016, 93, 165210.
15. N. A. Hill, J. Phys. Chem. B, 2000, 104, 6694–6709.
16. N. A. Spaldin, S.-W. Cheong and R. Ramesh, Phys. Today, 2010, 63, 38–43.
17. X. Xu, Y. Ma, T. Zhang, C. Lei, B. Huang and Y. Dai, Nanoscale Horiz., 2020, 5, 1386–1393.
18. I. I. Argatov, R. Guinovart-Díaz and F. J. Sabina, Int. J. Eng. Sci., 2012, 54, 42–57.
19. J. B. Choi and R. S. Lakes, J. Mater. Sci., 1992, 27, 5375–5381.
20. J. Choi and R. Lakes, Int. J. Fract., 1996, 80, 73–83.
21. M. Mir, M. N. Ali, J. Sami and U. Ansari, Adv. Mater. Sci. Eng., 2014, 753496.
22. Y. Prawoto, Comput. Mater. Sci., 2012, 58, 140–153.
23. T. Allen, N. Martinello, D. Zampieri, T. Hewage, T. Senior, L. Foster and A. Alderson, Procedia Eng., 2015, 112, 104–109.
24. R. Peng, Y. Ma, Q. Wu, B. Huang and Y. Dai, Nanoscale, 2019, 11, 11413–11428.
25. R. Peng, Y. Ma, Z. He, B. Huang, L. Kou and Y. Dai, Nano Lett., 2019, 19, 1227–1233.
26. Y. Chen, X. Liao, X. Shi, H. Xiao, Y. Liu and X. Chen, Phys. Chem. Chem. Phys., 2019, 21, 5916–5924.
27. G. Kresse and J. Furthmüller, Comput. Mater. Sci., 1996, 6, 15–50.
28. G. Kresse and J. Furthmüller, Phys. Rev. B: Condens. Matter Mater. Phys., 1996, 54, 11169–11186.
29. G. Kresse and J. Hafner, Phys. Rev. B: Condens. Matter Mater. Phys., 1994, 49, 14251–14269.
30. J. P. Perdew, M. Ernzerhof and K. Burke, J. Chem. Phys., 1996, 105, 9982–9985.
31. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865.
32. G. Kresse and D. Joubert, Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, 1758–1775.
33. S. Grimme, J. Comput. Chem., 2006, 27, 1787–1799.
34. A. V. Krukau, O. A. Vydrov, A. F. Izmaylov and G. E. Scuseria, J. Chem. Phys., 2006, 125, 224106.
35. A. Togo and I. Tanaka, Scr. Mater., 2015, 108, 1–5.
36. G. Mills, H. Jónsson and G. K. Schenter, Surf. Sci., 1995, 324, 305–337.
37. H. C. Andersen, J. Chem. Phys., 1980, 72, 2384–2393.
38. S. Becker and B. G. Müller, Angew. Chem., Int. Ed. Engl., 1990, 29, 406–407.
39. J. H. Jung, C.-H. Park and J. Ihm, Nano Lett., 2018, 18, 2759–2765.
40. B. Zhang, L. Fan, J. Hu, J. Gu, B. Wang and Q. Zhang, Nanoscale, 2019, 11, 7857–7865.
41. B. Zhang, G. Song, J. Sun, J. Leng, C. Zhang and J. Wang, Nanoscale, 2020, 12, 12490–12496.
42. F. Mouhat and F.-X. Coudert, Phys. Rev. B: Condens. Matter Mater. Phys., 2014, 90, 224104.
43. H. Ai, X. Song, S. Qi, W. Li and M. Zhao, Nanoscale, 2019, 11, 1103–1110.
44. G. H. O. Daalderop, P. J. Kelly and M. F. H. Schuurmans, Phys. Rev. B: Condens. Matter Mater. Phys., 1990, 41, 11919–11937.
45. L. Hu, X. Wu and J. Yang, Nanoscale, 2016, 8, 12939–12945.
46. R. G. Burns and R. G. Burns, Mineralogical applications of crystal field theory, Cambridge university press, Cambridge, 2nd edn, 1993.
47. M. A. Halcrow, Chem. Soc. Rev., 2013, 42, 1784–1795.
48. J. B. Goodenough and A. L. Loeb, Phys. Rev., 1955, 98, 391–408.
49. J. B. Goodenough, Magnetism and the Chemical Bond, John Wiley & Sons, Inc., New York, London, 1st edn, 1963.
50. J. Kanamori, J. Phys. Chem. Solids, 1959, 10, 87–98.
51. F. Zhang, Y.-C. Kong, R. Pang, L. Hu, P.-L. Gong, X.-Q. Shi and Z.-K. Tang, New J. Phys., 2019, 21, 053033.
52. W. Li and J. Li, Nat. Commun., 2016, 7, 10843.
53. Z. Tu and M. Wu, Sci. Bull., 2020, 65, 147–152.
54. H. Wang, X. Li, P. Li and J. Yang, Nanoscale, 2017, 9, 850–855.

---

Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0tc04846a

---