# Magnetic 2D materials

Magnetic 2D materials are two-dimensional materials that display ordered magnetic properties such as antiferromagnetism or ferromagnetism. After the discovery of graphene in 2004, the family of 2D materials has grown rapidly. Since then, there have been several reports of related materials, except for magnetic materials. But then there was a report of what this then unreported class of materials might hold in store.[1] This new class of materials has since been known as magnetic van der Waals materials, van der Waals magnets or more romantically magnetic graphene.

The first report of van der Waals magnets were made on antiferromagnetic materials of TMPS3 with TM = Fe and Ni in 2016.[2][3] Then in 2017 the first 2D ferromagnets, CrI3.[4] and Cr2Ge2Te6,[5] were fabricated.[6] One reason for this is that thermal fluctuations tend to destroy magnetic order easier compared to 3D bulk.[7] Low dimensional materials have different magnetic properties compared to bulk, and thus transition from 3D to 2D magnetism can be measured. This transition has been observed in Fe3GeTe2.[8]

Although the field has been around only a few years, it has become one of the most active field in condensed matter physics and materials science and engineering. There have been several review articles written up to highlight its future and promise.[9][10][11][7]

## Theory

Magnetic materials have their (spins) aligned over a macroscopic length scale. Alignment of the spins is typically driven by exchange interaction between neighboring spins. While at absolute zero (${\displaystyle T=0}$) the alignment can always exist, thermal fluctuations misalign magnetic moments at temperatures above the Curie temperature (${\displaystyle T_{C}}$), causing a phase transition to a non-magnetic state. Whether ${\displaystyle T_{C}}$ is above the absolute zero depends heavily on the dimensions of the system.

For a 3D system, the Curie temperature is always above zero, while a one-dimensional system can only be in a ferromagnetic state at ${\displaystyle T=0}$[12]

For 2D systems, the transition temperature depends on the spin dimensionality (${\displaystyle n}$).[7] In system with ${\displaystyle n=1}$, the planar spins can be oriented either in or out of plane. A spin dimensionality of two means that the spins are free to point in any direction parallel to the plane. A system with a spin dimensionality of three means there are no constraints on the direction of the spin. A system with ${\displaystyle n=1}$ is described by the 2D Ising model. Onsager's solution to the model demonstrates that ${\displaystyle T_{C}>0}$, thus allowing magnetism at obtainable temperatures. On the contrary, a system where ${\displaystyle n=3}$, described by the isotropic Heisenberg model, never displays magnetism at any finite temperature. The long range ordering of the spins is prevented by the Mermin-Wagner theorem stating that spontaneous symmetry breaking required for magnetism is not possible in isotropic two dimensional magnetic systems. Spin waves in this case have finite density of states and are gapless and are therefore easy to excite, destroying magnetic order. Therefore, an external source of magnetocrystalline anisotropy, such as external magnetic field, is required for materials with ${\displaystyle n=3}$ to demonstrate magnetism.

The 2D ising model describes the behavior of CrI3.[4] and Fe3GeTe2,[8] while Cr2Ge2Te6[5] behaves like isotropic Heisenberg model. The intrinsic anisotropy in CrI3 and Fe3GeTe2 is caused by strong spin-orbit coupling, allowing them to remain magnetic down to a monolayer, while Cr2Ge2Te6 has only exhibit magnetism as a bilayer or thicker.

The XY model describes the case where ${\displaystyle n=2}$. In this system, there is no transition between the ordered and unordered states, but instead the system undergoes a so called Kosterlitz–Thouless transition at finite temperature ${\displaystyle T_{KT}}$, where at temperatures below ${\displaystyle T_{KT}}$ the system has quasi-long-range magnetic order.

The above systems can be described by a generalized Heisenberg spin Hamiltonian:

${\displaystyle H=-{\frac {1}{2}}\sum _{}(J\mathbf {S} _{i}\cdot \mathbf {S} _{j}+\Lambda S_{j}^{z}S_{i}^{z})-\sum _{i}A(S_{i}^{z})^{2}}$,

Where ${\displaystyle J}$ is the exchange coupling between spins ${\displaystyle \mathbf {S} _{i}}$ and ${\displaystyle \mathbf {S} _{j}}$, and ${\displaystyle A}$ and ${\displaystyle \Lambda }$ are on-site and inter-site magnetic anisotropies, respectively. Setting ${\displaystyle A\rightarrow \pm \infty }$ recoveres the 2D Ising model and the XY model. (positive sign for ${\displaystyle n=1}$ and negative for ${\displaystyle n=2}$), while ${\displaystyle A\approx 0}$ and ${\displaystyle \Lambda \approx 0}$ recovers the Heisenberg model (${\displaystyle n=3}$). Along with the idealized models described above, the spin Hamiltonian can be used for most experimental setups,[13] and it can also model dipole-dipole interactions by renormalization of the parameter ${\displaystyle A}$.[7] However, sometimes including further neighbours or using different exchange coupling, such as antisymmetric exchange, is required.[7]

## Measuring two-dimensional magnetism

Magnetic properties of two-dimensional materials are usually measured using Magneto-optic Kerr effect, Magnetic circular dichroism or Anomalous Hall effect techniques.[7] The dimensionality of the system can be determined by measuring the scaling behaviour of magnetization (${\displaystyle M}$), susceptibility (${\displaystyle \chi }$) or correlation length (${\displaystyle \xi }$) as a function of temperature. The corresponding critical exponents are ${\displaystyle \beta }$, ${\displaystyle \gamma }$ and ${\displaystyle v}$ respectively. They can be retrieved by fitting

${\displaystyle M(T)\propto (1-T/T_{\text{C}})^{\beta }}$,
${\displaystyle \chi (T)\propto (1-T/T_{\text{C}})^{-\gamma }}$ or
${\displaystyle \xi (T)\propto (1-T/T_{\text{C}})^{-v}}$

to the data. The critical exponents depend on the system and its dimensionality, as demonstrated in Table 1. Therefore, an abrupt change in any of the critical exponents indicates a transition between two models. Furthermore, the Curie temperature can be measured as a function of number of layers (${\displaystyle N}$). This relation for a large ${\displaystyle N}$ is given by[14]

${\displaystyle T_{\text{C}}(N)/T_{\text{C}}^{\text{3D}}=1-(C/N)^{\frac {1}{v}}}$,

where ${\displaystyle C}$ is a material dependent constant. For thin layers, the behavior changes to ${\displaystyle T_{\text{C}}\propto N}$[15]

Table 1: Critical exponents for two and three dimensional Ising models
Model ${\displaystyle \beta }$ ${\displaystyle \gamma }$ ${\displaystyle v}$
2D Ising 0.125 1.75 1
3D Ising 0.3265 1.237 0.630

## Applications

Magnetic 2D materials can be used as a part of van der Waals heterostructures. They are layered materials consisting of different 2D materials held together by van der Waals forces. One example of such structure is a thin insulating/semiconducting layer between layers of 2D magnetic material, producing a magnetic tunnel junction. This structure can have significant spin valve effect,[16] and thus they can have many applications in the field of spintronics.

## References

1. ^ Je-Geun, Park (2016). "Opportunities and challenges of two-dimensional magnetic van der Waals materials: magnetic graphene?". J. Phys. Condens. Matter. 28: 301001.
2. ^ Kuo, Cheng-Tai; et al. (2016). "Exfoliation and Raman Spectroscopic Fingerprint of Monolayer and Few-Layer NiPS3 Van der Waals Crystals". Scientific Reports. 6: 20904. doi:10.1038/srep20904.
3. ^ Jae-Ung, Lee; et al. (2016). "Ising-Type Magnetic Ordering in Atomically Thin FePS3". Nano Lett. 16: 7433. doi:10.1021/acs.nanolett.6b03052.
4. ^ a b Huang, B.; et al. (2017). "Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit". Nature. 546 (7657): 270–273. arXiv:1703.05892. Bibcode:2017Natur.546..270H. doi:10.1038/nature22391. PMID 28593970. S2CID 4456526.
5. ^ a b Gong, C.; et al. (2017). "Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals". Conference on Lasers and Electro-Optics (CLEO), San Jose, CA. 546 (7657): 1–2. arXiv:1703.05753. Bibcode:2017Natur.546..265G. doi:10.1038/nature22060. PMID 28445468. S2CID 2633044.
6. ^ Samarth, N. (2017). "Magnetism in flatland". Nature. 546 (7657): 216–217. doi:10.1038/546216a. PMID 28593959.
7. Gibertini, M.; et al. (2019). "Magnetic 2D materials and heterostructures". Nature Nanotechnology. 14 (5): 408–419. arXiv:1910.03425. Bibcode:2019NatNa..14..408G. doi:10.1038/s41565-019-0438-6. PMID 31065072. S2CID 205568917.
8. ^ a b Fei, Z.; et al. (2018). "Two-dimensional itinerant ferromagnetism in atomically thin Fe3GeTe2". Nature Materials. 17 (9): 778–782. arXiv:1803.02559. Bibcode:2018NatMa..17..778F. doi:10.1038/s41563-018-0149-7. PMID 30104669. S2CID 51972811.
9. ^ Burch, Kenneth; Mandrus, David; Park, Je-Geun (2018). "Magnetism in two-dimensional van der Waals materials". Nature. 563: 47. doi:10.1038/s41586-018-0631-z.
10. ^ M., Gibertini (2019). "Magnetic 2D materials and heterostructures". Nature Nanotechnology. 14: 408. doi:10.1038/s41565-019-0438-6.
11. ^ Cheng, Gong (2019). "Two-dimensional magnetic crystals and emergent heterostructure devices". Science. 363: 4450. doi:10.1126/science.aav4450.
12. ^ Peierls, R. (1936). "On Ising's model of ferromagnetism". Proceedings of the Cambridge Philosophical Society. 32 (3): 477–481. Bibcode:1936PCPS...32..477P. doi:10.1017/S0305004100019174.
13. ^ de Jongh, L. J. (1990). Magnetic Properties of Layered Transition Metal Compounds (Vol. 9, 1 ed.). Netherlands: Springer. ISBN 978-94-009-1860-3.
14. ^ Fisher, M. E.; Barber, M. N. (1972). "Scaling theory for finite-size effects in critical region". Phys. Rev. Lett. 28 (23): 1516–1519. Bibcode:1972PhRvL..28.1516F. doi:10.1103/PhysRevLett.28.1516.
15. ^ Zhang, R. J.; Willis, R. F. (2001). "Thickness-dependent Curie temperatures of ultrathin magnetic films: Effect of the range of spin-spin interactions". Phys. Rev. Lett. 86 (12): 2665–2668. Bibcode:2001PhRvL..86.2665Z. doi:10.1103/PhysRevLett.86.2665. PMID 11290006.
16. ^ Wang, Z.; et al. (2018). "Tunneling spin valves based on Fe3GeTe2/hBN/Fe3GeTe2 van der Waals heterostructures". Nano Lett. 18 (7): 4303–4308. arXiv:1806.05411. Bibcode:2018NanoL..18.4303W. doi:10.1021/acs.nanolett.8b01278. PMID 29870263. S2CID 206747719.