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Long Range Intrinsic Ferromagnetism in Two Dimensional Materials and Dissipationless Future Technologies

The inherent susceptibility of low-dimensional materials to thermal fluctuations has long been expected to poses a major challenge to achieving intrinsic long-range ferromagnetic order in two-dimensional materials. The recent explosion of interest in atomically thin materials and their assembly into van der Waals heterostructures has renewed interest in two-dimensional ferromagnetism, which is interesting from a fundamental scientific point of view and also offers a missing ingredient necessary for the realization of spintronic functionality in van der Waals heterostructures. Recently several atomically thin materials have been shown to be robust ferromagnets. Such ferromagnetism is thought to be enabled by magneto crystalline anisotropy which suppresses thermal fluctuations. In this article, we review recent progress in two-dimensional ferromagnetism in detail and predict new possible two-dimensional ferromagnetic materials. We also discuss the prospects for applications of atomically thin ferromagnets in novel dissipationless electronics, spintronics, and other conventional magnetic technologies. Particularly atomically thin ferromagnets are promising to realize time reversal symmetry breaking in two-dimensional topological systems, providing a platform for electronic devices based on the quantum anomalous Hall Effect showing dissipationless transport. Our proposed directions will assist the scientific community to explore novel two-dimensional ferromagnetic families which can spawn new technologies and further improve the fundamental understanding of this fascinating area.

providing a platform for electronic devices based on the quantum anomalous Hall Effect showing dissipationless transport. Our proposed directions will assist the scientific community to explore novel two-dimensional ferromagnetic families which can spawn new technologies and further improve the fundamental understanding of this fascinating area. Ferromagnetism (FM) is a physical phenomenon realized in certain materials that retain a permanent magnetic moment in the absence of any external magnetic field as long as the temperature is below a critical value, generally known as the Curie temperature "T c ".
Ferromagnetic materials have long been recognized for their technological importance, beginning with the compass, and are still essential today in electric motors and generators and non-volatile information storage. While ferromagnetism in three dimensions (3D) can be achieved with T c much greater than room temperature, the spontaneous rotational symmetry breaking associated with isotropic ferromagnetism is strictly forbidden in two dimensions (2D). While certain anisotropic models are known to exhibit stable ferromagnetism in 2D, it is generally recognized that the increased instability of 2D materials to thermal fluctuations poses challenges to realizing ferromagnetism with high T c .
The past one and a half decades have seen an explosion of interest in atomically thin materials since the isolation of graphene, an atom-thick plane of carbon 1 . This class of materials is also known as van der Waals materials due to the weak bonding between layers, making these materials compatible with a variety of substrates and also opening a vast new field of "van der Waals heterostructures" which combine different atomically thin materials to form new structures with new properties 2 . Atomically thin materials have been discovered with a wide range of properties: metals, semiconductors, insulators, superconductors, charge-density-wave materials, and ferroelectrics. Ferromagnetism was notably absent from this list until recent discoveries of atomically thin ferromagnets Cr 2 Ge 2 Te 6 3 , CrI 3 4 , and VSe 2 5 .
The discovery of intrinsic ferromagnetism in 2D materials is vital for understanding spin behavior in low dimensions and could enable novel spintronics applications ranging from molecular quantum devices and sensing to high-density data storage ultrathin devices [6][7][8][9] . 2D ferromagnets may also be essential components in proposed heterostructures which can exhibit the quantum anomalous Hall effect at high temperatures, highly desirable for applications in dissipationless electronics and spintronics.
Since the discovery of graphene 10 , several techniques have been proposed to extrinsically induce long-range magnetic order into 2D materials such as defect engineering 11 , absorption of magnetic ions 12 or proximity effect 13 . In defect engineering, the simplest strategy for realizing magnetism in non-magnetic materials is to create unpaired electrons by modifying its electronic structure via vacancies 14,15 , adatoms 12 , grain boundaries 16 or edges 17  On the other hand, proximity induced ferromagnetic systems have more advantages and promise preservation of the original crystalline structure, better controllability of the electronic states, uniform magnetization, and uniformly distributed band gap in space, etc.
However, proximity induced magnetic responses are limited in 2D materials. These potential scarce in externally induced ferromagnetism motivates to find ferromagnetism originating intrinsically from the parent 2D lattice.
However, little attention has been drawn to investigate the intrinsic ferromagnetism in two-dimensional "2D" materials because the long-range magnetic order in 2D systems could be destroyed by thermal fluctuations, for example the Mermin-Wagner theorem indicates that long-range magnetic order is impossible at any finite temperature in an isotropic 2D spin system [21][22][23] . To suppress the thermal fluctuations and achieve long range intrinsic ferromagnetism in 2D materials, magnetic anisotropy is necessary. Based on various forms of magnetic anisotropies, recent groundbreaking experimental realizations of intrinsic ferromagnetism in Cr 2 Ge 2 Te 6, CrI 3 , and especially VSe 2 at room temperature have been achieved. Surprisingly, monolayer VSe 2 has been revealed to be ferromagnetic while 3D VSe 2 is paramagnetic, pointing out that mono-layers or two-dimensional (2D) structures can reveal distinctive properties in contrast to their multi-layers or three dimensional (3D) structures.
This review focuses on intrinsic ferromagnetism in 2D materials. By "intrinsic ferromagnetism", we intend to limit the discussion to ferromagnetism occurring in a stoichiometric compound or heterostructure of stoichiometric compounds, as opposed to defect or dopant induced ("dilute") ferromagnetism. By "2D material" we intend those materials which can be stably isolated in few atom thick layers; typically these materials consist of covalently bonded layers which interact with each other through weaker van der Waals bonding. This paper also explores the role of 2D ferromagnetism in various topological phases of quantum matter. The quantum anomalous Hall effect, which enables robust dissipationless electronic transport over macroscopic distances, is entangled with long-range intrinsic ferromagnetism in 2D materials 24,25 . The realization of quantum anomalous Hall effect through magnetically doped topological insulators will be discussed within the Landau framework of phase transitions, especially emphasizing time reversal symmetry breaking and the associated ferromagnetic order.
Finally, this timely review will help the community to discover novel magnetic materials that can lead to ultrathin 2D devices which will be more efficient, compact, and could be governed by using precise and reliable systems. Due to many available reviews about ferromagnetism in 3D and/or their other associated phenomena in literature 9,23,[26][27][28] , this review focuses solely on intrinsic ferromagnetism in 2D materials, and especially draws attention to proposed materials, i.e. which are mainly predicted by density functional theory (DFT) calculations, as well as experimentally realized 2D materials and future directions.

II. ROLE OF MAGNETIC ANISOTROPY IN 2D FERROMAGNETIC MATERIALS
Magnetic anisotropy is the dependence of the magnetic properties on the relative orientation of the applied magnetic field with respect to the crystal lattice. Magnetic anisotropy is an essential property of magnetic materials, having significance ranging from fundamental to technological, for example, magnetic anisotropy is essential for aligning the magnetic moments in magnetic storage media. Magnetic anisotropy is also a prerequisite to observe ferromagnetism in 2D materials 29 . There are different types of magnetic anisotropy such as shape anisotropy, exchange anisotropy, magnetoelastic anisotropy and most importantly magnetocrystalline anisotropy 29 . Magnetocrystalline anisotropy is caused by the spin-orbit coupling which couples the spin orientation to the orientation of the electron orbitals 30 . The structure becomes important when the average magnetization of a specimen as a function of an applied external field is considered since the magnetization depends on the orientation of the magnetic field with respect to the crystal axes. Additionally, external factors such as pressure which is commonly used to tune magnetic properties [31][32][33][34][35] can also modify the magnetic anisotropy 36 .
While magnetically isotropic materials can exhibit long range ferromagnetic order in 3D, such ordering is prohibited in 2D due to the thermal fluctuations. To achieve long-range ferromagnetism in 2D systems, magnetic anisotropies need to be incorporated with exchange interaction: where the first term represents Heisenberg isotropic model, second term represents magnetocrystalline anisotropy whereas the last term is the interaction of local moments with an external magnetic field B. Here S i is the spin operator on site i, J ij is the exchange coupling between spins on site i and j whose positive and negative values represent ferromagnetic and antiferromagnetic ordering, A is the single-ion anisotropy, whereas g and B  are landau g-factor and Bohr magneton respectively.  Figure 1(A) shows a graph for various proposed 2D ferromagnetic materials with their respective T c , which were mainly predicted via DFT calculations. The 2D ferromagnetism is mostly predicted in transition metal chalcogenides and transition metal halides, therefore, their general crystal structures including top and side views are presented in Fig 1(B).

III. PROPOSED 2D FERROMAGNETIC MATERIALS AND THEIR EXPERIMENTAL REALIZATION
Transition metal trichalcogenides (TMT) are compounds of the formula ABX 3 where A is a 3d metal (Cr), B can be metalloid or metal (such as Si, Ge, Sn) and X is a chalcogen (S, Se, Te) [41][42][43][44][45] . Generally, the transition metal atoms form a honeycomb lattice structure within each layer, and each of them is octahedrally coordinated by six chalcogenide atoms from its three neighboring (B 2 X 6 ) ligands, with the centers of the hexagons occupied by the B 2 groups 41 .
Magnetic order is predicted in the single-layer limit of these van der Waals materials, and the configuration of the magnetic phase is sensitive to the transition metal/chalcogen element combinations. Likewise, transition metal dichalcogenides (TMD) are 2D materials of the formula AX 2 , with A a transition metal (e.g., V, Cr, and Mn) and X a chalcogen (e.g. S and Se) [46][47][48] . Ferromagnetism is predicted in T phase TMDs, where each transition metal adopts an octahedrally coordinated structure with respect to the six chalcogenide atoms around it and each chalcogenide atom is bonded to three nearby transition metal atoms 46 [55][56][57][58][59] . Here, the superexchange mechanism contributes to the magnetism. Furthermore, ferromagnetism has also been predicted in several other materials (denoted by OM in Table 1) including materials such as MnO 2 , FeC 2 etc [60][61][62][63][64][65][66][67][68][69][70][71] . It is also important to mention here that Mounet and co-workers have also theoretically identified a group of ferromagnetic metals and semiconductors using data-mining algorithms and advanced, high-throughput electronic structure theory 72 . Table 1   Huang and colleagues demonstrated the intrinsic FM in a mono layer of CrI 3 with a suppressed T c of 45 K 4 , which they explain based on Ising exchange interaction. Interestingly they also observe a strong dependence of the ferromagnetism in CrI 3 on the number of layers as can be seen in figure 2. The observed remnant magnetization in a single layer is absent in a bilayer, indicating antiferromagnetism i.e. the two layers having oppositely oriented spins.
Interestingly, the net magnetization is again recovered in a trilayer. This indicates the strong layer-dependent interplay between different mechanisms that stabilize magnetic ordering in the atomically thin limit. The intrinsic 2D ferromagnetism in CrI 3 and Cr 2 Ge 2 Te 6 is observed only far below room temperature (i.e. <293 K) and a challenge remained to realize room temperature intrinsic 2D ferromagnetism until the very recent remarkable discovery of the strong ferromagnetic ordering in monolayer VSe 2 5 . Notably, the FM in VSe 2 monolayer with a large magnetic moment persists to above room temperature, which is a major breakthrough in this field and that could lead to potential spintronic devices. Magnetic properties of VSe 2 are illustrated in figure 3.

V. TOPOLOGICAL INSULATING PHASE
To show how TRS and its breaking are associated with the TI phase and the QAH phase respectively, we discuss here a four-band model which serves as a guiding principle for generating QAH phase in magnetically doped TIs [122][123][124][125][126] . In a suitably chosen basis, the time reversal invariant four-band mode model written as where the two-band component blocks The material parameters A, B, C, D, and M depend on the QW geometry. This four-band model describes TI phase found in both type-I HgTe and type-II InAs/GaSb semiconducting quantum wells; hybridization gap generated through coupling A between electron and hole sub-bands in type-II QW is relatively smaller than that in type-I QW.
Similarly, this four-band model also represents the two-dimensional limit of 3D TI in and t (and b) represent the top (and bottom) surface states while ↑, ↓ represent the spin up and down states respectively. Here, represents the tunneling effect between the top and bottom surface states at the surface of 3D TI with normal vector ẑ Here f v is the Fermi velocity. Bulk energy bands of the type-I semiconducting quantum well are shown in figure (5) while the local density approximation for the band structure of three-dimensional bulk and the ARPES measurements on the two-dimensional surface (111) of Bi 2 Se 3 is shown in figure (6).

VI. ORIGIN OF FERROMAGNETISM IN TOPOLOGICAL INSULATORS
At present, no intrinsically ferromagnetic topological 2D material is known.
Ferromagnetism has successfully been introduced into topological materials through substitutional doping. In analogous dilute magnetic semiconductors, the magnetic order can

VII. IMPORTANCE OF FERROMAGNETISM IN REALIZATION OF "QAH" PHASES
The QAH insulating phase can be realized in systems with spontaneous magnetization and/or strong spin-orbit-interaction. The presence of both collectively gives rise to a topologically non-trivial band structure characterized by non-zero Chern number. Since the observation of the TI phase in 2D semiconducting QW and 3D tetradymite semiconductors is the consequence of strong SOC, the key ingredient to observe QAH phase is to have long range ferromagnetic order in TIs.
The role of a ferromagnetic order associated with TR symmetry breaking can be formulated in terms of TR breaking four-band mode model as  (8,9).   Moreover, difficulty in achieving both a high T c and a large band gap opening simultaneously in magnetically doped TIs limits the applications of QAHE at room temperature. Room-temperature ferromagnetism is the key requirement for the realization of QAHE and its utilization in related spintronics and low-energy electronics technologies. In the QAH phase, quantized transverse resistance and extremely low longitudinal resistance allow possible future dissipationless spintronic and electronic devices. However, the highest temperature at which QAH phase is observed so far is less than 100 mK. For practical applications, the prime goal is to achieve dissipationless chiral edge states at room temperature. To achieve this, the key factor is intrinsic ferromagnetic order sustainable against thermal fluctuations at high temperature in the two-dimensional limit. In short, the discovery of intrinsic ferromagnetic order with high T C in atomically thin materials with topological non-trivial band structure would offer a much better platform to realize the QAHE without random magnetic doping.

VIII. FUTURE DIRECTIONS
The discovery of intrinsic 2D ferromagnetism could revolutionize modern day technologies in many ways from both comprehending the underlying physics of electronic and spin processes, and applications ranging from sensing to information, and communication technologies 180   transition metal compound monolayers such as 1T-MnSe 2 189, 190 , Fe 3 GeTe 2 191 , and transition metal dihydride MH 2 (M=Sc, Co) 192 host magnetic states and the van der Waals substrate stabilizes the former and leads toward 2D itinerant ferromagnetism. More interestingly, the hetro-interface plays key role in deriving a metal-to-insulator transition in the TMDC monolayers by inducing a band gap up several meV and leads to CDW phase in transition metal compounds 193,194 . This discussion has also been extended to electric field controlled (Gate-tuning effects on) magnetism in a few layered van der Waals ferromagnetic Cr 2 Ge 2 Te 6 195 , and antiferromagnetic CrI 3 semiconductors 196,197 .