
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. There have since been reports of several related materials, all except for magnetic materials. But then there was a paper in 2016 describing what this then unreported class of materials might hold in store. 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 = Mn, Fe and Ni in 2016. In the following year, the first 2D ferromagnets, CrI3. and Cr2Ge2Te6,
were fabricated. One reason for this is that thermal fluctuations tend to destroy magnetic order easier compared to 3D bulk. 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.
Although the field has been only around since 2016, it has become one of the most active fields in condensed matter physics and materials science and engineering. There have been several review articles written up to highlight its future and promise.
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 (formula_1) the alignment can always exist, thermal fluctuations misalign magnetic moments at temperatures above the Curie temperature (formula_2), causing a phase transition to a non-magnetic state. Whether formula_2 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 formula_1
For 2D systems, the transition temperature depends on the spin dimensionality (formula_5). In system with formula_6, 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 formula_6 is described by the 2D Ising model. Onsager's solution to the model demonstrates that formula_8, thus allowing magnetism at obtainable temperatures. On the contrary, a system where formula_9, 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 formula_9 to demonstrate magnetism.
The 2D ising model describes the behavior of FePS3, CrI3. and Fe3GeTe2, while Cr2Ge2Te6 and MnPS3 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 formula_11. 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 formula_12, where at temperatures below formula_12 the system has quasi-long-range magnetic order. It was reported that the theoretical predictions of the XY model are consistent with those experimental observations of NiPS3. The Heisenberg model describes the case where formula_9. In this system, there is no transition between the ordered and unordered states because of the Mermin-Wagner theorem. The experimental realization of the Heisenberg model was reported using MnPS3.
The above systems can be described by a generalized Heisenberg spin Hamiltonian:
Where formula_16 is the exchange coupling between spins formula_17 and formula_18, and formula_19 and formula_20 are on-site and inter-site magnetic anisotropies, respectively. Setting formula_21 recovered the 2D Ising model and the XY model. (positive sign for formula_6 and negative for formula_11), while formula_24 and formula_25 recovers the Heisenberg model (formula_9). Along with the idealized models described above, the spin Hamiltonian can be used for most experimental setups, and it can also model dipole-dipole interactions by renormalization of the parameter formula_19. However, sometimes including further neighbours or using different exchange coupling, such as antisymmetric exchange, is required.
Measuring two-dimensional magnetism.
Magnetic properties of two-dimensional materials are usually measured using Raman spectroscopy, Magneto-optic Kerr effect, Magnetic circular dichroism or Anomalous Hall effect techniques. The dimensionality of the system can be determined by measuring the scaling behaviour of magnetization (formula_28), susceptibility (formula_29) or correlation length (formula_30) as a function of temperature. The corresponding "critical exponents" are formula_31, formula_32 and formula_33 respectively. They can be retrieved by fitting
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 (formula_37). This relation for a large formula_37 is given by
where formula_40 is a material dependent constant. For thin layers, the behavior changes to formula_41
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, and thus they can have many applications in the field of spintronics. Another newly emerging direction came from the rather unexpected observation of magnetic exciton in NiPS3.