A ferroelectric crystal exhibits a stable and switchable electrical polarization that is manifested in the form of cooperative atomic displacements. A ferromagnetic crystal exhibits a stable and switchable magnetization that arises through the quantum mechanical phenomenon of exchange. There are very few 'multiferroic' materials that exhibit both of these properties, but the 'magnetoelectric' coupling of magnetic and electrical properties is a more general and widespread phenomenon. Although work in this area can be traced back to pioneering research in the 1950s and 1960s, there has been a recent resurgence of interest driven by long-term technological aspirations.

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Multiferroic and magnetoelectric materials

Magneto-optical Kerr effect microscopy is used to show that monolayer chromium triiodide is an Ising ferromagnet with out-of-plane spin orientation. The question of what happens to the properties of a material when it is thinned down to atomic-scale thickness has for a long time been a largely hypothetical one. In the past decade, new experimental methods have made it possible to isolate and measure a range of two-dimensional structures, enabling many theoretical predictions to be tested. But it has been a particular challenge to observe intrinsic magnetic effects, which could shed light on the longstanding fundamental question of whether intrinsic long-range magnetic order can robustly exist in two dimensions. In this issue of Nature, two groups address this challenge and report ferromagnetism in atomically thin crystals. Xiang Zhang and colleagues measured atomic layers of Cr2Ge2Te6 and observed ferromagnetic ordering with a transition temperature that, unusually, can be controlled using small magnetic fields. Xiaodong Xu and colleagues measured atomic layers of CrI3 and observed ferromagnetic ordering that, remarkably, was suppressed in double layers of CrI3, but restored in triple layers. The two studies demonstrate a platform with which to test fundamental properties of purely two-dimensional magnets. Since the discovery of graphene1, the family of two-dimensional materials has grown, displaying a broad range of electronic properties. Recent additions include semiconductors with spin–valley coupling2, Ising superconductors3,4,5 that can be tuned into a quantum metal6, possible Mott insulators with tunable charge-density waves7, and topological semimetals with edge transport8,9. However, no two-dimensional crystal with intrinsic magnetism has yet been discovered10,11,12,13,14; such a crystal would be useful in many technologies from sensing to data storage15. Theoretically, magnetic order is prohibited in the two-dimensional isotropic Heisenberg model at finite temperatures by the Mermin–Wagner theorem16. Magnetic anisotropy removes this restriction, however, and enables, for instance, the occurrence of two-dimensional Ising ferromagnetism. Here we use magneto-optical Kerr effect microscopy to demonstrate that monolayer chromium triiodide (CrI3) is an Ising ferromagnet with out-of-plane spin orientation. Its Curie temperature of 45 kelvin is only slightly lower than that of the bulk crystal, 61 kelvin, which is consistent with a weak interlayer coupling. Moreover, our studies suggest a layer-dependent magnetic phase, highlighting thickness-dependent physical properties typical of van der Waals crystals17,18,19. Remarkably, bilayer CrI3 displays suppressed magnetization with a metamagnetic effect20, whereas in trilayer CrI3 the interlayer ferromagnetism observed in the bulk crystal is restored. This work creates opportunities for studying magnetism by harnessing the unusual features of atomically thin materials, such as electrical control for realizing magnetoelectronics12, and van der Waals engineering to produce interface phenomena15.

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Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit

This is a tutorial review on the Potts model aimed at bringing out in an organized fashion the essential and important properties of the standard Potts model. Emphasis is placed on exact and rigorous results, but other aspects of the problem are also described to achieve a unified perspective. Topics reviewed include the mean-field theory, duality relations, series expansions, critical properties, experimental realizations, and the relationship of the Potts model with other lattice-statistical problems.

The Potts model

The purpose of this critical review is to give a representative and comprehensive overview of the arising developments in the field of magnetic metal–organic frameworks, in particular those containing cobalt(II). We examine the diversity of magnetic exchange interactions between nearest-neighbour moment carriers, covering from dimers to oligomers and discuss their implications in infinite chains, layers and networks, having a variety of topologies. We progress to the different forms of short-range magnetic ordering, giving rise to single-molecule-magnets and single-chain-magnets, to long-range ordering of two- and three-dimensional networks (323 references).

Magnetic metal-organic frameworks.

Although known since the late 19th century, organic–inorganic perovskites have recently received extraordinary research community attention because of their unique physical properties, which make them promising candidates for application in photovoltaic (PV) and related optoelectronic devices. This review will explore beyond the current focus on three-dimensional (3-D) lead(II) halide perovskites, to highlight the great chemical flexibility and outstanding potential of the broader class of 3-D and lower dimensional organic-based perovskite family for electronic, optical, and energy-based applications as well as fundamental research. The concept of a multifunctional organic–inorganic hybrid, in which the organic and inorganic structural components provide intentional, unique, and hopefully synergistic features to the compound, represents an important contemporary target.

Organic–Inorganic Perovskites: Structural Versatility for Functional Materials Design

“…. For the truth of the conclusions of physical science, observation is the supreme Court of Appeal….” (Sir Arthur Eddington, The Philosophy of Physical Science.) In this paper we shall review the theoretical and experimental results obtained on simple magnetic model systems. We shall consider the Heisenberg, XY and Ising type of interaction (ferro and antiferromagnetic), on magnetic lattices of dimensionality 1, 2 and 3. Particular attention will be paid to the approximation of these model systems in real crystals, viz. how they can be realized or be expected to exist in nature. A large number of magnetic compounds which, according to the available experimental information, meet the requirements set by one or the other of the various models are considered and their properties discussed. Many examples will be given that demonstrate to what extent experiments on simple magnetic systems support theoretical descriptions of magnetic ordering phenomena and contribute to their understanding. It will a...

Experiments on simple magnetic model systems

Magnetic measurements on (C2H5NH3)2 CuCl4: Ferromagnetic layers coupled by a very weak antiferromagnetic interaction

A number of experiments on magnetic insulating systems is reviewed which bear upon the following subjects: (i) the magnetic XY model, (ii) the quadratic Heisenberg magnet, (iii) magnetic double‐layers, (iv) lattice‐dimensionality crossovers, (v) singlet–ground state systems, (vi) magnetic systems doped with magnetic or nonmagnetic impurities.

The ordering temperatures of the magnetically diluted antiferromagnetic systems K2MnpMg1−pF4 and KMnpMg1−pF3 are presented. From a magnetic point of view these systems can be looked upon as being simple quadratic and simple cubic, respectively. Near p=1, (d/dp)[TN(p)/TN(1)] is clearly larger than 1, as is theoretically expected for diluted Heisenberg systems. The curves for TN as a function of the concentration show that the critical concentrations for ordering at T=0 are clearly different for the two magnetic systems; they may be equal to the values predicted for the percolation problem. The low‐field perpendicular susceptibility strongly depends on p, at the lower temperatures it can be approximately described by a kind of Curie‐Weiss law. Extrapolating the Curie‐Weiss behavior down to T=0, one obtains values for the inverse perpendicular susceptibility that, if plotted as a function of the concentration of magnetic atoms, show a clear similarity to the concentration dependence of the ordering temperature itself. Some results of the two‐dimensional nearly Ising system K2CopMg1−pF4 are also presented.

A.R. Miedema

Papers

Experiments on simple magnetic model systems

Magnetic measurements on (C2H5NH3)2 CuCl4: Ferromagnetic layers coupled by a very weak antiferromagnetic interaction

Experiments on simple magnetic model systems

Ordering in Two‐ and Three‐Dimensional Diluted Heisenberg Antiferromagnets

Some experiments on magnetically diluted antiferromagnets