Multiferroics are those materials with more than one ferroic order, and magnetoelectricity refers to the mutual coupling between magnetism (spins and/or magnetic field) and electricity (electric dipoles and/or electric field). In spite of the long research history in the whole twentieth century, the discipline of multiferroicity has never been so highly active as that in the first decade of the twenty-first century, and it has become one of the hottest disciplines of condensed matter physics and materials science. A series of milestones and steady progress in the past decade have enabled our understanding of multiferroic physics substantially comprehensive and profound, which is further pushing forward the research frontier of this exciting area. The availability of more multiferroic materials and improved magnetoelectric performance are approaching to make the applications within reach. While seminal review articles covering the major progress before 2010 are available, an updated review addressing the new achievements since that time becomes imperative. In this review, following a concise outline of the basic knowledge of multiferroicity and magnetoelectricity, we summarize the important research activities on multiferroics, especially magnetoelectricity and related physics in the last six years. We consider not only single-phase multiferroics but also multiferroic heterostructures. We address the physical mechanisms regarding magnetoelectric coupling so that the backbone of this divergent discipline can be highlighted. A series of issues on lattice symmetry, magnetic ordering, ferroelectricity generation, electromagnon excitations, multiferroic domain structure and domain wall dynamics, and interfacial coupling in multiferroic heterostructures, will be revisited in an updated framework of physics. In addition, several emergent phenomena and related physics, including magnetic skyrmions and generic topological structures associated with magnetoelectricity will be discussed. The review is ended with a set of prospectives and forward-looking conclusions, which may inevitably reflect the authors' biased opinions but are certainly critical.

The key physical property of multiferroic materials is the existence of coupling between magnetism and polarization, i.e. magnetoelectricity. The origin and manifestations of magnetoelectricity can be very different in the available plethora of multiferroic systems, with multiple possible mechanisms hidden behind the phenomena. In this review, we describe the fundamental physics that causes magnetoelectricity from a theoretical viewpoint. The present review will focus on mainstream physical mechanisms in both single-phase multiferroics and magnetoelectric heterostructures. The most recent tendencies addressing possible new magnetoelectric mechanisms will also be briefly outlined.

Recent research on piezoelectric materials is predominantly devoted to enhancing the piezoelectric coefficient, but overlooks its sign, largely because almost all of them exhibit positive longitudinal piezoelectricity. The only experimentally known exception is ferroelectric polymer poly(vinylidene fluoride) and its copolymers, which condense via weak van der Waals (vdW) interaction and show negative piezoelectricity. Here we report quantitative determination of giant intrinsic negative longitudinal piezoelectricity and electrostriction in another class of vdW solids—two-dimensional (2D) layered ferroelectric CuInP₂S₆. With the help of single crystal x-ray crystallography and density-functional theory calculations, we unravel the atomistic origin of negative piezoelectricity in this system, which arises from the large displacive instability of Cu ions coupled with its reduced lattice dimensionality. Furthermore, the sizable piezoelectric response and negligible substrate clamping effect of the 2D vdW piezoelectric materials warrant their great potential in nanoscale, flexible electromechanical devices.

Two-dimensional (2D) ferromagnetic semiconductors (FMSs) exhibit novel spin-dependent electronic and optical properties, opening up exciting opportunities for nanoscale spintronic devices. However, experimentally confirmed 2D FMSs based on transition metal ions are rather limited and their performances are not satisfactory, e.g. typically with low Curie temperatures and small magnetic signals. Different from most known 2D magnets based on d-electrons, here an exotic 2D FMS based on rare-earth ions with f-electrons, a GdI₂ monolayer, is predicted to have a large magnetization (8 μ_B f.u.⁻¹), whose ferromagnetism can survive near room temperature (241 K). In addition, with a small exfoliation energy from its layered van der Waals (vdW) bulk, this GdI₂ monolayer holds excellent dynamical and thermal stabilities, making our prediction promising in experiments. Our prediction not only offers a compelling FMS for spintronics, but also provides an alternative route to acquire more high-performance 2D FMSs, going beyond pure d-electron compounds.