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 n...

Multiferroic materials and magnetoelectric physics: symmetry, entanglement, excitation, and topology

Multiferroics are those materials with more than one ferroic order, and magnetoelectricity refers to the mutual coupling between magnetism and electricity. 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.

Induced multiferroics, where ferroelectricity arises through the magnetic order, have attracted significant interest, despite maximum Curie temperatures of only 40 K. The discovery of multiferroic coupling up to 230 K in CuO therefore represents a major advance towards high-TC multiferroics. Materials that combine coupled electric and magnetic dipole order are termed ‘magnetoelectric multiferroics’1,2,3,4. In the past few years, a new class of such materials, ‘induced-multiferroics’, has been discovered5,6, wherein non-collinear spiral magnetic order breaks inversion symmetry, thus inducing ferroelectricity7,8,9. Spiral magnetic order often arises from the existence of competing magnetic interactions that reduce the ordering temperature of a more conventional collinear phase10. Hence, spiral-phase-induced ferroelectricity tends to exist only at temperatures lower than ∼40 K. Here, we propose that copper(II) oxides (containing Cu2+ ions) having large magnetic superexchange interactions11 can be good candidates for induced-multiferroics with high Curie temperature (TC). In fact, we demonstrate ferroelectricity with TC=230 K in cupric oxide, CuO (tenorite), which is known as a starting material for the synthesis of high-Tc (critical temperature) superconductors. Our result provides an important contribution to the search for high-temperature magnetoelectric multiferroics.

Cupric oxide as an induced-multiferroic with high-TC.

In recent years, the study of ladder materials has developed into a well-established area of research within the general context of strongly correlated electrons. This effort has been triggered by an unusual cross-fertilization between theory and experiments. In this paper, the main experimental results obtained in the context of ladders are reviewed from the perspective of a theorist. Emphasis is given to the many similarities between the two-dimensional high-Tc cuprates and the two-leg ladder compounds, including Sr14-xCaxCu24O41 ([14-24-41]) which has a superconducting phase at high pressure and a small hole density. Examples of these similarities include regimes of linear resistivity versus temperature in metallic ladders and a normal state with spin gap or pseudogap characteristics. It is remarked that the ladder [14-24-41] is the first superconducting Cu oxide material with a non-square-lattice layered arrangement, and certainly much can be learned from a careful analysis of this compound. A short summary of the main theoretical developments in this field is also included, as well as a brief description of the properties of non-Cu-oxide ladders. Suggestions by the author on possible experiments are described in the text. Overall, it is concluded that the enormous experimental effort carried out on ladders has already unveiled quite challenging and interesting physics that adds to the rich behaviour of electrons in transition metal-oxides, and in addition contributes to the understanding of the two-dimensional cuprates. However, considerable work still needs to be carried out to fully understand the interplay between charge and spin degrees of freedom in these materials.

https://iopscience.iop.org/article/10.1088/0034-4885/62/11/202/pdf

Experiments on ladders reveal a complex interplay between a spin-gapped normal state and superconductivity

A three-dimensional system of interacting spins typically develops static long-range order when it is cooled. If the spins are quantum (S=1/2), however, novel quantum paramagnetic states may appear. The most highly sought state among them is the resonating-valence-bond state, in which every pair of neighbouring quantum spins forms an entangled spin singlet (valence bonds) and these singlets are quantum mechanically resonating among themselves. Here we provide an experimental indication for such quantum paramagnetic states existing in frustrated antiferromagnets, Zn(x)Cu(4-x)(OD)(6)Cl(2), where the S=1/2 magnetic Cu2+ moments form layers of a two-dimensional kagome lattice. We find that in Cu(4)(OD)(6)Cl(2), where distorted kagome planes are weakly coupled, a dispersionless excitation mode appears in the magnetic excitation spectrum below approximately 20 K, whose characteristics resemble those of quantum spin singlets in a solid state, known as a valence-bond solid, that breaks translational symmetry. Doping with non-magnetic Zn2+ ions reduces the distortion of the kagome lattice, and weakens the interplane coupling but also dilutes the magnetic occupancy of the kagome lattice. The valence-bond-solid state is suppressed, and for ZnCu(3)(OD)(6)Cl(2), where the kagome planes are undistorted and 90% occupied by the Cu2+ ions, the low-energy spin fluctuations become featureless.

Quantum-spin-liquid states in the two-dimensional kagome antiferromagnets ZnxCu4-x(OD)6Cl2.

The electronic states and magnetic properties for the copper oxides containing edge-sharing Cu-O chains such as ${\mathrm{Li}}_{2}{\mathrm{CuO}}_{2}$, ${\mathrm{La}}_{6}{\mathrm{Ca}}_{8}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$, and ${\mathrm{CuGeO}}_{3}$ are systematically studied. The optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ and the temperature dependence of the magnetic susceptibility $\ensuremath{\chi}(T)$ for single-crystalline samples ${\mathrm{Li}}_{2}{\mathrm{CuO}}_{2}$ are measured as a reference system and analyzed by using the exact diagonalization method for small Cu-O clusters. It is shown that the spectral distribution of $\ensuremath{\sigma}(\ensuremath{\omega})$ is different between edge-sharing and corner-sharing Cu-O-Cu bonds. The charge transfer gap in edge-sharing chains is larger than that of high-${T}_{\mathrm{c}}$ cuprates. The exchange interaction between nearest-neighbor copper ions in edge-sharing chains ${J}_{1}$ depends sensitively on the Cu-O-Cu bond angles. In addition to ${J}_{1}$, the exchange interaction between next-nearest-neighbor copper ions ${J}_{2}$ has sufficient contribution to the magnetic properties. We calculate ${J}_{1}$ and ${J}_{2}$ for all the copper oxides containing edge-sharing Cu-O chains and discuss the magnetic properties.

Electronic states and magnetic properties of edge-sharing cu-o chains

We examine the electronic states of Sr 14- x Ca x Cu 24 O 41 by the ionic and cluster model approach. It is found that self-doped holes are likely to stay on the chain at x =0 and Ca substitution drives the holes to move to the ladder. This feature is caused by changes of the positions of (Sr,Ca) layers, which results in enhancement of the electrostatic potentials in the chain. The optical conductivity is calculated with application of the exact diagonalization method to small Cu-O clusters. It is shown that the excitations below 2 eV are governed by the ladder, giving rise to the spectral weight transfer from the charge-transfer excitation at around 2 eV to low energy Drude excitation, while the contribution from the chain mainly emerges in a higher energy region showing large spectral weight at around 3 eV. These findings provide a consistent explanation for the results of a recent experiment.

Electronic States of Doped Spin Ladders (Sr,Ca)14Cu24O41

We examine the electronic states of Sr$_{14-x}$Ca$_x$Cu$_{24}$O$_{41}$ by the ionic and cluster model approach. It is found that self-doped holes are likely to stay on the chain at $x=0$ and the Ca substitution drives the holes to move to the ladder. This feature is caused by the change of the positions of (Sr,Ca) layers, which enhances the electrostatic potentials in the chain. The optical conductivity is calculated applying the exact diagonalization method to small Cu-O clusters. It is shown that the excitations below 2 eV are governed by the ladder, giving rise to the spectral weight transfer from the charge-transfer excitation around 2 eV to low energy Drude excitation, while the contribution from the chain mainly emerges in higher energy region showing large weight around 3 eV. These provide consistent explanation to recent experiment.

Electronic States of doped spin ladders (Sr,Ca)14Cu24O41

We examine two-spin (2S) and cyclic four-spin (4S) magnetic interactions in insulating ladder and two-dimensional (2D) cuprates. By a comparison of eigenstates between d-p and Heisenberg models, we evaluate magnitudes of these interactions. We find that the 4S interaction is ∼10 % of nearest neighboring 2S interaction, and a diagonal 2S interaction is considerably small. The 4S interaction for a ladder cuprate is larger than that for 2D one, and plays an important role in the low-energy excitation. The Heisenberg ladder with the obtained 2S and 4S interactions reproduces very well the experimental result of the temperature dependence of the magnetic susceptibility.

Magnetic Interaction in Insulating Cuprates

We examine two-spin (2S) and cyclic four-spin (4S) magnetic interactions in insulating ladder and two-dimensional (2D) cuprates. By a comparison of eigenstates between d-p and Heisenberg models, we evaluate magnitudes of these interactions. We find that the 4S interaction is $\sim$10% of nearest neighboring 2S interaction, and a diagonal 2S interaction is considerably small. The 4S interaction for a ladder cuprate is larger than that for 2D one, and plays an important role in the low-energy excitation. The Heisenberg ladder with the obtained 2S and 4S interactions reproduces very well the experimental result of the temperature dependence of the magnetic susceptibility.

Yoshiaki Mizuno

Papers

Electronic states and magnetic properties of edge-sharing cu-o chains

Electronic States of Doped Spin Ladders (Sr,Ca)14Cu24O41

Electronic States of doped spin ladders (Sr,Ca)14Cu24O41

Magnetic Interaction in Insulating Cuprates

Magnetic Interaction in Insulating Cuprates