First principles study of improper ferroelectricity in TbMnO3

Origins of multi-sublattice magnetism and superexchange interactions in double-double perovskite CaMnCrSbO6

The multi-sublattice magnetism and electronic structure in double-double perovskite compound CaMnCrSbO6is explored using density functional theory. The bulk magnetization and neutron diffraction suggest a ferrimagnetic order (TC∼49 K) between between Mn2+and Cr3+spins. Due to the non-equivalent Mn atoms (labelled as Mn(1) and Mn(2) which have tetrahedral and planar oxygen coordinations, respectively) and the Cr atom in the centre of distorted oxygen octahedron in the unit cell, the exchange interactions are more complex than that expected from a two sublattice magnetic system. The separations between the on-site energies of thed-orbitals of Mn(1), Mn(2) and Cr obtained from Wannier function analysis are in agreement with their expected crystal field splitting. While the DOS obtained from non spin-polarized calculations show a metallic character, starting from HubbardU = 0 eV the spin-polarized electronic structure calculations yield a ferrimagnetic insulating ground state. The band gap increases withUeff(U - J), thereby showing a Mott-Hubbard nature of the system. The inclusion of anti-site disorder in the calculations show decrease in band-gap and also reduction in the total magnetic moment. Due to the ∼90∘superexchange, nearest neighbour exchange constants obtained from DFT are an order of magnitude smaller than those reported for various magnetic perovskite and double-perovskite compounds. The Mn(1)-O-Mn(2) (out of plane and in-plane), Mn(1)-O-Cr and Mn(2)-O-Cr superexchange interactions are found to be anti-ferromagnetic, while the Cr-O-O-Cr super-superexchange is found to be ferromagnetic. The Mn(2)-O-Cr superexchange is weaker than the Mn(1)-O-Cr super-exchange, thus effectively resulting in ferrimagnetism. From a simple 3-site Hubbard model, we derived expressions for the antiferromagnetic superexchange strengthJAFMand also for the weaker ferromagneticJFM. The relative strengths ofJAFMfor the various superexchange interactions are in agreement with those obtained from DFT. The expression for Cr-O-O-Cr super-superexchange strength (J~SS), which has been derived considering a 4-site Hubbard model, predicts a ferromagnetic exchange in agreement with DFT. Finally, our mean field calculations reveal that assuming a set of four magnetic sub-lattices for Mn2+spins and a single magnetic sublattice for Cr3+spins yields a much improvedTC, while a simple two magnetic sublattice model yields a much higherTC.

Spin-Chirality-Driven Multiferroicity in van der Waals Monolayers

Driven by the expected contribution of two-dimensional multiferroic systems with strong magnetoelectric coupling to the development of multifunctional nanodevices, here we propose, by means of first-principles calculations, vanadium-halide monolayers as a new class of spin-chirality-driven van der Waals multiferroics. The frustrated 120-deg magnetic structure in the triangular lattice induces a ferroelectric polarization perpendicular to the spin-spiral plane, whose sign is switched by a spin-chirality change. It follows that, in the presence of an applied electric field perpendicular to the monolayers, one magnetic chirality can be stabilized over the other, thereby allowing the long-sought electrical control of spin textures. Moreover, we demonstrate the remarkable role of spin-lattice coupling on magnetoelectricity, which adds to the expected contribution of spin-orbit interaction determined by an anion. Indeed, such compounds exhibit sizeable spin-driven structural distortions, thereby promoting the investigation of multifunctional spin-electric-lattice couplings.

Synthesis and Magnetic Properties of the Multiferroic [C(NH2)3]Cr(HCOO)3 Metal-Organic Framework: The Role of Spin-Orbit Coupling and Jahn-Teller Distortions

We report for the first time the synthesis of [C(NH2)3]Cr(HCOO)3 stabilizing Cr2+ in formate perovskite, which adopts a polar structure and orders magnetically below 8 K. We discuss in detail the magnetic properties and their coupling to the crystal structure based on first-principles calculations, symmetry, and model Hamiltonian analysis. We establish a general model for the orbital magnetic moment of [C(NH2)3]M(HCOO)3 (M = Cr, Cu) based on perturbation theory, revealing the key role of the Jahn-Teller distortions. We also analyze their spin and orbital textures in k-space, which show unique characteristics.

Molecular Ferroelectric Crystals with Superior Pyroelectricity, Plasticity, and Recyclability

The pyroelectric effect is used in a wide range of applications such as infrared (IR) detection and thermal energy harvesting, which require the pyroelectric materials to simultaneously have a high pyroelectric coefficient and a low dielectric constant for high figures of merit. However, in conventional proper ferroelectrics, the positive correlation between the pyroelectric coefficient and the dielectric constant imposes an insurmountable challenge in upgrading the figures of merit. Here, we explored superior pyroelectricity in [(CH3)4N][FeCl4] (TMA-FC) and [(CH3)4N][FeCl3Br] (TMA-FCB) molecular ferroelectric plastic crystals, which could decouple this positive correlation due to the nature of improper polarization behavior. Therefore, TMA-FC and TMA-FCB derive a high pyroelectric coefficient and a low dielectric constant simultaneously, yielding record-high figures of merit around room temperature. Furthermore, the favorable plasticity enables ferroelectric crystals to attach surfaces with different shapes for device design and integration. More interestingly, the molecular ferroelectrics could be softened and reshaped at elevated temperatures without decay in pyroelectricity, making them recyclable for cost savings and e-waste reduction. Combined with the facile fabrication process, the findings of this work would open avenues for employing molecular ferroelectric plastic crystals in the manufacture of high-performance pyroelectric devices.

Magnetically Induced Polarization in Centrosymmetric Bonds

We reveal the microscopic origin of electric polarization P[over →] induced by noncollinear magnetic order. We show that in Mott insulators, such P[over →] is given by all possible combinations of position operators r[over →][over ^]_{ij}=(r[over →]_{ij}^{0},r[over →]_{ij}) and transfer integrals t[over ^]_{ij}=(t_{ij}^{0},t_{ij}) in the bonds, where r[over →]_{ij}^{0} and t_{ij}^{0} are spin-independent contributions in the basis of Kramers doublet states, while r[over →]_{ij} and t_{ij} stem solely from the spin-orbit interaction. Among them, the combination t_{ij}^{0}r[over →]_{ij}, which couples to the spin current, remains finite in the centrosymmetric bonds, thus yielding finite P[over →] in the case of noncollinear arrangement of spins. The form of the magnetoelectric coupling, which is controlled by r[over →]_{ij}, appears to be rich and is not limited to the phenomenological law P[over →]∼ε_{ij}×[e_{i}×e_{j}] with ε_{ij} being the bond vector connecting the spins e_{i} and e_{j}. Using density-functional theory, we illustrate how the proposed mechanism works in the spiral magnets CuCl_{2}, CuBr_{2}, CuO, and α-Li_{2}IrO_{3}, providing a consistent explanation for the available experimental data.

Non-collinear magnetism & multiferroicity: the perovskite case

The most important types of non-collinear magnetic orders that are realized in simple perovskite oxides are outlined in relation to multiferroicity. These orders are classified and rationalized in terms of a mimimal spin Hamiltonian, based on which the notion of spin-driven ferroelectricity is illustrated. These concepts find direct application in reference materials such as BiFeO3, GdFeO3 and TbMnO3 whose multiferroic properties are briefly reviewed.

Femtometer atomic displacement, the root cause for multiferroic behavior of CuO unearthed through polarized Raman spectroscopy

Recently, CuO has been proposed as a potential multiferroic material with high transition temperature. Competing models based on spin current and ionic displacements are invoked to explain ferroelectricity in CuO. The theoretical model based on ionic displacement predicted very small displacement (∼10^-5Å) along thebaxis. Experimentally detecting displacements of such a small amplitude in a particular direction is extremely challenging. Through our detailed angle resolved polarized Raman spectroscopy study on single crystal of CuO, we have validated the theoretical study and provided direct evidence of displacement along thebaxis. Our study provides important contribution in the high temperature multiferroic compounds and showed for the first time, the use of the polarized Raman scattering in detecting ionic displacements at the femtometer scale.

Magneto-electric multiferroics: designing new materials from first-principles calculations

In parallel with the revival of interest for magneto-electric multiferroic materials in the beginning of the century, first-principles simulations have grown incredibly in efficiency during the last two decades. Density functional theory calculations, in particular, have so become a must-have tool for physicists and chemists in the multiferroic community. While these calculations were originally used to support and explain experimental behaviour, their interest has progressively moved to the design of novel magneto-electric multiferroic materials. In this article, we mainly focus on oxide perovskites, an important class of multifunctional material, and review some significant advances to which contributed first-principles calculations. We also briefly introduce the various theoretical developments that were at the core of all these advances.

Realizing Magnetoelectric Coupling with Hydrogen Intercalation

Materials with a coexistence of magnetic and ferroelectric order (i.e., multiferroics) provide an efficient route for the control of magnetism by electric fields. Unfortunately, a long-sought room temperature multiferroic with strongly coupled ferroelectric and ferromagnetic (or ferrimagnetic) orderings is still lacking. Here, we propose that hydrogen intercalation in antiferromagnetic transition-metal oxides is a promising way to realize multiferroics with strong magnetoelectric coupling. Taking brownmillerite SrCoO_{2.5} as an example, we show that hydrogen intercalated SrCoO_{2.5} displays strong ferrimagnetism and large electric polarization in which the hydroxide acts as a new knob to simultaneously control the magnetization and polarization at room temperature. We expect that ion intercalation will become a general way to design magnetoelectric and spintronic functional materials.

Spin-coupling-induced Improper Polarizations and Latent Magnetization in Multiferroic BiFeO3

Multiferroic BiFeO₃ (BFO) that exhibits a gigantic off-centering polarization (OCP) is the most extensively studied material among all multiferroics. In addition to this gigantic OCP, the BFO having R3c structural symmetry is expected to exhibit a couple of parasitic improper polarizations owing to coexisting spin-polarization coupling mechanisms. However, these improper polarizations are not yet theoretically quantified. Herein, we show that there exist two distinct spin-coupling-induced improper polarizations in the R3c BFO on the basis of the Landau-Lifshitz-Ginzburg theory: ΔP_LF arising from the Lifshitz gradient coupling in a cycloidal spin-density wave, and ΔP_ms originating from the biquadratic magnetostrictive interaction. With the help of ab initio calculations, we have numerically evaluated magnitudes of these improper polarizations, in addition to the estimate of all three relevant coupling constants. We further predict that the magnetic susceptibility increases substantially upon the transition from the bulk R3c BFO to the homogeneous canted spin state in a constrained epitaxial film, which satisfactorily accounts for the experimental observation. The present study will help us understand the magnetoelectric coupling and shed light on design of BFO-based materials with improved multiferroic properties.

Interface-Induced Phenomena in Magnetism

This article reviews static and dynamic interfacial effects in magnetism, focusing on interfacially-driven magnetic effects and phenomena associated with spin-orbit coupling and intrinsic symmetry breaking at interfaces. It provides a historical background and literature survey, but focuses on recent progress, identifying the most exciting new scientific results and pointing to promising future research directions. It starts with an introduction and overview of how basic magnetic properties are affected by interfaces, then turns to a discussion of charge and spin transport through and near interfaces and how these can be used to control the properties of the magnetic layer. Important concepts include spin accumulation, spin currents, spin transfer torque, and spin pumping. An overview is provided to the current state of knowledge and existing review literature on interfacial effects such as exchange bias, exchange spring magnets, spin Hall effect, oxide heterostructures, and topological insulators. The article highlights recent discoveries of interface-induced magnetism and non-collinear spin textures, non-linear dynamics including spin torque transfer and magnetization reversal induced by interfaces, and interfacial effects in ultrafast magnetization processes.

Non-collinear magnetism in multiferroic perovskites

We present an overview of the current interest in non-collinear magnetism in multiferroic perovskite crystals. We first describe the different microscopic mechanisms giving rise to the non-collinearity of spins in this class of materials. We discuss, in particular, the interplay between non-collinear magnetism and ferroelectric and antiferrodistortive distortions of the perovskite structure, and how this can promote magnetoelectric responses. We then provide a literature survey on non-collinear multiferroic perovskites. We discuss numerous examples of spin cantings driving weak ferromagnetism in transition metal perovskites, and of spin-induced ferroelectricity as observed in the rare-earth based perovskites. These examples are chosen to best illustrate the fundamental role of non-collinear magnetism in the design of multiferroicity.

Pressure and strain effects of hexagonal rare-earth manganites: a first-principles study

We have investigated the structural, electrical and magnetic properties as well as the phonon modes of hexagonal rare-earth manganites (RMnO3, R  =  Pr, Nd, Pm, Sm, Gd, Tb, Dy, Ho, Er, Tm and Lu) under chemical pressure, hydrostatic pressure and epitaxial strain by first-principles calculations. The magnetic ground state of RMnO3 is found to have Γ4 magnetic configuration and to be stable under all considered external conditions. In contrast, the K3 phonon mode, which is the primary order parameter and responsible for the 'improper ferroelectricity', is greatly influenced by pressure and epitaxial strain. Consequently, the electric polarization is enhanced by 56.7% when the chemical pressure increases from R  =  Pr to R  =  Lu. The hydrostatic pressure can also improve the polarization to a certain degree, e.g. by 14.7% from 0 GPa to 40 GPa in LuMnO3. Finally, the dependence of polarization on the epitaxial strain is also given, revealing that the compressive strain could promote the ferroelectricity while tensile strain will suppress it.

Anharmonic lattice interactions in improper ferroelectrics for multiferroic design

The design and discovery of new multiferroics, or materials that display both ferroelectricity and long-range magnetic order, is of fundamental importance for new electronic technologies based on low-power consumption. Far too often, however, the mechanisms causing these properties to arise are incompatible or occur at ordering temperatures below room temperature. One design strategy which has gained considerable interest is to begin with a magnetic material, and find novel ways to induce a spontaneous electric polarization within the structure. To this end, anharmonic interactions coupling multiple lattice modes have been used to lift inversion symmetry in magnetic dielectrics. Here we provide an overview of the microscopic mechanisms by which various types of cooperative atomic displacements result in ferroelectricity through anharmonic multi-mode coupling, as well as the types of materials most conducive to these lattice instabilities. The review includes a description of the origins of the displacive modes, a classification of possible non-polar lattice modes, as well as how their coupling can produce spontaneous polarizations. We then survey the recent improper ferroelectric literature, and describe how the materials discussed fall within a proposed classification scheme, offering new directions for the theoretical design of magnetic ferroelectrics. Finally, we offer prospects for the future discovery of new magnetic improper ferroelectrics, as well as detail remaining challenges and open questions facing this exciting new field.

Multiferroics of spin origin

Multiferroics, compounds with both magnetic and ferroelectric orders, are believed to be a key material system to achieve cross-control between magnetism and electricity in a solid with minute energy dissipation. Such a colossal magnetoelectric (ME) effect has been an issue of keen interest for a long time in condensed matter physics as well as a most desired function in the emerging spin-related electronics. Here we begin with the basic mechanisms to realize multiferroicity or spin-driven ferroelectricity in magnetic materials, which have recently been clarified and proved both theoretically and experimentally. According to the proposed mechanisms, many families of multiferroics have been explored, found (re-discovered), and newly developed, realizing a variety of colossal ME controls. We overview versatile multiferroics from the viewpoints of their multiferroicity mechanisms and their fundamental ME characteristics on the basis of the recent advances in exploratory materials. One of the new directions in multiferroic science is the dynamical ME effect, namely the dynamical and/or fast cross-control between electric and magnetic dipoles in a solid. We argue here that the dynamics of multiferroic domain walls significantly contributes to the amplification of ME response, which has been revealed through the dielectric spectroscopy. Another related issue is the electric-dipole-active magnetic resonance, called electromagnons. The electromagnons can provide a new stage of ME optics via resonant coupling with the external electromagnetic wave (light). Finally, we give concluding remarks on multiferroics physics in the light of a broader perspective from the emergent electromagnetism in a solid as well as from the possible application toward future dissipationless electronics.

Magnetic susceptibility and heat capacity measurements of single crystal TbMnO3

Measurements of the magnetic susceptibility χ and heat capacity C on single crystals of the multiferroic TbMnO3 are presented. A non-magnetic isostructural compound, LaGaO3, was used to isolate the magnetic component of the heat capacity. An anisotropic magnetic susceptibility, deviations from Curie-Weiss behaviour and a significant magnetic entropy above the antiferromagnetic ordering temperature TN1 = 41 K are attributed to a combination of crystal-field effects and short-range order between the Mn moments. Heat capacity in a magnetic field applied along the a axis confirms the saturation of Tb(3+) moments in 90 kOe. A hyperfine contribution from the Tb and Mn nuclear moments that may be convolved with a contribution from low-lying Tb crystal-field levels leads to a low-temperature rise in C(T)/T.

Room-temperature spin-spiral multiferroicity in high-pressure cupric oxide

Multiferroic materials, in which ferroelectric and magnetic ordering coexist, are of fundamental interest for the development of multi-state memory devices that allow for electrical writing and non-destructive magnetic readout operation. The great challenge is to create multiferroic materials that operate at room temperature and have a large ferroelectric polarization P. Cupric oxide, CuO, is promising because it exhibits a significant polarization, that is, P~0.1 μC cm(-2), for a spin-spiral multiferroic. Unfortunately, CuO is only ferroelectric in a temperature range of 20 K, from 210 to 230 K. Here, by using a combination of density functional theory and Monte Carlo calculations, we establish that pressure-driven phase competition induces a giant stabilization of the multiferroic phase of CuO, which at 20-40 GPa becomes stable in a domain larger than 300 K, from 0 to T>300 K. Thus, under high pressure, CuO is predicted to be a room-temperature multiferroic with large polarization.

Evidence of antiferromagnetic and ferromagnetic superexchange interactions in bulk TbMn(1-x)Cr(x)O(3)

Powder samples of solid solution TbMn1-xCrxO3 (0 ≤ x ≤ 1) were synthesized via a facile solution route. The substitution of non-Jahn–Teller active Cr3+ for Mn3+ in TbMnO3 was found to decrease the unit cell volume and orthorhombic distortion. TbMn1-xCrxO3 with low Cr content (x ≤ 0.33) exhibited magnetic behavior similar to the pure TbMnO3 sample. However, ferromagnetic-like Mn–Cr interactions were introduced in these samples and maximum magnetic field coercivity and remanence were found at x ~0.33. For x ≥ 0.5, signatures of a canted G-type antiferromagnetic ordering similar to pure TbCrO3 were observed. The Mn3+/Cr3+ spins rotate from parallel to the a-axis to parallel to the c-axis with increasing Cr content. Based on the magnetization results, a magnetic phase diagram for bulk solid solution TbMn1-xCrxO3 has been proposed for the first time.

Charge carriers and small-polaron migration as the origin of intrinsic dielectric anomalies in multiferroic TbMnO3 polycrystals

Temperature-dependent and frequency-dependent dielectric investigations have been performed in TbMnO3 polycrystals sintered in either oxidative or reductive atmospheres. The results revealed the occurrence of two dielectric anomalies above 100 K, which are caused by the thermal activation of charge carriers and their motion in grain cores and grain boundaries. The temperature dependence of the bulk dc conductivity was also analysed and indicates that charge carriers move between inequivalent sites according to a variable-range-hopping mechanism. Also, a strong correlation between dielectric properties and crystalline structure was observed. Furthermore, a low-temperature dielectric relaxation, commonly reported in rare-earth manganite crystals, was observed in both samples. This relaxation follows the empirical Cole-Cole model and was attributed to small-polaron tunnelling. Polaron motion was observed to be affected by the magnetic transitions, structural properties and intrinsic anisotropies in TbMnO3. It is also worth mentioning that the dielectric anomaly due to motion of charge carriers in grain boundaries is the only one of extrinsic origin, while the anomalies related to carrier motion in grain cores and small-polaron tunnelling are intrinsic to TbMnO3.

Melting of chiral order in terbium manganate (TbMnO3) observed with resonant x-ray Bragg diffraction

Resonant Bragg diffraction of soft, circularly polarized x-rays has been used to observe directly the temperature dependence of chiral-order melting in a motif of Mn ions in terbium manganate. The underlying mechanism uses the b-axis component of a cycloid, which vanishes outside the polar phase. Melting is witnessed by the first and second harmonics of a cycloid, and we explain why the observed temperature dependence differs in the two harmonics. Conclusions follow from an exact treatment of diffraction by using atomic multipoles in a circular cycloid, since a standard treatment of the diffraction, based on a single material-vector identified with the magnetic dipole, does not reproduce correctly observations at the second harmonic.

Microscopic origin of large negative magnetoelectric coupling in Sr(1/2)Ba(1/2)MnO3

With a combined ab initio density functional and model Hamiltonian approach we establish that in the recently discovered multiferroic phase of the manganite Sr(1/2)Ba(1/2)MnO3 the polar distortion of Mn and O ions is stabilized via enhanced in-plane Mn-O hybridizations. The magnetic superexchange interaction is very sensitive to the polar bond-bending distortion, and we find that this dependence directly causes a strong magnetoelectric coupling. This novel mechanism for multiferroicity is consistent with the experimentally observed reduced ferroelectric polarization upon the onset of magnetic ordering.

Strong Dzyaloshinskii-Moriya interaction and origin of ferroelectricity in Cu2OSeO3

By performing density functional calculations, we investigate the origin of the Skyrmion state and ferroelectricity in Cu2OSeO3. We find that the Dzyaloshinskii-Moriya interactions between the two different kinds of Cu ions are extremely strong and induce the helical ground state and the Skyrmion state in the absence and presence of a magnetic field, respectively. On the basis of the general model for the spin-order induced polarization, we propose that the ferroelectric polarization of Cu2OSeO3 in the collinear ferrimagnetic state arises from an unusual mechanism, i.e., the single-spin-site contribution due to the spin-orbit coupling.

Valley-selective circular dichroism of monolayer molybdenum disulphide

A two-dimensional honeycomb lattice harbours a pair of inequivalent valleys in the k-space electronic structure, in the vicinities of the vertices of a hexagonal Brillouin zone, K_±. It is particularly appealing to exploit this emergent degree of freedom of charge carriers, in what is termed 'valleytronics'. The physics of valleys mimics that of spin, and will make possible devices, analogous to spintronics, such as valley filter and valve, and optoelectronic Hall devices, all very promising for next-generation electronics. The key challenge lies with achieving valley polarization, of which a convincing demonstration in a two-dimensional honeycomb structure remains evasive. Here we show, using first principles calculations, that monolayer molybdenum disulphide is an ideal material for valleytronics, for which valley polarization is achievable via valley-selective circular dichroism arising from its unique symmetry. We also provide experimental evidence by measuring the circularly polarized photoluminescence on monolayer molybdenum disulphide, which shows up to 50% polarization.

The monolayer transition-metal dichalcogenide molybdenum disulphide has recently attracted attention owing to its distinctive electronic properties. Cao and co-workers present numerical evidence suggesting that circularly polarized light can preferentially excite a single valley in the band structure of this system.

Giant ferroelectric polarization of CaMn7O12 induced by a combined effect of Dzyaloshinskii-Moriya interaction and exchange striction

By extending our general spin-current model to noncentrosymmetric spin dimers and performing density functional calculations, we investigate the causes for the helical magnetic order and the origin of the giant ferroelectric polarization of CaMn7O12. The giant ferroelectric polarization is proposed to be caused by the symmetric exchange striction due to the canting of the Mn4+ spin arising from its strong Dzyaloshinskii-Moriya interaction. Our study suggests that CaMn7O12 may exhibit a novel magnetoelectric coupling mechanism in which the magnitude of the polarization is governed by the exchange striction, but the direction of the polarization by the chirality of the helical magnetic order.

General theory for the ferroelectric polarization induced by spin-spiral order

The ferroelectric polarization of triangular-lattice antiferromagnets induced by helical spin-spiral order is not explained by any existing model of magnetic-order-driven ferroelectricity. We resolve this problem by developing a general theory for the ferroelectric polarization induced by spin-spiral order and then by evaluating the coefficients needed to specify the general theory on the basis of density functional calculations. Our theory correctly describes the ferroelectricity of triangular-lattice antiferromagnets driven by helical spin-spiral order and incorporates known models of magnetic-order-driven ferroelectricity as special cases.

Dynamical magnetoelectric effects in multiferroic oxides

Multiferroics with coexistent ferroelectric and magnetic orders can provide an interesting laboratory to test unprecedented magnetoelectric (ME) responses and their possible applications. One such example is the dynamical and/or resonant coupling between magnetic and electric dipoles in a solid. As examples of such dynamical ME effects, (i) the multiferroic domain wall dynamics and (ii) the electric dipole active magnetic responses are discussed with an overview of recent experimental observations.

Femtoscale Magnetically Induced Lattice Distortions in Multiferroic TbMnO3

Magneto-electric multiferroics exemplified by TbMnO3 possess both magnetic and ferroelectric long-range order. The magnetic order is mostly understood, whereas the nature of the ferroelectricity has remained more elusive. Competing models proposed to explain the ferroelectricity are associated respectively with charge transfer and ionic displacements. Exploiting the magneto-electric coupling, we used an electric field to produce a single magnetic domain state, and a magnetic field to induce ionic displacements. Under these conditions, interference between charge and magnetic x-ray scattering arose, encoding the amplitude and phase of the displacements. When combined with a theoretical analysis, our data allow us to resolve the ionic displacements at the femtoscale, and show that such displacements make a substantial contribution to the zero-field ferroelectric moment.

Chiral order and electromagnetic dynamics in one-dimensional multiferroic cuprates

We show by unbiased numerical calculations that the ferromagnetic nearest-neighbor exchange interaction stabilizes a vector spin chiral order against the quantum fluctuation in a frustrated spin-1/2 chain relevant to multiferroic cuprates, LiCu2O2 and LiCuVO4. Our realistic semiclassical analyses for LiCu2O2 resolve controversies on the helical magnetic structure and unveil the pseudo-Nambu-Goldstone modes as the origin of experimentally observed electromagnons.

Temperature-dependent magnetoelectric effect from first principles

We show that nonrelativistic exchange interactions and spin fluctuations can give rise to a linear magnetoelectric effect in collinear antiferromagnets at elevated temperatures that can exceed relativistic magnetoelectric responses by more than 1 order of magnitude. We show how symmetry arguments, ab initio methods, and Monte Carlo simulations can be combined to calculate temperature-dependent magnetoelectric susceptibilities entirely from first principles. The application of our method to Cr2O3 gives quantitative agreement with experiment.

Spin model of magnetostrictions in multiferroic Mn perovskites

We theoretically study origins of the ferroelectricity in the multiferroic phases of the rare-earth (R) Mn perovskites, RMnO(3), by constructing a realistic spin model including the spin-phonon coupling, which reproduces the entire experimental phase diagram in the plane of temperature and Mn-O-Mn bond angle for the first time. Surprisingly we reveal a significant contribution of the symmetric (S·S)-type magnetostriction to the ferroelectricity even in a spin-spiral-based multiferroic phase, which can be larger than the usually expected antisymmetric (S×S)-type contribution. This explains well the nontrivial behavior of the electric polarization. We also predict the noncollinear deformation of the E-type spin structure and a wide coexisting regime of the E and spiral states, which resolve several experimental puzzles.

Multiferroics with spiral spin orders

Cross correlation between magnetism and electricity in a solid can host magnetoelectric effects, such as magnetic (electric) induction of polarization (magnetization). A key to attain the gigantic magnetoelectric response is to find the efficient magnetism-electricity coupling mechanisms. Among those, recently the emergence of spontaneous (ferroelectric) polarization in the insulating helimagnet or spiral-spin structure was unraveled, as mediated by the spin-exchange and spin-orbit interactions. The sign of the polarization depends on the helicity (spin rotation sense), while the polarization direction itself depends on further details of the mechanism and the underlying lattice symmetry. Here, we describe some prototypical examples of the spiral-spin multiferroics, which enable some unconventional magnetoelectric control such as the magnetic-field-induced change of the polarization direction and magnitude as well as the electric-field-induced change of the spin helicity and magnetic domain.

Magnetoelectric response of multiferroic BiFeO3 and related materials from first-principles calculations

We present a first-principles scheme for computing the magnetoelectric response of multiferroics. We apply our method to BiFeO3 (BFO) and related compounds in which Fe is substituted by other magnetic species. We show that under certain relevant conditions--i.e., in the absence of incommensurate spin modulation, as in BFO thin films and some BFO-based solid solutions--these materials display a large linear magnetoelectric response. Our calculations reveal the atomistic origin of the coupling and allow us to identify the most promising strategies to enhance it.

Multiferroicity in rare-earth nickelates RNiO3

We show that charge ordered rare-earth nickelates of the type RNiO3 (R = Ho, Lu, Pr and Nd) are multiferroic with very large magnetically-induced ferroelectric (FE) polarizations. This we determine from first principles electronic structure calculations. The emerging FE polarization is directly tied to the long-standing puzzle of which kind of magnetic ordering is present in this class of materials: its direction and size indicate the type of ground-state spin configuration that is realized. Vice versa, the small energy differences between the different magnetic orderings suggest that a chosen magnetic ordering can be stabilized by cooling the system in the presence of an electric field.

First principles studies of multiferroic materials

Multiferroics, materials where spontaneous long-range magnetic and dipolar orders coexist, represent an attractive class of compounds, which combine rich and fascinating fundamental physics with a technologically appealing potential for applications in the general area of spintronics. Ab initio calculations have significantly contributed to recent progress in this area, by elucidating different mechanisms for multiferroicity and providing essential information on various compounds where these effects are manifestly at play. In particular, here we present examples of density-functional theory investigations for two main classes of materials: (a) multiferroics where ferroelectricity is driven by hybridization or purely structural effects, with BiFeO(3) as the prototype material, and (b) multiferroics where ferroelectricity is driven by correlation effects and is strongly linked to electronic degrees of freedom such as spin-, charge-, or orbital-ordering, with rare-earth manganites as prototypes. As for the first class of multiferroics, first principles calculations are shown to provide an accurate qualitative and quantitative description of the physics in BiFeO(3), ranging from the prediction of large ferroelectric polarization and weak ferromagnetism, over the effect of epitaxial strain, to the identification of possible scenarios for coupling between ferroelectric and magnetic order. For the second class of multiferroics, ab initio calculations have shown that, in those cases where spin-ordering breaks inversion symmetry (e.g. in antiferromagnetic E-type HoMnO(3)), the magnetically induced ferroelectric polarization can be as large as a few µC cm(-2). The examples presented point the way to several possible avenues for future research: on the technological side, first principles simulations can contribute to a rational materials design, aimed at identifying spintronic materials that exhibit ferromagnetism and ferroelectricity at or above room temperature. On the fundamental side, ab initio approaches can be used to explore new mechanisms for ferroelectricity by exploiting electronic correlations that are at play in transition metal oxides, and by suggesting ways to maximize the strength of these effects as well as the corresponding ordering temperatures.

Relaxations as key to the magnetocapacitive effects in the perovskite manganites

We present a detailed dielectric study of the relaxation effects that occur in several perovskite rare-earth manganites, including the multiferroics TbMnO(3) and DyMnO(3). We demonstrate that the strong magnetocapacitive effects, observed for electrical fields E parallel c, are nearly completely governed by magnetic-state induced changes of the relaxation parameters. The multiferroic materials, which undergo a transition into a spiral magnetic state, show qualitatively different relaxation behavior than those compounds transferring into an A-type antiferromagnetic state. We ascribe the relaxations in both cases to the off-center motion of the manganese ions, which in the multiferroic systems also leads to the ferroelectric ordering.

Epitaxial TbMnO(3) thin films on SrTiO(3) substrates: a structural study

TbMnO(3) films have been grown under compressive strain on (001)-oriented SrTiO(3) crystals. They have an orthorhombic structure and display the (001) orientation. With increasing thickness, the structure evolves from a more symmetric (tetragonal) to a less symmetric (bulk-like orthorhombic) structure, while keeping constant the in-plane compression, thereby leaving the out-of-plane lattice spacing unchanged. The domain microstructure of the films is also revealed, showing an increasing number of orthorhombic domains as the thickness is decreased: we directly observe ferroelastic domains as narrow as 4 nm. The high density of domain walls may explain the induced ferromagnetism observed in the films, while both the decreased anisotropy and the small size of the domains could account for the absence of a ferroelectric spin spiral phase.

Polarization and adiabatic pumping in inhomogeneous crystals

We develop a general theory of electric polarization in crystals with inhomogeneous order. We show that the inhomogeneity-induced polarization can be classified into two parts: a perturbative contribution stemming from a correction to the basis functions and a topological contribution described in terms of the Chern-Simons form of the Berry gauge fields. The latter is determined up to an uncertainty quantum, which is the second Chern number in appropriate units. Our theory provides an exhaustive link between microscopic models and the macroscopic polarization.

Magnetically induced ferroelectricity in TbMnO(3): inverse Goodenough-Kanamori interaction

Improper ferroelectricity in magnets, as induced by non-centrosymmetric spin-, charge- or orbital-ordering, is a branch of the field of multiferroics having fascinating physics and a potentially important technological outcome. We focus here on ferroelectricity in orthorhombic TbMnO(3), where the magnetic field along the a-axis produces a polar collinear spin-arrangement with a rather large in-plane electric polarization. The mechanism, similar to that occurring in orthorhombic HoMnO(3) in the AFM-E phase, is efficiently driven by a large modification of the structural properties (such as MnO bond-lengths and Mn-O-Mn bond-angles) to favor e(g) electron hopping between Mn with parallel spins. A similar mechanism where the t(2g) states are involved is examined through a hypothetical collinear spin-structure, resulting in a weaker out-of-plane ferroelectric polarization.

Spin-orbit coupling and ion displacements in multiferroic TbMnO3

The magnetic and ferroelectric (FE) properties of TbMnO3 are investigated on the basis of relativistic density functional theory calculations. We show that, due to spin-orbit coupling, the spin-spiral plane of TbMnO3 can be either the bc or ab plane, but not the ac plane. As for the mechanism of FE polarization, our work reveals that the "pure electronic" model by Katsura, Nagaosa, and Balatsky is inadequate in predicting the absolute direction of FE polarization. Our work indicates that to determine the magnitude and the absolute direction of FE polarization in spin-spiral states, it is crucial to consider the displacements of the ions from their centrosymmetric positions.