Session D12: Theory of Ferroelectric, Multiferroic, and Other Structural Transitions

Dynamics of domains switching in epitaxial BaTiO$_3$/SrTiO$_3$ superlattices from first principles

Superlattices (SL) consisting of alternating layers of perovskite oxides can possess properties that are dramatically different from those of bulk ferroelectrics. [BaTiO$_3$]$_n$/[SrTiO$_3$]$_n$ (BT/ST) SL with relatively large periods exhibit novel nanostripe domains for some specific epitaxial strains and within a particular temperature window [1]. Here, an effective Hamiltonian approach is used within molecular dynamics method to predict the evolution of these nanostripe domains in BT/ST SL under an {\it ac} electric field applied along the SL growth direction. For any investigated frequency, four different regions occur, depending on the magnitude of the electric field: Region I that consists of nanostripe domains in both BT and ST layers; Region II that exhibits nanostripe domains in BT layers while possessing monodomains in ST layers; Region III where bubble domains in BT layers coexist with monodomains in ST layers; and Region IV where monodomains form in both BT and ST layers. The dependency of the domain velocities, activation and critical fields on the field frequency is revealed. [1] Lisenkov et al., {\em Phys. Rev. B}, {\bf 76}, 020102(R) (2007).   

Ferromagnetic-like closure domains in ferroelectric ultrathin films: First principles simulation

We simulate from first-principles the energetic, structural, and electronic properties of ferroelectric domains in ultrathin capacitors made of a few unit cells of BaTiO$_3$ between two metallic SrRuO$_3$ electrodes in short circuit. The domains are stabilized down to two unit cells, adopting the form of a domain of closure, common in ferromagnets but only recently detected experimentally in ferroelectric thin films. The domains are closed by the in-plane relaxation of the atoms in the first SrO layer of the electrode, that behaves more like SrO in highly polarizable SrTiO$_3$ than in metallic SrRuO$_3$. Even if small, these lateral displacements are essential to stabilize the domains, and might provide some hints to explain why some systems break into domains while others remain in a monodomain configuration. An analysis of the electrostatic potential reveals preferential points of pinning for charged defects at the ferroelectric-electrode interface, possibly playing a major role in films fatigue.   

The theory of domain patterns in ferroelastics

The theory of ferroelastic domain patterns is well developed and it explains qualitatively the domain patterns observed experimentally in many materials. It is known that much of the complexity of these patterns, which differ remarkably from those in conventional order-parameter systems, results from the differential rotation associated with domain walls. But there has been no direct confrontation between theory and experiment at a more quantitative level. I propose here such a confrontation. Specifically, I shall present predictions (obtained from numerical work) of the atomic displacements at the collision of orthogonal domain walls in tetragonal-orthorhombic ferroelastics and suggest that they be compared with HREM images of these materials.   

Peculiar ordering in flat ferroelectric nanoparticles driven by lattice misfit

Flat \emph{free-standing} ferroelectric (FE)\ nanopartilcles tend to have a vortex-like polarization ordering with the in-plane polarization that curls around an out-of-plane vortex core axis. The question is, if such a structure is still a ground state in presence of noticeable misfit strains induced by a substrate, and whether the 180$^{\circ}$ stripe domains may form, similarly to the case of ultra thin FE films? Here we perform an \textit{ab initio} based study of disk- shaped Pb(Zr$_{1-x} $Ti$_{x}$)O$_{3}$ and BaTiO$_{3}$ nanoparticles that have the vortex ground state when no stress is applied. Our study leads to the following findings for the disks having circular and square footprint: (i) under strong enough compressive strains the curling state is no longer stable and yields to a multi-domain structure with an out-of-plane polarization, and different possible in-plane domains:\ triangle, stripe-like, or ``bubble'' in shape, (ii) each separate domain, regardless of its shape, runs through the entire thickness of a disk, and (iii) the 180$^{\circ}$ stripes occur only under special conditions depending on the shape and chemical composition of the nanostructures. Further, we discovered that starting with the vortex state and then increasing the compressive strains may lead to a metastable bi- or multi-domain phase different from the ground state obtained by gradual cooling at a fixed strain. This leads to a novel hysteretic behavior as a function of the misfit strain.   

Static and Dynamic Properties of Ferroelectric Nanostructures and Multiferroic Bulk Systems: A Multiscale Approach

Ferroelectric nanostructures and multiferroic bulk systems are studied in a multiscale approach. The excitation energy, associated damping of ferroelectric modes and polarization are presented as a function of temperature, defect concentration, size and shape. The softening of the mode is strongly influenced by the kind of doping ions, the surface configuration and the defect composition. The analysis is based on a modified Ising model in a transverse field. A Green's function technique in real space provides the static and dynamic properties, which differ significantly from the bulk behavior. Additionally, a mesoscopic approach is carried out similar to the Landau-Lifshitz equation with Gilbert damping for ferromagnets. The temperature dependence of the damping parameters is discussed. The analysis is extended to multiferroic bulk systems, where the magnetic moments interact via the Heisenberg model and the multiferroic coupling term differs for hexagonal and orthorhombic materials. We present the dielectric function and the dynamic properties of the coupled model by applying previous methods.   

A new order parameter in complex dipolar structures

Microscopic models have been used to reveal the existence of a new order parameter that is associated with many complex dipolar structures in magnets and ferroelectrics. This overlooked order parameter involves a double cross product of the local dipoles with their positions. It provides a measure of subtle microscopic features, such as the helicity of the two domains inherent to onion states, curvature of the dipolar pattern in flower states or characteristics of set of vortices with opposite chirality (e.g., distance between vortices' centers and/or magnitude of their local dipoles). This work is mostly supported by DOE grant DE-FG02-05ER46188. We also acknowledge support from ONR grant N00014-04-1-0413 and NSF grants DMR-0701558, DMR-0404335 and DMR-0080054 (C-SPIN). Some computations were made possible thanks to the MRI Grants 0421099 and 0722625 from NSF.   

Microscopic origin of Magnetic Ferroelectrics in Non-collinear Multiferroics

We propose a microscopic mechanism to understand the origin of magnetoelectric coupling in the best known multiferroic family $R$MnO$_3$ ($R$=Tb, Dy...). The mechanism lies in the vanishing of electric current in an insulator. A spontaneous electric polarization is thus necessary, as it causes an electric current through a spin-orbit coupling to cancel another nonzero local electric current induced by a non-collinear modulated magnetic structure. Within this counter-balance mechanism, the magnitude of the ferroelectric order is determined by the magnetic order parameter and the spin-orbit coupling strength. Based on the theory, we predict a general physical limit for the value of ferroelectricity.   

A theory for the multiferroic compound LiCu$_{2}$O$_{2}$

We investigate the possible coupling between ferroelectricity and magnetic structure in the zig-zag spin chain compound LiCu$_{2}$O$_{2}$. Based on a group theory analysis, we construct a multi-order parameter phenomenological model and show that a coupling involving the inter-chain magnetic structures and ferroelectricity is necessary in order to understand the experimental results of Park \emph{et. al}. The model is able to account for the electric polarization flip through $\pi /2$ and explain the occurrence of an electric polarization parallel to an applied external magnetic field.   

Phase-field model of strain-induced grain-boundary premelting

Grain-boundary premelting depends in a complex way on the relative magnitude of the solid-liquid interfacial free-energy and grain boundary energy as well as temperature and strain. We study this dependence in a bicrystal geometry using a phenomenological three-order parameter phase-field model. This model describes the short scale attractive or repulsive interaction between crystal-melt interfaces and macroscopic linear elasticity including the important effect of the density contrast between solid and liquid. The model exhibits a rich behavior characterized by single or multiple premelting transitions between dry or wet grain boundaries with different liquid layer thicknesses as a function of applied tensile stress. The results have important implications for the phenomenon of liquid metal embrittlement associated with the stress-driven penetration of nanometric liquid films along grain boundaries.   

Incorporation of plasticity into the Landau-Ginzburg theory of martensitic phase transformations

The Landau-Ginzburg theory of martensitic phase transformations has been utilized to reproduce the evolution of elastic texture in defect-free materials undergoing structural phase transformations. Generalizations of this theory to phase transformations that are accompanied by significant plastic distortions (as in U$_6$Nb) have been little studied. We propose a simple model that demonstrates how to incorporate plasticity into the Landau-Ginzburg theory. In the presence of topological defects such as dislocations, the usual Saint Venant compatibility constraint becomes an incompatibility constraint and this is represented by a tensor field $\eta_{ij}$. In our case, the components of $\eta_{ij}$ are expressed as gradients of the components of the Nye tensor that represent the dislocation density. The presence of dislocations induces large internal stresses in certain regions of the material, and these act as initiation sites for plastic deformation. When the external loading is applied, dislocations moving from these regions cause strain hardening that is detectable in experimental uniaxial measurements. This model serves as a starting point for further development of the framework of three-dimensional rate-independent theory of plasticity within the Landau-Ginzburg formalism.   

Phonon Driven BCC to Orthorhombic Transformation in U-Nb Alloys

The martensitic transformation in uranium alloys is of great strategic importance. We study the crystallography and model the well characterized BCC to orthorhombic phase tranformation in the shape memory alloy U-Nb for low Nb concentrations. Our predictions are consistent with the experimentally observed orientation relationship between the BCC and orthorhombic phases. We find that this temperature induced transformation is driven by a specific zone boundary phonon that couples to a particular shear mode. We also obtain a Landau free energy for this transformation. In addition, we compare our results with a similar shuffle based mechanism in a related martensitic alloy AuZn.   

First-Principles Study of the Jahn-Teller Distortion in the Ti$_{1-X}$V$_{X}$H$_{2}$ and Zr$_{1-X}$Nb$_{x}$H$_{2}$ Alloys

The transition metal dihydrides TiH$_{2}$ and ZrH$_{2}$ present the fluorite structure (CaF$_{2})$ at high temperature but undergoes a tetragonal distortion with c/a$<$1 at low temperature. Electronic band structure calculations have shown that TiH$_{2}$ and ZrH$_{2}$ in the cubic phase display a very flat band at the Fermi level. Thus the low temperature tetragonal distortion has been associated to a Jahn-Teller effect. In order to understand the role of band filling in controlling the structural instability of the transition metal dihydrides, we have performed a first-principles total energy study of the Ti$_{1-X}$V$_{x}$H$_{2}$ and Zr$_{1-x}$Nb$_{x}$H$_{2}$ alloys. The calculations were performed using FP-LAPW method within the (DFT) and we use the GGA for exchange correlation functional energy. The critical concentration for which the Jahn-Teller effect is suppressed, was determined from the evolution of the tetragonal-cubic energy barrier. We discuss the electronic mechanism of the structural-instability, in terms of the band filling. From the obtained results we conclude that the tetragonal distortion in TiH$_{2}$ and ZrH$_{2}$ is not produced only by a Jahn-Teller Effect. This research was supported by Consejo Nacional de Ciencia y Tecnolog\'{\i}a (Conacyt) under Grant No. 43830-F.   

Density functional analysis of long range order

A density functional analysis of order-disorder transitions in alloys shows that ordering energy is stored in superlattice wavevectors. Thermal vibrations play a key role and lower the transition temperature, T$_{c}$, to the experimental value (741 K) from the mean-field value (933 K) in the Bragg-Williams model for beta brass, $\beta $-CuZn. An isotope effect with 4 K difference in T$_{c}$ is predicted for $^{63}$Cu$^{64}$Zn and $^{65}$Cu$^{68}$Zn in a Modified Bragg-Williams model. The above conclusions are shown to be applicable in magnetic transitions as well. Theoretical analysis shows that thermal vibrations alter the exchange and total magnetic ordering energy. Every microscopic theory of magnetic and alloy phase transitions must satisfy the twin criteria that ordering energy is a) stored in superlattice wavevectors and b) a function of temperature due to thermal vibrations. An isotope effect is predicted to be a universal feature of alloy and magnetic phase transitions. The nuclear-nuclear energy term, $E_{n-n}$, converges without artificial parameters if zero point vibrations are included unlike the Ewald sum technique.   

Ab-initio study of the structural and magnetic properties for the fcc Fe-Co alloy.

We have studied the structural and magnetic properties of the fcc Fe-Co alloy by means of first-principles calculations. For modeling the alloy we have used the ab-initio self-consistent Virtual Crystal Approximation. The ground state properties was calculated with the Fixed Spin Moment methodology and the Full-Potential LAPW method. For the exchange-correlation potential we have used the Generalized Gradient Approximation. For ferromagnetic fcc-Fe we find an anti-invar behavior (ELS$<$EHS) with the co-existence of two ferromagnetic states (metamagnetism). For the fcc-FeCo alloy we find a progressive evolution of the metamagnetism with the Co concentration. Using the calculated total-energy vs the lattice parameter and the Boltzman distribution function, we have obtained the lattice parameter as function of the temperature, in order to determine the thermal expansion coefficient $\alpha$ as function of the Co-concentration. We find that Fe65Co35 show an invar behavior.