Strain Order Parameters Research Articles

Ferroelectric HfO2 and the importance of strain

Ferroelectric oxides based on ${\mathrm{HfO}}_{2}$ show tremendous promise for the next generation of memory and logic devices. The ferroelectric polymorph is one of several that can be derived from the high symmetry cubic fluorite structure of ${\mathrm{HfO}}_{2}$. A single grain of ${\mathrm{HfO}}_{2}$ may consist of a coherent mixture of multiple orientational and translational variants of different polymorphs. Here, we use symmetry-adapted strain-order parameters to elucidate the relationship between the different ${\mathrm{HfO}}_{2}$ polymorphs and their symmetrically equivalent variants. We use first-principles electronic structure methods to identify minimum energy pathways and map them in subspaces of the symmetry-adapted strain order parameters. We next investigate the atomic structure of domain boundaries that separate coexisting variants of ferroelectric ${\mathrm{HfO}}_{2}$. We rely on Gibbsian excess quantities and a precise specification of mechanical boundary conditions to describe the thermodynamic properties of domain boundaries. Our first-principles calculations show that the O and Hf shuffle arrangement within a domain boundary is closely related to the intermediate shuffle patterns of the homogeneous pathways between ferroelectric variants. Furthermore, the preferred structure within a boundary is very sensitive to local strain constraints imposed by the adjacent ferroelectric variants, leading to highly anisotropic domain boundary energies.

Octahedral rotations in Ruddlesden-Popper layered oxides under pressure from first principles

The combination of reduced dimensionality and tunable structural distortions in layered perovskite oxides makes these materials ideal platforms for designing novel properties and functionalities. One example is hybrid improper ferroelectricity in $n=2$ Ruddlesden-Popper oxides, where the combination of a layered crystal structure and rotations of the metal-oxide octahedra break symmetry and induce a polarization. Precisely controlling the octahedral rotation distortions, for example by the application of hydrostatic pressure, provides a pathway to tune and optimize the properties of these materials. We combine group theoretic methods, density functional theory calculations, and Landau theory analysis to investigate how octahedral rotations respond to pressure in the hybrid improper ferroelectrics ${\mathrm{Sr}}_{3}{\mathrm{Zr}}_{2}{\mathrm{O}}_{7}, {\mathrm{Ca}}_{3}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$, and ${\mathrm{Sr}}_{3}{\mathrm{Sn}}_{2}{\mathrm{O}}_{7}$. We find that factors that are known to control the pressure response of $\mathrm{A}\mathrm{B}{\mathrm{O}}_{3}$ perovskites---the formal charge of the $A$- and $B$-site cations, tolerance factor, and $B$-site chemistry---also impact the pressure response of these layered perovskites. We also show that coupling between the octahedral rotation and strain order parameters plays a key role in determining the overall pressure response. Despite some similarities, we find that these layered perovskites display a distinct pressure response compared to their $\mathrm{A}\mathrm{B}{\mathrm{O}}_{3}$ perovskite analogs. By identifying trends and underlying mechanisms that control octahedral rotations in Ruddlesden-Popper oxides under pressure, this work lays the foundation for tailoring the structure and properties of these materials.

Structural and thermal transport properties of ferroelectric domain walls in GeTe from first principles

Ferroelectric domain walls are boundaries between regions with different polarization orientations in a ferroelectric material. Using first-principles calculations, we characterize all different types of domain walls forming on ($11\overline{1}$), (111), and ($1\overline{1}0$) crystallographic planes in thermoelectric GeTe. We find large structural distortions in the vicinity of most of these domain walls, which are driven by polarization variations. We show that such strong strain-order parameter coupling will considerably reduce the lattice thermal conductivity of GeTe samples containing domain walls with respect to a single crystal. Our results thus suggest that domain engineering is a promising path for enhancing the thermoelectric figure of merit of GeTe.

Intermittent dynamics in externally driven ferroelastics and strain glasses

The interplay of elastic anisotropy and disorder dictates many of the properties of ferroic materials, specifically martensites. We use a phase-field model for ferroelastic athermal materials to study their response to an increasing external stress that couples to the strain order parameter. We show that these systems evolve through avalanches and study the avalanche-size distribution for ferroelastic systems (large anisotropy and/or small disorder) and for the strain glass (small anisotropy and/or large disorder) using various statistical analysis techniques, including the maximum likelihood method. The model predicts that in the former case the distribution is subcritical or power law (in agreement with experimental observations), whereas in the latter case it becomes supercritical. Our results are consistent with experiments on martensitic materials, and we predict specific avalanche behavior that can be tested and used as an alternative means to characterize strain glasses.

Octahedral tilting instabilities in inorganic halide perovskites

Dynamic instabilities, stabilized by anharmonic interactions in cubic and tetragonal halide perovskites at high temperature, play a role in the electronic structure and optoelectronic properties of halide perovskites. In particular, inorganic and hybrid perovskite materials undergo structural phase transitions associated with octahedral tilts of the metal-halide octahedra. We investigate the structural instabilities present in inorganic $\mathrm{Cs}M{X}_{3}$ perovskites with Pb or Sn on the metal site and Br or I on the $X$ site. Defining primary order parameters in terms of symmetry-adapted collective displacement modes and secondary order parameters in terms of symmetrized Hencky strain components, we unravel the coupling between octahedral tilt modes and macroscopic strains as well as the role of $A$-site displacements in perovskite phase stability. Symmetry-allowed secondary strain order parameters are enumerated for the 14 unique perovskite tilt systems. Using first-principles calculations to explore the Born-Oppenheimer energy surface in terms of symmetrized order parameters, we find coupling between octahedral tilting and $A$-site displacements is necessary to stabilize $Pnma$ ground states. Additionally, we show that the relative stability of an inorganic halide perovskite tilt system correlates with the volume decrease from the high-symmetry cubic phase to the low-symmetry distorted phase.

Phase and structural stability in Ni-Al systems from first principles

We report on a comprehensive first-principles study of phase stability in the Ni-Al binary, both at zero Kelvin and at finite temperature. First-principles density functional theory calculations of the energies of enumerated orderings on fcc and the sublattices of B2 not only predict the stability of known phases, but also reveal the stability of a family of ordered phases that combine features of $\mathrm{L}{1}_{2}$ and $\mathrm{L}{1}_{0}$ in different ratios to adjust their overall composition. The calculations also confirm the stability of vacancy ordered B2 derivatives that are stable in the Al-rich half of the phase diagram. We introduce strain order parameters to systematically analyze instabilities with respect to the Bain path connecting the fcc and bcc lattices. Many unstable orderings on both fcc and bcc are predicted around compositions of ${x}_{\text{Ni}}=0.625$, where a martensitic phase transformation is known to occur. Cluster expansion techniques together with Monte Carlo simulations were used to calculate a finite-temperature-composition phase diagram of the Ni-Al binary. The calculated phase diagram together with an analysis of Bain instabilities reveals the importance of anharmonicity in determining the phase bounds between the B2 based $\ensuremath{\beta}$ phase and the $\mathrm{L}{1}_{2}$ based ${\ensuremath{\gamma}}^{\ensuremath{'}}$ phase, as well as properties related to martensitic transformations that are observed upon quenching Ni-rich $\ensuremath{\beta}$.

Nanoscaled Martensitic Domains in Ferroelastic Systems: Strain Glass

Strain glass, a frozen disordered ferroelastic state characterized by randomly distributed nanoscaled, non-ergodic martensitic domains, was recently found in ferroelastic TiNi-based alloys. Point defects like anti-site defects or dopants are found essential for the formation of the strain glass, but their role has remained obscure. In this paper, phase field method is employed to simulate how point defects change a normal ferroelastic system into a strain glass and as the result to clarify the role of point defects. We found that the local symmetry-breaking caused by the anisotropic local strain field of point defects plays a central role in the formation of strain glass. Such point-defect-induced local symmetry-breaking hinders the formation of long-range strain-ordering (twinned martensite) and results in a decrease in transition temperature Tc and a reduction of ferroelastic domain size. Above a critical defect concentration, the system transforms to a frozen state of nano-sized ferroelastic domains, i.e., strain glass. In addition, our simulation shows a gradual vanishing of heat capacity peak at the transition temperature with increasing defect concentration, and it also reproduces a characteristic non-ergodic ZFC/FC behavior of the glass state, which is in good agreement with recent experimental observations. Our results also indicate that the concentration fluctuation effect of point defects cannot stabilize a strain glass to very low temperature; thus the direct interaction between the random anisotropic strain field caused by point defects and the strain order parameter of the system is crucial for the formation of strain glass. Keywords: Martensitic phase transition, nanoscaled martensite, phase field, point defect, strain glass.

Ferroelasticity in a metal–organic framework perovskite; towards a new class of multiferroics

A metal–organic framework perovskite, [(CH2)3NH2][Mn(HCOO)3], exhibits a weakly first order ferroelastic phase transition at ∼272K, from orthorhombic Pnma to monoclinic P21/n, and a further transition associated with antiferromagnetic ordering at ∼8.5K. The main structural changes, through the phase transition, are orientational ordering of the azetidium groups and associated changes in hydrogen bonding. In marked contrast to conventional improper ferroelastic oxide perovskites, the driving mechanism is associated with the X-point of the cubic Brillouin zone rather than being driven by R- and M-point octahedral tilting. The total ferroelastic shear strain of up to ∼5% is substantially greater than found for typical oxide perovskites, and highlights the potential of the flexible framework to undergo large relaxations in response to local structural changes. Measurements of elastic and anelastic properties by resonant ultrasound spectroscopy show some of the characteristic features of ferroelastic materials. In particular, acoustic dissipation below the transition point can be understood in terms of mobility of twin walls under the influence of external stress with relaxation times on the order of ∼10−7s. Elastic softening as the transition is approached from above is interpreted in terms of coupling between acoustic modes and dynamic local ordering of the azetidium groups. Subsequent stiffening with further temperature reduction is interpreted in terms of classical strain–order parameter coupling at an improper ferroelastic transition which is close to being tricritical. By way of contrast, there are no overt changes in elastic or anelastic properties near 9K, implying that any coupling of the antiferromagnetic order parameter with strain is weak or negligible.

Band Jahn–Teller effect on the density of states of the magnetic high-Tc superconductors: A model study

We report here a mean-field study of competing antiferromagnetism, superconductivity and lattice strain phases and their effect on the local density of states of the cuprate system. Our model Hamiltonian incorporating these interactions is reported earlier [G.C. Rout et al., Physica C, 2007]. The analytic expression for superconducting, antiferromagnetism and lattice strain order parameters are calculated and solved self-consistently. The interplay of these order parameters is investigated considering the calculated density of states (DOSs) of the conduction electrons. The DOS displays multiple gap structures with multiple peaks. It is suggested that the tunneling conductance data obtained from the scanning tunneling microscopy (STM) measurements could be interpreted by using the quasi-particle bands calculated from our model Hamiltonian. We have discussed the mechanism to calculate the order parameters from the conductance data.

Modeling Abnormal Strain States in Ferroelastic Systems: The Role of Point Defects

Recent experiments have revealed a rich variety of strain states in doped ferroelastic systems. We study the origin of two abnormal strain states; precursory tweed and strain glass, and their relationship with the well-known austenite and martensite (the para- and ferroelastic states). A Landau free energy model is proposed, which assumes that point defects alter the global thermodynamic stability of martensite and create local lattice distortions that interact with the strain order parameters and break the symmetry of the Landau potential. Phase field simulations based on the model have predicted all the important signatures of a strain glass found in experiment. Moreover, the generic "phase diagram" constructed from the simulation results shows clearly the relationships among all the strain states, which agrees well with experimental measurements.

Theoretical model of the coupled magnetostructural phase transitions in Heusler Ni-Mn-In alloys by Monte Carlo simulation

On the basis of classical Monte Carlo simulations, we investigate the temperature and magnetic field dependencies of magnetization, strain order parameter, specific heat and entropy of Ni-Mn-In Heusler alloy, in which the part of the Mn atoms interact antiferromagnetically. It is shown that this antiferromagnetic exchange is responsible for existence of several magnetic phase transitions in the Heusler alloys. It is shown that with the help of the model Hamiltonian we obtain coupled magnetostructural phase transition. Results of simulations are in good qualitatively agreement with the experimental data.

Two-dimensional model for ferromagnetic martensites

We consider a recently introduced 2-D square-to-rectangle martensite model that explains several unusual features of martensites to study ferromagnetic martensites. The strain order parameter is coupled to the magnetic order parameter through a 4-state clock model. Studies are carried out for several combinations of the ordering of the Curie temperatures of the austenite and martensite phases and, the martensite transformation temperature. We find that the orientation of the magnetic order which generally points along the short axis of the rectangular variant, changes as one crosses the twin or the martensite-austenite interface. The model shows the possibility of a subtle interplay between the growth of strain and magnetic order parameters as the temperature is decreased. In some cases, this leads to qualitatively different magnetization curves from those predicted by earlier mean field models. Further, we find that strain morphology can be substantially altered by the magnetic order. We have also studied the dynamic hysteresis behavior. The corresponding dissipation during the forward and reverse cycles has features similar to the Barkhausen's noise.

Charge ordering and elastic anomalies inPr0.65Ca0.35MnO3

Ultrasonic sound velocity and attenuation at 15 MHz, the Youngmodulus at 5 kHz and the shear modulus at 1 Hz were measured forpolycrystalline Pr0.65Ca0.35MnO3 in thetemperature range 100-300 K. Upon cooling from room temperaturea common behaviour can be observed for the different moduli: asmall softening above the charge-ordering temperature TCO~230 K, followed by a large hardening belowTCO. A corresponding attenuation peak is observed atTCO in ultrasound measurements. The modulus effect isrelated to a strain-order parameter coupling proportional toQ2, where Q is the order parameter. The critical exponentβ estimated from the elastic constant behaviour is muchsmaller than the mean-field one.

Attenuation of low-frequency acoustic energy in Bi-based superconductors

The attenuation of low-frequency acoustic energy in stripes of the low-temperature tetragonal phase of Bi-2212 ceramics is studied taking into account the dx2−y2-symmetry of the superconducting order parameter (SOP) and the dynamic analog of phase separation. The contribution to the energy density of the strain–order parameter coupling leads to a finite value of attenuation in superconductors with a peculiar SOP. The frequency and temperature dependences of attenuation are analyzed and compared with the available experimental results. It is shown that these data are an indirect evidence of the SOP symmetry: the dx2−y2-symmetry for the Bi-2212 phase and the s-symmetry for the Bi–2223 phase. The existence of an ordered low-temperature state responsible for the low-frequency attenuation peak and other anomalies in the properties of Bi-based ceramics in the temperature range 20–40 K is proposed.

Calibration of excess thermodynamic properties and elastic constant variations associated with the alpha <-->beta phase transition in quartz

Spontaneous strains for the a ↔ b transition in quartz were determined from lattice parameter data collected by X-ray powder diffraction and neutron powder diffraction over the temperature range ;5‐1340 K. These appear to be compatible with previous determinations of the order parameter variation in a quartz only if there is a non-linear relationship between the individual strains and the square of the order parameter. An expanded form of the 2-4-6 Landau potential usually used to describe the phase transition was developed to account for these strains and to permit calculation of the elastic constant variations. Calibration of the renormalized coefficients of the basic 2-4-6 potential, using published heat capacity data, provides a quantitative description of the excess free energy, enthalpy, entropy, and heat capacity. Values of the unrenormalized coefficients in the Landau expansion that include all the strain-order parameter coupling coefficients were used to calculate variations of the elastic constants. Values of the bare elastic constants were extracted from published elasticity data for b quartz. Calculated variations of C11 and C12 match their observed variations closely, implying that the extended Landau expansion provides a good representation of macroscopic changes within the (001) plane of quartz. Agreement was not as close for C33, suggesting that other factors may influence the strain parallel to [001]. The geometrical mechanism for the transition involves both rotations and shearing of SiO4 tetrahedra, with each coupled differently to the driving order parameter. Only the shearing part of the macroscopic distortions appears to show the same temperature dependence as other properties that scale with Q 2 . Coupling between the strain and the order parameter provides the predominant stabilization energy for a quartz and is also responsible for the first-order character of the transition.

Computer simulation of domain formation in the order-disorder phase transition of the KSCN model.

The tetragonal-orthorhombic improper ferroelastic phase transition and domain formation of the order-disorder KSCN model is studied by molecular-dynamics simulation. A two-dimensional model of either 105\ifmmode\times\else\texttimes\fi{}105 or 49\ifmmode\times\else\texttimes\fi{}49 unit cells represents a layer of this crystal. The simulation determines the orientational and strain order parameters and transition temperature. The mean-field theory gives excellent qualitative agreement of the calculated and simulated phase transition temperature. On quenching from the disorder phase and subsequent annealing, a tweed texture is obtained, which later evolves to a stripe phase with well oriented ferroelastic domain walls and arbitrarily oriented antiphase boundaries.

High-pressure ultrasonic study of vibrational anharmonicity in bcc Cu-Al-Be alloys.

To quantify the vibrational anharmonicity of the long-wavelength acoustic modes of bcc ${\mathrm{Cu}}_{74.1}$${\mathrm{Al}}_{23.1}$${\mathrm{Be}}_{2.8}$ near its martensitic transition temperature ${\mathit{M}}_{\mathit{s}}$ (261 K), the hydrostatic pressure derivatives (\ensuremath{\partial}${\mathit{C}}_{\mathit{I}\mathit{J}}$/\ensuremath{\partial}P${)}_{\mathit{P}=0}$ of the elastic stiffness moduli have been measured. The Gr\uneisen parameters at 268 K (just above ${\mathit{M}}_{\mathit{s}}$), especially of longitudinal modes, which become smaller than those of the shear modes, are quite different from those at 295 K: the anharmonicity changes markedly in the vicinity of the transition. Similar trends are noted for ${\mathrm{Cu}}_{66.5}$${\mathrm{Al}}_{12.7}$${\mathrm{Zn}}_{20.8}$. Experimental data near ${\mathit{M}}_{\mathit{s}}$ are used to estimate cubic invariants in the strain order parameters in a Landau formalism.

Sound propagation in heavy fermion compounds

Sound propagation experiments are used to investigate the superconducting (sc) phases of Heavy Fermion (HF) compounds. Velocity measurements lead to accurate B− T phase diagrams and results for the temperature dependent attenuation of all modes in UPt 3 are presented. In addition to the strain-order parameter interaction for longitudinal modes there is a Lorentz force coupling of transverse modes to the normal electrons. It leads to a B-dependence of velocity and attenuation from which information about both London and skin penetration depth in URu 2Si 2 and UBe 13 may be obtained. In UBe 13 magnetoacoustic quantum oscillations are observed which prove the coexistence of light and heavy electrons in this compound.

Elastic properties of NiTi

The authors present measurements of the elastic constants on a single crystal of the shape-memory alloy NiTi in a wide temperature range. Step-like anomalies and hysteresis are detected at the austenitic-martensitic phase transition which provide evidence for the strain-order parameter coupling in the pre-martensitic phases. The results are discussed in the framework of existing Landau theory models.

Elastic constant anomalies and the superconducting B-T phase diagrams of UPt3 and URu2Si2

Abstract We present elastic constant measurements for the heavy fermion superconductors UPt3 and URu2Si2. The temperature and field dependence of elastic anomalies is studied and used to construct superconducting B-T phase diagrams for these compounds. Ultrasonic attenuation and AC susceptibility measurements are also discussed. The results are interpreted within a phenomenological model assuming unconventional doubly degenerate E1g(UPt3) and Eg(URu2Si2) superconducting order parameters. Both compounds exhibit a spin density wave (SDW) already far above the superconducting transition. The coupling of SDW to the superconducting order parameters, its consequences and problems are discussed. The elastic constant anomalies are investigated within a phenomenological model for the strain-order parameter coupling. It is found that similar to the normal state behaviour Gruneisen parameter coupling to longitudinal modes is the dominant mechanism. Estimates of coupling parameters are given and compared to results from the pressure dependence of specific heat results. The double transition observed in both materials is intrinsic only for UPt3. In URu2Si2 it is not allowed by symmetry but is due to phase inhomogeneities.