Ultrasonic Studies Near Structural Phase Transitions

Elastic Moduli: a Tool for Understanding Chemical Bonding and Thermal Transport in Thermoelectric Materials

The elastic behavior of a material can be a powerful tool to decipher thermal transport. In thermoelectrics, measuring the elastic moduli-directly tied to sound velocity-is critical to understand trends in lattice thermal conductivity, as well as study bond anharmonicity and phase transitions, given the sensitivity of elastic moduli to the chemical bonding. In this review, we introduce the basics of elasticity and explain the origin of high-temperature lattice softening from a bonding perspective. We then review elasticity data throughout classes of thermoelectrics, and explore trends in sound velocity, anharmonicity, and thermal conductivity. We reveal how experimental sound velocities can improve the accuracy of common thermal conductivity models and present a critical discussion of Grüneisen parameter estimates from elastic moduli. Readers will be equipped with tools to leverage elasticity measurements or calculations to accurately interpret thermal transport trends.

Ultrasound measurement technique for the single-turn-coil magnets

Ultrasound is a powerful means to study numerous phenomena of condensed-matter physics as acoustic waves couple strongly to structural, magnetic, orbital, and charge degrees of freedom. In this paper, we present such a technique combined with single-turn coils (STCs) that generate magnetic fields beyond 100 T with the typical pulse duration of 6 µs. As a benchmark of this technique, the ultrasound results for MnCr₂S₄, Cu₆[Si₆O₁₈]·6H₂O, and liquid oxygen are shown. The resolution for the relative sound-velocity change in the STC is estimated as Δv/v ∼ 10^-3, which is sufficient to study various field-induced phase transitions and critical phenomena.

Magnetoelastic coupling associated with vacancy ordering and ferrimagnetism in natural pyrrhotite, Fe7S8

Magnetoelastic coupling associated with the hexagonal-monoclinic transition in a natural sample of the mineral pyrrhotite, Fe7S8, has been analysed in terms of separate coupling of spontaneous strains with two discrete order parameters,qvfor Fe/vacancy ordering andqmfor magnetic ordering. Coupling of the two order parameters separately with strain gives rise to two terms for coupling between them,λqm2qv2andλqm2qv8, and a pattern of evolution in whichqvvaries continuously andqmdiscontinuously through a single transition point. The transition is ferrimagnetic and ferroelastic but the relatively slow relaxation rate for Fe/vacancy ordering, in comparison with magnetic ordering, results in elastic and anelastic properties which are quite different from those observed in other ferroic or multiferroic materials with two instabilities. Instead of classical elastic softening, there is stiffening of the elastic constants which scales withqm2andqv2. Instead of the normal pattern of acoustic loss associated with the mobility and subsequent freezing for ferroelastic twin walls, the loss is consistently low throughout the temperature range 300 K-875 K.

Strain relaxation behaviour of vortices in a multiferroic superconductor

The elastic and anelastic properties of a single crystal of Co-doped pnictide Ba(Fe_0.957Co_0.043)₂As₂ have been determined by resonant ultrasound spectroscopy in the frequency range 10-500 kHz, both as a function of temperature through the normal-superconducting transition (T _c  ≈  12.5 K) and as a function of applied magnetic field up to 12.5 T. Correlation with thermal expansion, electrical resistivity, heat capacity, DC and AC magnetic data from crystals taken from the same synthetic batch has revealed the permeating influence of strain on coupling between order parameters for the ferroelastic (Q _E) and superconducting (Q _SC) transitions and on the freezing/relaxation behaviour of vortices. Elastic softening through T _c in zero field can be understood in terms of classical coupling of the order parameter with the shear strain e ₆, λe ₆ [Formula: see text], which means that there must be a common strain mechanism for coupling of the form λ [Formula: see text] Q _E. At fields of ~5 T and above, this softening is masked by Debye-like stiffening and acoustic loss processes due to vortex freezing. The first loss peak may be associated with the establishment of superconductivity on ferroelastic twin walls ahead of the matrix and the second is due to the vortex liquid-vortex glass transition. Strain contrast between vortex cores and the superconducting matrix will contribute significantly to interactions of vortices both with each other and with the underlying crystal structure. These interactions imply that iron-pnictides represent a class of multiferroic superconductors in which strain-mediated coupling occurs between the multiferroic properties (ferroelasticity, antiferromagnetism) and superconductivity.

Elastic and magnetoelastic relaxation behaviour of multiferroic (ferromagnetic + ferroelectric + ferroelastic) Pb(Fe0.5Nb0.5)O3 perovskite

Resonant Ultrasound Spectroscopy has been used to characterize elastic and anelastic anomalies in a polycrystalline sample of multiferroic Pb(Fe(0.5)Nb(0.5))O(3) (PFN). Elastic softening begins at ~550 K, which is close to the Burns temperature marking the development of dynamical polar nanoregions. A small increase in acoustic loss at ~425 K coincides with the value of T(*) reported for polar nanoregions starting to acquire a static or quasi-static component. Softening of the shear modulus by ~30-35% through ~395-320 K, together with a peak in acoustic loss, is due to classical strain/order parameter coupling through the cubic → tetragonal → monoclinic transition sequence of ferroelectric/ferroelastic transitions. A plateau of high acoustic loss below ~320 K is due to the mobility under stress of a ferroelastic microstructure but, instead of the typical effects of freezing of twin wall motion at some low temperature, there is a steady decrease in loss and increase in elastic stiffness below ~85 K. This is attributed to freezing of a succession of strain-coupled defects with a range of relaxation times and is consistent with a report in the literature that PFN develops a tweed microstructure over a wide temperature interval. No overt anomaly was observed near the expected Néel point, ~145 K, consistent with weak/absent spin/lattice coupling but heat capacity measurements showed that the antiferromagnetic transition is actually smeared out or suppressed. Instead, the sample is weakly ferromagnetic up to ~560 K, though it has not been possible to exclude definitively the possibility that this could be due to some magnetic impurity. Overall, evidence from the RUS data is of a permeating influence of static and dynamic strain relaxation effects which are attributed to local strain heterogeneity on a mesoscopic length scale. These, in turn, must have a role in determining the magnetic properties and multiferroic character of PFN.

Static and dynamic strain coupling behaviour of ferroic and multiferroic perovskites from resonant ultrasound spectroscopy

Resonant ultrasound spectroscopy (RUS) provides a window on the pervasive influence of strain coupling at phase transitions in perovskites through determination of elastic and anelastic relaxations across wide temperature intervals and with the application of external fields. In particular, large variations of elastic constants occur at structural, ferroelectric and electronic transitions and, because of the relatively long interaction length provided by strain fields in a crystal, Landau theory provides an effective formal framework for characterizing their form and magnitude. At the same time, the Debye equations provide a robust description of dynamic relaxational processes involving the mobility of defects which are coupled with strain. Improper ferroelastic transitions driven by octahedral tilting in KMnF3, LaAlO3, (Ca,Sr)TiO3, Sr(Ti,Zr)O3 and BaCeO3 are accompanied by elastic softening of tens of % and characteristic patterns of acoustic loss due to the mobility of twin walls. RUS data for ferroelectrics and ferroelectric relaxors, including BaTiO3, (K,Na)NbO3,Pb(Mg1/3Nb2/3)O3 (PMN), Pb(Sc1/2Ta1/2)O3 (PST), (Pb(Zn1/3Nb2/3)O3)0.955(PbTiO3)0.045 (PZN-PT) and (Pb(In1/2Nb1/2)O3)0.26(Pb(Mg1/3Nb2/3)O3)0.44(PbTiO3)0.30 (PIN-PMN-PT) show similar patterns of softening and attenuation but also have precursor softening associated with the development of polar nano regions. Defect-induced ferroelectricity occurs in KTaO3, without the development of long range ordering. By way of contrast, spin-lattice coupling is much more variable in strength, as reflected in a greater range of softening behaviour for Pr0.48Ca0.52MnO3 and Sm0.6Y0.4MnO3 as well as for the multiferroic perovskites EuTiO3,BiFeO3, Bi0.9Sm0.1FeO3, Bi0.9Nd0.1FeO3, (BiFeO3)0.64(CaFeO2.5)0.36, (Pb(Fe0.5Ti0.5)O3)0.4(Pb(Zr0.53Ti0.47)O3)0.6. A characteristic feature of transitions in which there is a significant Jahn-Teller component is softening as the transition point is approached from above, as illustrated by PrAlO3, and this is suppressed by application of an external magnetic field in the colossal magnetoresistive manganite Pr0.48Ca0.52MnO3 or by reducing grain size in La0.5Ca0.5MnO3. Spin state transitions for Co(3+) in LaCoO3, NdCoO3 and GdCoO3 produce changes in the shear modulus that scale with a spin state order parameter, which is itself coupled with the order parameter(s) for octahedral tilting in a linear-quadratic manner. A new class of phase transitions in perovskites, due to orientational or conformational ordering of organic molecules on the crystallographic A-site of metal organic frameworks, is illustrated for [(CH3)2NH2]Co(HCOO)3 and [(CH2)3NH2]Mn(HCOO)3 which also display elastic and anelastic anomalies due to the influence of intrinsic and extrinsic strain relaxation behaviour.

Elastic properties and acoustic dissipation associated with a disorder–order ferroelectric transition in a metal–organic framework

Elastic properties and acoustic dissipation associated with the disorder–order ferroelectric transition in a single crystal metal–organic framework (MOF), [NH₄][Zn(HCOO)₃], have been investigated using resonant ultrasound spectroscopy (RUS) in the temperature range between 10 K and 300 K.

Strain relaxation mechanisms of elastic softening and twin wall freezing associated with structural phase transitions in (Ca,Sr)TiO3 perovskites

Resonant ultrasound spectroscopy has been used to measure the bulk modulus (K), shearmodulus (G) and acoustic dissipation of polycrystalline perovskite samples across theCaTiO(3)–SrTiO(3) solid solution in the temperature range ∼10–1350 K. A remarkable pattern of up to ∼25% softening of G as a function of both temperature and composition is due to coupling of shear strain with order parameters for the Pm3m ↔ I 4/mcm, I 4/mcm↔Pnmaand I 4/mcm↔Pbcm transitions. Anomalies in K associated with the phase transitions are small, consistent with only weak coupling of octahedral tilting order parameter(s) with volume strain. A change from tricritical character for the Pm3m ↔ I 4/mcm transition towards second order character at Sr-rich compositions appears to be due to changing properties of the soft optic mode rather than to changes in magnitude of strain/order parameter coupling coefficients. Precursor softening of G ahead of the Pm3m ↔ I 4/mcm transition, due tof luctuations or clustering, occurs over a temperature interval of up to ∼200 K, and also changes character at the most Sr-rich compositions. The tetragonal structure with Sr-rich compositions is characterized by additional softening with falling temperature which is most likely related to the proximity of a ferroelectric instability. The I 4/mcm↔Pnma transition is accompanied by stiffening, which is attributed to the effects of strong coupling between order parameters for M-point and R-point tilting. The pattern of attenuation at RUS frequencies in the tetragonal phase can be understood in terms of the mobility of twin walls which be come pinned below ∼500 K, and the loss mechanism most likely involves local bowing of the walls by lateral motion of ledges rather than the advance and retraction of needle tips. Twin wall mobility is suppressed in the orthorhombic structure.

Strain heterogeneity and magnetoelastic behaviour of nanocrystalline half-doped La, Ca manganite, La0.5Ca0.5MnO3

Elastic and anelastic properties of La0.5Ca0.5MnO3 determined by resonant ultrasound spectroscopy in the frequency range ∼100-1200 kHz have been used to evaluate the role of grain size in determining the competition between ferromagnetism and Jahn-Teller/charge order of manganites which show colossal magneto resistance. At crystallite sizes of ∼75 and ∼135 nm the dominant feature is softening of the shear modulus as the charge order transition point, Tco (∼225 K), is approached from above and below, matching the form of softening seen previously in samples with 'bulk' properties. This is consistent with a bilinear dominant strain/order parameter coupling, which occurs between the tetragonal shear strain and the Jahn-Teller (Γ3(+)) order parameter. At crystallite sizes of ∼34 and ∼42 nm the charge ordered phase is suppressed but there is still softening of the shear modulus, with a minimum near Tco. This indicates that some degree of pseudoproper ferroelastic behaviour is retained. The primary cause of the suppresion of the charge ordered structure in nanocrystalline samples is therefore considered to be due to suppression of macroscopic strain, even though MnO6 octahedra must develop some Jahn-Teller distortions on a local length scale. This mechanism for stabilizing ferromagnetism differs from imposition of either an external magnetic field or a homogeneous external strain field (from a substrate), and is likely to lead both to local strain heterogeneity within the nanocrystallites and to different tilting of octahedra within the orthorhombic structure. An additional first order transition occurs near 40 K in all samples and appears to involve some very small strain contrast between two ferromagnetic structures.

Multiferroic (ferroelastic/ferromagnetic/ferrimagnetic) aspects of phase transitions in RCo2 Laves phases

Magnetic phase transitions in RCo2 Laves phases with R as a rare earth element are accompanied by changes in crystallographic space group. For purely structural transitions they would be described as improper ferroelastic and therefore fulfil the condition for multiferroic phase transitions in combining two out of three properties, ferro/antiferromagnetism, ferroelectricity and ferroelasticity. Here lattice parameter data from the literature and new measurements of elastic and anelastic properties, by resonant ultrasound spectroscopy, for NdCo2 and ErCo2 have been analysed from this perspective. The temperature dependence of symmetry-breaking shear strains is consistent with the cubic ↔ tetragonal transition in NdCo2 being close to tricritical in character and the cubic ↔ rhombohedral transition in ErCo2 being first order. Elastic softening and acoustic loss within the stability ranges of the ferroelastic phases can be understood in terms of a combination of intrinsic softening due to strain/order parameter coupling and ferroelastic twin-wall motion. Softening ahead of the transitions does not fit with standard macroscopic descriptions of dynamic effects from other systems but, rather, in the case of NdCo2, might be attributed to the involvement of a second zone centre order parameter related to a separate instability driven by cooperative Jahn-Teller distortions. In ErCo2, acoustic loss in the temperature interval above the transition point is discussed in terms of a possible tweed microstructure associated with strain coupling to local magnetic ordering. The overall multiferroic behaviour can be understood in terms of a single magnetic order parameter (irrep mΓ+4 of magnetic space group Fd3m1') which couples with a structural order parameter (irrep Γ+3 or Γ+5). The coupling is linear/quadratic which, in the case of two separate instabilities, causes them to combine in a single multiferroic phase transition.

Elastic and anelastic anomalies associated with the antiferromagnetic ordering transition in wüstite, FexO

The elastic and anelastic properties of three different samples of Fe(x)O have been determined in the frequency range 0.1-2 MHz by resonant ultrasound spectroscopy and in the range 0.1-50 Hz by dynamic mechanical analysis in order to characterize ferroelastic aspects of the magnetic ordering transition at T(N) ~ 195 K. No evidence was found of separate structural and magnetic transitions but softening of the shear modulus was consistent with the involvement of bilinear coupling, λe(4)q, between a symmetry-breaking strain, e(4), and a structural order parameter, q. Unlike a purely ferroelastic transition, however, C(44) does not go to zero at the critical temperature, T*(c), due to the intervention of the magnetic ordering at a higher temperature. The overall pattern of behaviour is nevertheless consistent with what would be expected for a system with separate structural and magnetic instabilities, linear-quadratic coupling between the structural (q) and magnetic (m) driving order parameters, λqm(2), and T(N) > T*(c). Comparison with data from the literature appears to confirm the same pattern in MnO and NiO, with a smaller difference between T(N) and T*(c) in the former and a larger difference in the latter. Strong attenuation of acoustic resonances at high frequencies and a familiar pattern of attenuation at low frequencies suggest that twin walls in the rhombohedral phase have typical ferroelastic properties. Acoustic dissipation in the stability field of the cubic phase is tentatively attributed to anelastic relaxations of the defect ordered structure of non-stoichiometric wüstite or of the interface between local regions of wüstite and magnetite, with a rate controlling step determined by the diffusion of iron.

Elastic and anelastic relaxations in the relaxor ferroelectric Pb(Mg1/3Nb2/3)O3: I. Strain analysis and a static order parameter

The structural evolution of Pb(Mg(1/3)Nb(2/3))O(3) (PMN) has been reviewed in terms of characteristic temperatures, length scales and timescales, with a view to considering the overall relaxor behaviour from the perspectives of strain and elasticity. A conventional analysis of lattice parameter data in terms of spontaneous strain and strain/order parameter coupling shows that even though a normal phase transition does not occur the relaxor ordering process is accompanied by a significant volume strain which follows the pattern of a static order parameter evolving according to that expected for a tricritical phase transition with T(c) ≈ 350 K. This matches the evolution of the intensity of the elastic central peak in neutron scattering spectra, and reflects the development of static (or quasistatic) polar nanoregions (PNRs) as if by a mean-field phase transition. Use of a Landau free energy expansion, which includes Γ4(-) order parameter components to describe ferroelectric contributions and an R1(+) order parameter to describe cation ordering together with their formal coupling with strain, then allows the pattern of elastic softening expected for a cubic → rhombohedral phase transition to be anticipated. The extent to which observed softening differs from this static mean-field pattern serves to highlight the additional roles of local heterogeneity and relaxation dynamics in determining the relaxor properties of PMN.

Elastic and anelastic relaxations in the relaxor ferroelectric Pb(Mg1/3Nb2/3)O3: II. Strain-order parameter coupling and dynamic softening mechanisms

Elastic and anelastic behaviour of single crystal and ceramic samples of Pb(Mg(1/3)Nb(2/3))O(3) has been investigated at frequencies of ~0.1-1.2 MHz through the temperature interval 10-800 K by resonant ultrasound spectroscopy (RUS). Comparison with data from the literature shows that softening of the shear modulus between the Burns temperature and the freezing interval is independent of frequency. The softening is attributed to coupling between acoustic modes and the relaxation mode(s) responsible for central peaks in Raman and neutron scattering spectra below the Burns temperature, and can be described with Vogel-Fulcher parameters. Shear elastic compliance and dielectric permittivity show similar patterns of temperature dependence through the freezing interval, demonstrating strong coupling between ferroelectric polarization and strain such that the response to applied stress is more or less the same as the response to an applied electric field, with a frequency dependence consistent with Vogel-Fulcher-like freezing in both cases. Differences in detail show, however, that shearing induces flipping between different twin orientations, in comparison with the influence of an electric field, which induces 180° flipping: the activation energy barrier for the former appears to be higher than for the latter. Below the freezing interval, the anelastic loss also has a similar pattern of evolution to the dielectric loss, signifying again that essentially the same mechanism is involved in the freezing process. Overall softening at low temperatures is attributed to the contributions of strain relaxations due to coupling with the local ferroelectric order parameter and of coupling between acoustic modes and continuing relaxational modes of the polar nanostructure. Dissipation is attributed to movement of boundaries between PNRs or between correlated clusters of PNRs. Overall, strain coupling is fundamental to the development of the characteristic strain, dielectric and elastic properties of relaxors.

Anelastic loss behaviour of mobile microstructures in SrZr(1-x)Ti(x)O3 perovskites

Anelastic loss mechanisms associated with phase transitions in SrZr(1-x)Ti(x)O(3) perovskites (x = 0.375, 0.450, 0.550, 0.775) have been investigated by dynamic mechanical analysis between 128 and 723 K at frequencies of 0.1-50 Hz. Distinctive patterns of changes in the elastic moduli due to octahedral tilting transitions correlate closely with data for the shear modulus obtained previously by resonant ultrasound spectroscopy at high frequencies (∼0.5 MHz). The I4/mcm <--> Imma transition is first order and has a characteristic minimum in the shear modulus and Young's modulus. For x = 0.450 and 0.550, a dissipation peak occurs at the transition temperature, the maximum of which varies with frequency according to a power law relationship of the form tanδ = Af(n), with n≈ - 0.3. Debye-like dissipation peaks in the stability field of the Imma structure at x = 0.375 have a frequency and temperature dependence consistent with twin wall pinning by defects with an activation energy ∼ 184 kJ mol(-1). These results indicate that there is diversity of pinning and relaxation processes for transformation twin walls and interfaces in different perovskites with I4/mcm, Imma and Pnma structures.

Elastic relaxations associated with the Pm3m-R3c transition in LaAlO(3): III. Superattenuation of acoustic resonances

Resonant ultrasound spectroscopy has been used to characterize elastic softening and anelastic dissipation processes associated with the Pm3m <--> R3c transition in single crystal and ceramic samples of LaAlO(3). Softening of the cubic structure ahead of the transition point is not accompanied by an increase in dissipation but follows different temperature dependences for the bulk modulus, (1/3)(C(11) + C(12)), and the shear components, (1/2)(C(11) + C(12)) and C(44), as if the tilting instability contains two slightly different critical temperatures. The transition itself is marked by the complete disappearance of resonance peaks (superattenuation), which then reappear below ∼700 K in spectra from single crystals. Comparisons with low frequency, high stress data from the literature indicate that the dissipation is not due to macroscopic displacement of needle twins. An alternative mechanism, local bowing of twin walls under low dynamic stress, is postulated. Pinning of the walls with respect to this displacement process occurs below ∼350 K. Anelasticity maps, analogous to plastic deformation mechanism maps, are proposed to display dispersion relations and temperature/frequency/stress fields for different twin wall related dissipation mechanisms. These allow comparisons to be made of anelastic loss mechanisms under mechanical stress with elastic behaviour observed by means of Brillouin scattering at high frequencies which might also be related to microstructure.

Elastic relaxations associated with the Pm3m-R3c transition in LaAlO(3): I. Single crystal elastic moduli at room temperature

Values for all six independent components of the 3m elastic modulus tensor of LaAlO(3) perovskite are reported. These were determined by means of Brillouin scattering measurements of acoustic velocities in single crystal plates cut parallel to (110) and (100), as defined with respect to the cubic parent structure, and by pure-mode longitudinal and transverse sound velocity measurements along [100], [110] and [111] directions using GHz pulse-echo ultrasonics. The crystals contained intimate intergrowths of twins arising from the Pm3m <--> R3c transition at higher temperature but, in combination with a careful analysis of twin orientation relationships, the two sets of data have allowed a unique solution to be obtained for individual twin components. The new data set represents an important contribution to the characterization of LaAlO(3) single crystals which are widely used as the substrate for a plethora of different thin films with technological applications.

Elastic anomalies associated with transformation sequences in perovskites: I. Strontium zirconate, SrZrO(3)

Elastic behaviour associated with the hierarchy of tilting transitions in SrZrO(3) has been examined using resonant ultrasound spectroscopy on a ceramic sample at temperatures between 153 and 1531 K. Changes in slope of the evolution of resonance frequencies with temperature indicate that phase transitions occur at 1038 K ([Formula: see text]), 1122 K ([Formula: see text]), and 1367 K ([Formula: see text]). Strain analysis of previously recorded neutron diffraction data shows that the [Formula: see text] and [Formula: see text] transitions are close to tricritical in character, and that [Formula: see text] is first order. Deviations from the form of the elastic behaviour predicted by Landau theory are found. In particular, elastic softening in the vicinity of the [Formula: see text] transition suggests that local dynamical fluctuations between individual tilt systems occur, rather than a discontinuous switch from one phase to another. Determinations of the mechanical quality factor, Q, show that SrZrO(3) in the [Formula: see text] phase is a classically high-Q (i.e. non-dissipating) cubic material. I4/mcm and Imma phases both have much greater dissipation (low Q), which is tentatively attributed to the mobility of twin walls. The room temperature Pnma phase is unexpectedly much stiffer than both I4/mcm and Imma phases and has high Q. It appears that when two separate tilt systems operate, as in Pnma, they can interact to reduce strain/order parameter relaxations.

Alpha-plutonium's polycrystalline elastic moduli over its full temperature range

alpha-plutonium's volume-corrected polycrystal elastic moduli were measured between 18 K and the upper limit of its occurrence, near 400 K. The two independent moduli for a polycrystal-bulk and shear-behave smoothly, indicating no phase transition. Both moduli show the same 50% increase on cooling, an order of magnitude larger than in other metals. The Debye temperature obtained from low-temperature elastic moduli, 207 K, significantly exceeds most previous estimates. The Gruneisen parameter gamma=5.3, obtained from the temperature dependence of the bulk modulus, is intermediate among previous estimates using other approaches, alpha-plutonium's Poisson ratio nu is low: 0.18, nearly temperature independent, and its small decrease on warming opposes usual behavior. The high gamma, large but equal bulk modulus and shear modulus fractional stiffening on cooling, and near-temperature-invariant nu are attributed to a single mechanism: 5-f electron localization-delocalization.