Professor Michael Carpenter

Phase transitions, strain, elasticity and anelastic phenomena in minerals and functional materials

1. Phase transitions

Phase transitions are by definition collective phenomena. Some local instability in the atomic or electronic configuration of a structure develops correlations over a significant length scale, macroscopic symmetry is broken and a new long-range order is established. Changes in symmetry conform to group theoretical constraints, giving rise to the concept of order parameters and providing the rigorous underpinning of Landau theory.

2. Strain and elasticity

It is well understood that strain has a fundamental and pervasive influence on almost all types of phase transitions, either as the driving order parameter (acoustic mode instability) or by coupling with some other driving mechanism, which may be structural (soft mode, atomic ordering, hydrogen bonding….), ferroelectric (displacive, order/disorder, relaxor….), magnetic (ferro/antiferromagnetic, spin-glass….), or electronic (charge order, Jahn-Teller, spin state, superconducting, metal-insulator….).

The most overt implications are firstly that the correlation length of the order parameter takes on the generally longer length scale of strain fields, secondly that overlapping strains from otherwise separate order parameters can result in strong coupling between multiple instabilities and, thirdly, that transformation microstructures such as tweed and twin walls may interact strongly with defects. Some of the consequences relate to reduction of the Ginsburg temperature interval of critical fluctuations, an expectation that mean field models should provide effective descriptions of thermodynamic properties, a tendency for transitions which would otherwise be second order to become first order in character, interdependence of different physical properties such as ferroelectricity and magnetism, and control of the dynamics and mechanisms of switching by strain relaxation or defect pinning processes.

Understanding the behaviour of bulk samples also feeds into considerations for thin film technologies since a key variable is the coherency strain of the film with its substrate. The choice of substrate is a choice of imposed strain which, in turn, drives a strain coupled order parameter in the film to some desired configuration and magnitude. There will be subtle differences, however, since the imposed strain will be homogeneous, whereas a bulk material containing ferroelastic twin walls or polar nano regions, for example, will contain strain heterogeneities at a local, mesoscopic length scale.

In the context of multiferroic materials, direct magnetoelectric coupling tends to be weak, as represented by the small overlap of fields for ferro/antiferromagnetism and ferroelectricity, but if both the ferroelectric dipole and the magnetic order parameter each couple with a co-elastic or ferroelastic strain the (indirect) coupling may be substantially increased.

2. Elasticity measurements by Resonant Ultrasound Spectroscopy (RUS)

Most methods of measuring elastic moduli are dynamic and the best analogy is with measurements of dielectric properties. The dynamically applied field is stress (electric field) and the response has real and imaginary components which are typically used to evaluate the elastic compliance (dielectric permittivity) and acoustic loss (dielectric loss) as functions of temperature and frequency. However, while the frequency of an electric field can be adjusted continuously through many orders of magnitude, there is no single experimental method for measuring elastic properties over a wide frequency interval. Instead, a number of different methods are used in relatively narrow frequency windows.

3. Elastic anomalies associated with phase transitions

An illustration of the data obtained by automatic collection of RUS spectra through a sequence of phase transitions is provided for KMnF3. Individual spectra from an irregularly shaped single crystal, with mass 0.3354 g and approximate dimensions 10 x 10 x 1 mm3, are stacked in proportion to the temperature at which they were obtained. The sequence of transitions is Pmm (paramagnetic) – I4/mcm (paramagnetic) at ~185 K, I4/mcm (paramagnetic) – Cmcm (antiferromagnetic) at ~87 K, Cmcm (antiferromagnetic) – Pnma (canted antiferromagnetic) at ~83 K.

Substantial softening, evident as shifts to lower frequencies of individual resonance peaks, and an increase in loss in the stability field of the I4/mcm structure, evident as peak broadening, are confirmed in the variations of f2 and Q-1. Softening of the elastic constants by up to ~40% is due to the development of shear strains associated with the octahedral tilting transitions but there is no overt evidence of contributions from magnetoelastic relaxations. The Debye-like loss peak centred on ~130 K has been attributed to freezing of ferroelastic twin walls due to pinning by F vacancies or dumbbell pairs of F interstitials

 4. Magnetoelastic coupling

Magnetic transitions are typically accompanied by strains which are often referred to as spin-lattice coupling. The strength of such coupling is enormously variable. One example is provided by small but significant static and dynamic strain coupling effects in Cu2OSeO3 associated with magnetic phase transitions observed as a function of temperature (1.5-150 K) and magnetic field (0-300 mT). The magnetic transition near 60 K is accompanied by a small increase in single crystal elastic constants which can be understood in terms of biquadratic coupling between shear strain and the magnetic order parameter, even though the shear strain itself is negligibly small. The conical – collinear transition is associated with distinct minima in the elastic properties, while weaker anomalies at lower fields may be related to changes in the configuration of magnetic domains. A distinctive acoustic loss peak at ~42 K, independent of magnetic field, is attributed to freezing of a defect which is coupled with shear strain, has an associated activation energy of ~5 kJ.mole-1 and may play some role in pinning the magnetic microstructures. Anomalies below ~10 K indicate the presence of some additional relaxation process which could signify a change in magnetic structure. (Evans et al, 2017, Phys Rev B 95, 094426).

5. Anelasticity associated with phase transitions

Anelastic losses occur in crystals when the response to an externally applied dynamic stress involves some relaxation which is coupled to a strain and which has a relaxation time that is concomitant with the frequency of the applied stress. The strain may be coupled to an intrinsic process, such as proton ordering in lawsonite, CaAl2Si2O7(OH)2.H2O. Alternatively, the relaxation process could be extrinsic, such as displacements of impurity atoms or transformation microstructures.

The influence of twin walls due to ferroelastic transitions is illustrated by the case of LaAlO3. The octahedral tilting transition is marked by an abrupt disappearance of resonance peaks from spectra collected below Tc (817 K) and is attributed to the motion under external stress of ferroelastic twin walls in the rhombohedral structure. This complete attenuation (“superattenuation”) implies maximum values of Q-1 greater than ~0.02 and continues until the resonance peaks gradually reappear below ~600 K when the mobile twin walls become pinned by interaction with defects.

6. Resonant Piezoelectric Spectroscopy (RPS)

A new adaptation of the RUS method for characterising ferroic materials and their possible adaptive states is to stimulate resonances of the samples by an ac electric field instead of mechanically by a piezoelectric transducer (Resonant Piezoelectric Spectroscopy, RPS). A schematic view of the experimental configuration is given below. If the sample is intrinsically piezoelectric or if it contains locally polar regions, mechanical resonances can be excited by the ac electric field. These are detected by the transducer and their frequencies should be the same as obtained by mechanical excitation in a conventional set up for RUS. RPS allows the detection of polarity in domain walls and within tweed or other nanodomain microstructures with high sensitivity.