The equilibrium domain arrangements of ferroelectric single crystals are significantly affected by loads and boundary conditions.
Domain structures evolve towards a minimum energy state. In this paper, a variational method, which minimizes
a functional based on free energy and dissipation, is developed to model the evolution of several typical rank-2 laminate
domain patterns in the tetragonal crystal system. Periodic laminates which satisfy compatibility at every domain wall are
studied. These domain patterns include herringbone and vortex array structures. The unit cells for both types of domain
pattern dictate a set of domain walls whose positions may vary while maintaining the same topology. The positions of
domain walls are treated as thermodynamic variables in the formulation, and the total dissipation rate is then a function of
the velocities of the domain walls. By using this model, many features normally observed in ferroelectric single crystals
can be reproduced, such as the dielectric hysteresis loop and butterfly loop. The characteristics of the hysteresis loop for
different topologies, as well as under different applied loads and boundary conditions are discussed. The model can readily
be extended to higher rank laminate structures and other crystal systems.
Imaging of domains is a key step in understanding the microstructure and hence the properties of ferroelectric single
crystals. This understanding is essential for exploiting engineered domain configurations to achieve enhanced
performance. In this paper, single crystals of Barium Titanate are observed by reflection topography using unfocussed
monochromatic synchrotron X-ray light. A 10 x 10 mm polished surface of an unpoled crystal was mapped to form a
composite image, indicating a fine structure of a- and c-domains. By making use of the angular separation of the
diffracted reflections and specimen rocking, the relative tilts between adjacent domains about two orthogonal axes were
found. Angular resolution better than 0.1mrad in tilt measurements allowed the local elastic curvature of lattice planes to
be observed. The resulting composite images show well defined boundaries between regions of distinct microstructure,
and give an indication of the proportion of the domain types present. Over large regions of the crystal the domain
structure was finer than the X-ray camera resolution of 6.5μm; AFM and SEM imaging of domains was then used to
confirm the typical domain spacing. The results are interpreted in the context of models of compatible microstructure in
tetragonal crystals using microscopy of etched crystals to assist the interpretation. The technique shows promise for
mapping fine microstructure in single crystals, through the use of high resolution X-ray cameras, and is successful in
revealing lattice orientation information that is not normally available in optical or AFM measurements.
The microstructure of ferroelectric single crystals is a crucial factor that determines macroscopic properties and poling
behaviour. Recent models of domain configuration, (such as that of Li & Liu, Journal of Mechanics and Physics of
Solids, 2004) employ multi-rank laminate structures that satisfy compatibility in an average sense. In general, these
models result in high-rank structures, corresponding to fine microstructure. However, minimum energy structures may
be expected to have low rank and to satisfy compatibility requirements at every domain wall exactly. In this paper, the
criteria of exact compatibility and average compatibility are defined and then used to determine energy minimizing
microstructure in the tetragonal crystal system. In addition, the lowest rank construction of compatible laminate structure
for a given macroscopic state of strain and polarization is found. Based on this, poling paths from unpoled to the fully-poled
state in the tetragonal system are found, which allow the structure to stay in the lowest possible rank while
maintaining exact compatibility. The application of the theory to a broader class of crystal structures is discussed.
The performance of ferroelectric random access memory devices (FeRAM) relies on the remnant polarisation.
For high performance, the remnant polarisation of a ferroelectric thin film memory capacitor is desired to be
as great as possible. However, the remnant polarisation in thin film form is typically only a third to a half of
its bulk value. The coercive field is also several times greater in a thin film than in its bulk counterpart. A
theoretical work is carried out in this study to explore the roles played by substrate and ferroelectric properties
in altering the remnant polarisation. A constitutive law based on the crystal plasticity theory and the finite
element method are used to model the ferroelectric switching behavior of a memory capacitor. In particular, it is
found that factors such as crystallographic orientation and the initial volume fractions of ferroelectric variants,
that are dependent on the type of substrate and film deposition method, can significantly alter the achievable
remnant polarisation. An explanation of these dependencies is given, suggesting approaches to the problem of
increasing the remnant polarisation of a thin film memory capacitor.
The technique of birefringence imaging was exploited to observe the evolution of creeping strain fields in transparent
PLZT 8/65/35 samples. PLZT samples with features that produce non-uniform fields were loaded with
constant voltage boundary conditions. The resulting birefringence contours evolve with time and can be related
to strain measurements. Three experimental arrangements are reported: partial surface electrodes producing
intense fields near an electrode tip, a round insulating hole producing local concentration of electric field, and
a thin, sharp crack producing crack tip fields. In each case, material was initially in the as-sintered (unpoled)
state, and was loaded with nominal electric field strengths that were well below the coercive field. However, the
birefringence imaging indicates significant remanent strain evolving over a time period of order 103s. The resulting
mean electric displacements are greatly enhanced relative to uniform field conditions at the same mean field
strength. The measurements show only weak interaction between thin cracks and the applied electric field, suggesting
that the thin cracks are effectively permeable. The results are of potential use in calibrating multi-axial
and time dependent material models.
The time-dependent remanent strain and polarization were measured in initially unpoled PZT-855 under electromechanical
loads. Various levels of constant compressive uniaxial stress combined with constant electric field
parallel to the stress axis were used to produce the creeping remanent strain and polarization. The remanent
quantities were deduced from measurements of total strain and electric displacement by subtracting the linear
(reversible) parts of strain and electric displacement, making use of previous measurements of the linear moduli
and accounting for their variation with material state. Mechanical compressive stresses alone produce remanent
strains that rapidly reach a saturated state. Under combined loading, increasing the compressive stress reduces
the observed creep rates caused by an additional electrical loading. The creep behaviour is most significant at
loads close to the coercive stress or electric field. At loads well above or below the coercive field levels, the
behaviour is nearly rate-independent.
In this study, measurements of the evolving linear elastic, dielectric and piezoelectric moduli of a soft ferroelectric
PZT are made during loadings of uniaxial compressive stress combined with an electric field. Using short pulses of
electric field and stress, the incremental remanent strain and polarization state of the material and the unloading
moduli were determined. The remanent quantities are treated as state variables, with a view to expressing the
moduli as functions of the material state. The piezoelectric moduli vary approximately linearly with polarization,
whilst the dielectric moduli and elastic compliances show more complex behaviour.
The dielectric and piezoelectric responses to a constant electric field have been measured on initially unpoled PZT-5H and PLZT 8/65/35. In particular, we are interested in the time dependent remanent strain and polarization due to unipolar electric field because they are associated with the domain switching behaviour. An experimental method has been developed to obtain the remanent strains and polarization by accounting for dielectric and piezoelectric effects. A linear dependence of dielectric and piezoelectric modulus on remanent polarization was found in PZT-5H, but not in PLZT 8/65/35. In PLZT 8/65/35, a transition was observed at a value of remanent polarization of 0.07 Cm-2 gradually evolving polarization to rapid switching. A change in the dominant mechanism of switching was also observed.
A non-linear bimorph device is proposed, in which the strain accompanying ferroelectric switching is used to drive curvature of a bilayer of ferroelectric material. An analysis of the proposed device suggests that a curvature about an order of magnitude greater than that readily obtained by conventional piezoelectric bimorphs is possible, whilst the peak tensile stresses in the device are comparable with those in conventional devices. The curvature achieved by the device upon poling is calculated approximately by using an analytical approach with idealized material behaviour. Two potential designs for the poling electrodes are suggested, and numerical calculations are used to optimize these designs.
Fatigue crack growth is studied in PZT-5H and PLZT 8/65/35 ferroelectric ceramics under purely electrical cyclic loading. The growing cracks are intergranular and exhibit features such as bifurcation, tunneling, arrest and bursts of rapid growth. Two ferroelectrics show different cracking behaviors once the crack growth commences - the crack growth rate in PZT-5H decreases with the number of cycles, whereas for PLZT 8/65/35, a period of rapid steady-state crack growth is typical. A band of damaged material forms around the crack and propagates through the ceramic. The thickness of this band of damaged material is directly proportional to the strength of the applied electric field. Crack growth measurements are presented for two ferroelectric compositions under varying load amplitude and with varying frequency and test geometry. Potential mechanisms underlying the fatigue behavior include cracking at local stress concentrations due to inhomogeneity at the grain length scale, a wedging mechanism in the crack wake, the action of the crack as a field intensifier, and the degradation of a actuating mechanism.
Samples of soft PZT-5H, hard PZT-4D and Barium Titanate were subjected to multi-axial loading in stress and electric field space. The loading paths were: (1) Poling with electric field, followed by repolarizing with electric field at an angle to the original poling direction. (2) Proportional loading with electric field and coaxial compressive stress, the proportions of stress and electric field being varied between tests. In case (1) the poled material was cut to produce faces angled to the original poling direction. The measured material responses are reported and initial switching surfaces are calculated based on an offset from linear response in electric displacement. The measurements are used to assess the features required in micromechanical or phenomenological models of switching.
The use of ferroelectrics in increasingly demanding roles as sensors and actuators motivates study of their fundamental constitutive behavior. This paper gives a preliminary report on measurements of the behavior of PZT-5H under multiaxial loading paths, and comparison with model predictions. The loading paths considered are loading of poled material with electric field at various angles (theta) to the poling direction, and loading with uniaxial compressive stress rotated through various angles (theta) to the poling direction. In each case the material response is measured by tracking electric displacement in the form of surface charge. Model predictions are made using a self-consistent crystal plasticity approach. The model is able to predict well the response to multiaxial electric field loading. There is good qualitative agreement in the case of mechanical loading.
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