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example, domain wall motion contributes substantially to the room temperature permittivity of BaTiO -
3
based multilayer ceramic capacitors. Similarly, up to 75% of the piezoelectric response of soft PbZr Ti O
x
1-x
3
ceramics arises from domain wall motion, albeit at the expense of increased hysteresis. The relative extent of
domain wall motion for a given material depends explicitly on the history of the sample, the sample
microstructure and orientation, and the excitation conditions. For example, the following factors are
[7,8]
critical to the observed extent of extrinsic contributions to the properties:
Electric field
Higher amplitudes of the AC electric field (E ) act to drive domain walls more strongly and can
AC 0-pk
substantially increase the extrinsic contributions to the properties. This increases the relative permittivity
and loss tangent with the AC field, which is called dielectric nonlinearity. Often, the increase in relative
permittivity is linear with AC electric field, producing the so-called Rayleigh regime (from a modified
[9]
version of the Rayleigh law originally developed for magnetic materials) , i.e.,

where ε' is the relative permittivity, ε' is the reversible Rayleigh coefficient that includes both the intrinsic
init
response and reversible motion of domain walls or phase boundaries, and α' is the irreversible Rayleigh
coefficient . The Rayleigh law is typically observed up to modest electric fields under conditions where the
[10]
domain structure is unchanged during the measurement and the distribution of the restoring forces for the
[11]
domain walls is Gaussian . In cases where the domain wall density is changed by the applied field, the
Rayleigh law will not be observed, and a more complex formalism, such as first-order reversal curves, needs
to be employed [12-16] . In many but not all ferroelectric materials, significant levels of domain nucleation are
observed at E exceeding ~a third to a half of the coercive field.
AC 0-pk

In contrast, because DC biases stabilize the domain state, they also act to suppress extrinsic contributions to
the properties. This produces a far stronger DC bias dependence of the relative permittivity than would be
[4]
predicted based on phenomenological descriptions of the field dependence of the intrinsic permittivity (e.g.,
intrinsic dielectric stiffening) [17,18] . This progressive loss of the extrinsic contributions to the permittivity is
problematic in multilayer ceramic capacitors where the voltage saturation significantly depresses the usable
capacitance . In contrast, the suppression is very helpful in some piezoelectric sensing applications, as it
[19]
increases the figure of merit = piezoelectric coefficient/relative permittivity .
[20]
Temperature
There is a strong coupling between temperature and extrinsic contributions to the properties. First, domain
wall motion is thermally activated such that domain walls become more mobile as temperature increases.
Therefore, extrinsic contributions typically rise as the Curie temperature is approached. Other ferroelectric-
ferroelectric phase transitions also tend to favor the motion of the mobile interfaces and will induce
increases in the extrinsic contributions. Secondly, because domain wall and phase boundary motions are
[21]
inherently dissipative, substantial levels of extrinsic contributions can increase the sample temperature .
Thirdly, changes in temperature can perturb the domain structure and so de-age the material, increasing the
extrinsic contributions temporarily.

Time
Domain structures, especially in ceramic materials, are rarely fully stable as a function of time. Thus, over
the course of time, the domain structure progressively seeks lower energy configurations. As this occurs, the
remaining domain walls tend to be less mobile. This is one of the contributions to the aging of the dielectric