In this work, we apply the thermal wave method and the thermal pulse method for non-destructive characterisation of the polarisation state of embedded piezoelectric transducers. Heating the sample with a square-wave modulated laser beam or a single laser pulse leads to a pyroelectric current recorded in the frequency or time domain, respectively. It carries information about the polarisation state. Analytical and numerical finite element models describe the pyroelectric response of the piezoceramic. Modelling and experimental results are compared for a simple lead–zirconate–titanate (PZT) plate, a low-temperature co-fired ceramics (LTCC)/PZT sensor and actuator, and a macro-fibre composite (MFC) actuator.

Piezoelectric smart structures are created by embedding piezoelectric transducers into structural components to make them controllable or responsive to their environment. These structures find applications, for instance, for health-monitoring of safety components, for reducing noise emission in automobile engineering, or for damping vibrations. Their mass-production requires control of the polarisation state due to mechanical and thermal loads appearing during device fabrication.

Non-destructive methods for obtaining polarisation profiles rely on an external excitation of the material leading to a local change of material properties (Mellinger et al., 2007). A thermal excitation in terms of thermal waves or thermal pulses gives rise to a pyroelectric current, which carries information on the polarisation profile. In the frequency domain, the laser intensity modulation method (LIMM) is well-established. Thereby temperature oscillations are generated by a periodically modulated laser beam (Lang and Das-Gupta, 1986). When thermal pulses are applied with a pulsed laser, the signal is recorded in the time domain (Collins, 1977). The advantage of the thermal pulse method is a higher pyroelectric signal in a shorter measuring time. In Pham et al. (2009) both methods are used to map the polarisation profiles in thin dielectric films with a high resolution. Pham et al. (2009) came to the conclusion that they provide similar results with the thermal pulse method being up to 50 times faster. The use of scanning LIMM to generate a polarisation map of the sample surface is described in more detail by Stewart and Cain (2009). In Stewart and Cain (2015), piezoelectric films of thicknesses down to 100 nm are measured with high-frequency LIMM.

In this work, we present simplified analytical and numerical finite element models to describe the pyroelectric response of the LIMM and the thermal pulse method, and we compare them to experimental results.

The pyroelectric current is described by a fundamental relation:

Assuming a homogeneous polarisation equivalent to

In this work, we use different models to determine the temperature
distribution

In the case of LIMM, we consider a harmonically heated piezoelectric plate
exhibiting heat losses to the environment, characterised by a thermal
relaxation time

For a continuous distribution of relaxation times (instead of a single time
constant) and a homogeneous polarisation with

Depending on the sample design, a superposition of several relaxation processes is also possible.

The temperature distribution after heating the piezoelectric plate by a
laser pulse can be described by a transient pulse model of Camia (1967). It
assumes an infinitely short laser pulse, a thermally isolated top surface
and thermal coupling of the backside to an ideal heat sink. The temperature
distribution is given by an infinite series of exponential terms:

Another analytical approach is a one-dimensional transient heat transfer
model introduced by Bloß et al. (2000). It additionally considers the thermal mass of
the electrode. The temperature distribution yields

The temperature distribution was also determined by a finite element model (FEM) solved by ANSYS 15.0.

The transient thermal analysis was performed by applying a heat flux in the
area of the laser spot on the top of the sample for the duration of the
laser pulse (0.5

Material properties of the PZT plate (PI Ceramic GmbH, 2015).

Two different types of embedded piezoelectric transducers were evaluated:

A low-temperature co-fired ceramics (LTCC)/lead–zirconate–titanate
(PZT) sensor and actuator consisting of an already sintered PZT plate
(CeramTec Sonox^{®} P53) with a size of (^{®} Tape-HL2000).
Sample fabrication is described in detail elsewhere (Flössel et al.,
2010). The sample capacitance was 30 nF, the dielectric loss tangent
amounted to about 2 % at 10 kHz.

A commercial M-8528-P2 macro-fibre composite (MFC) actuator (Smart Materials, Dresden, Germany) with an overall length of 105 mm, an active length of 85 mm, an active width of 28 mm, a thickness of about 0.3 mm, a sample capacitance of 170 nF, and a dielectric loss tangent of about 5 % at 10 kHz. PZT macro-fibres are embedded in epoxy resin. They are electrically contacted by copper electrode strips and covered by a Kapton film.

For LIMM measurements, the samples were periodically heated by an array of six laser diodes or a single laser diode (LCU98A041A, Laser Components GmbH, Olching, Germany) square-wave modulated with frequencies of up to 1 kHz each with a power of 14 mW at a wavelength of 980 nm. The complex pyroelectric current was determined by an impedance/gain-phase analyser (Solartron 1260, Solartron Analytical, Farnborough, UK) with DC coupling. In order to reduce noise, 30 measurement repetitions were used for averaging.

Thermal pulse measurements were carried out by heating the samples with a
pulsed laser diode (LC905D1S3J09UA, Laser Components GmbH, Olching, Germany)
at a wavelength of 905 nm with a maximal peak power of 75 W, a pulse width
of 0.15 or 0.5 ^{®} Xi-A oscilloscope (LeCroy, Chestnut
Ridge, USA).

Figure 1 shows the pyroelectric current spectrum of the non-embedded PZT
plate fitted to Eq. (5). The thermal relaxation time amounts to 0.8 s. The
fit shows a deviation from an ideal Debye-like model due to a slight time
distribution, which is attributed to the impact of the electrodes and the
electrical contact by a wire. The spectrum is best described by a
Cole–Davidson relaxation with

Pyroelectric current spectrum of a PZT plate in comparison to a fit to Eq. (5).

Figure 2 illustrates the pyroelectric current spectrum of a LTCC/PZT module
fitted to Eq. (5). Between 0.1 and 10 Hz the embedded PZT plate loses
heat to the LTCC layers with a thermal relaxation time of 0.16 s. A minor
decrease of the real part at higher frequencies is attributed to a slightly
inhomogeneous polarisation distribution, which is taken into account by

Pyroelectric current spectrum of a LTCC/PZT sensor–actuator
in comparison to a fit to Eq. (4) with

The MFC actuator is an example for a broad relaxation time distribution,
which is well described by a Cole–Cole function with

Pyroelectric current spectrum of a MFC actuator in comparison to a fit to Eq. (5).

The mean temperature of the sample and the resulting pyroelectric current were determined for the PZT plate with the analytical models and the FEM model as described above.

Figure 4 illustrates the mean temperature over a time period of 0.1 s.
The analytical models start at the temperature maximum whereas the FEM model
shows the increase of the temperature from room temperature to a maximum
value since the laser pulse has a given time duration of 0.5

Mean temperature within a PZT plate, determined by three different
models (cf. Sect. 2.2) with

The pyroelectric current is proportional to the time derivative of the mean temperature of the sample presented in Fig. 5 (cf. Eq. 2). In the analytical models, the current starts at a very high positive value caused by initial heating and falls down almost immediately to a negative value. On the other hand, the FEM model illustrates additionally the rise of the positive current during the heating period (cf. Fig. 5b). After reaching a minimum the current slowly returns to zero in all models; i.e. the sample returns to steady-state conditions.

Time derivative of the mean temperature (proportional to the
pyroelectric current) within a PZT plate, determined by three different
models (cf. Sect. 2.2) with

Figures 6 and 7 illustrate the measured pyroelectric current of the PZT plate and of the LTCC/PZT module and the MFC actuator. There is a short initial negative peak followed by a large positive signal.

Time dependence of the output voltage of the current amplifier
(10

Time dependence of the output voltage of the current amplifier
(10

Figure 6 illustrates that a longer pulse width leads to a higher heat input and thus a larger signal. The experimental data in Fig. 6 are in qualitative agreement with the simulation in Fig. 5b. Consequently, the experimental results manifest the initial heating of the piezoelectric plate. However, there is still a time shift between the signal maxima of the model and the experiment. One reason could be the thermal buffering effect of the top electrode (Bloß et al., 2000). It absorbs thermal energy during the short thermal pulse but only slowly transfers it to the piezoelectric since the heat transfer is limited by the piezoelectric's thermal diffusivity. Another reason could be the neglect of the rise and decay times of the laser pulse in the simulation. Both effects are now subject of further research. On the other hand, the signal of the cooling period at longer times is still too noisy for a quantitative analysis. Due to the very high gain of the current amplifier, mainly noise is present in the measured signal for measurement times of more than 0.5 ms. The signal shape is similar to the thermal pulse response of electret polymers obtained in Mellinger et al. (2005), where the high-gain signal decreases to zero for times exceeding 1 ms. For further measurements, a low-pass filter and a 50 Hz notch filter will be used to reduce the noise contribution.

In Fig. 7, the time shift of the maximum signal for the embedded piezoelectric plates is suspected to be caused by the heat transfer time through the embedding top layer. The attenuation of the absorbed heat in the top layer would result in a smaller maximum of the pyroelectric signal.

The pyroelectric response of an embedded piezoceramic plate for the LIMM has been described by an analytical model, which was successfully applied to integrated sensor–actuator modules. For the thermal pulse method, the temperature distribution and the resulting pyroelectric current have been characterised both by two analytical and one FEM model. Further improvement of the thermal pulse set-up is required to reduce the signal-to-noise ratio in the cooling period. In the next step, a Fourier transform will be performed to analyse the thermal pulse signal in the frequency domain. This enables the application of the LIMM models to a signal that was recorded in a much shorter measuring time.

This research is supported by the Deutsche Forschungsgemeinschaft (DFG) in context of the Collaborative Research Centre/Transregio 39 PT-PIESA, subproject C8. Edited by: A. Schütze Reviewed by: two anonymous referees